Coins from Reform (491-602)
While the transition from Roman to Byzantine is gradual and somewhat fluid when it comes to political criteria, it is pretty clearly defined numismatically; In 491 Anastasius I became Emperor, and in 498 AD he introduced a monetary reform.

The new system involved three denominations of gold (the solidus and its half and third) and five of copper – the follis, worth 40 nummi, and its fractions worth 20, 10, 5 and 1 nummus. The obverses featured a highly stylized portrait, while the reverse featured the value in Greek numerals; M=40, K=20, I=10, E=5.

Politically this period covers Anastasius and the Justinian Dynasty, and was characterized by efforts to reclaim lost territory – North Africa, Illyria, southern Spain, and Italy. The last Justinian emperor – Maurice – managed to make himself unpopular enough to trigger a revolt which cost him his life in 602 AD, and triggered a series of cataclysmic events which would undo the progress of the last century.
Reform (491-602)
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An AE Follis struck 491-518 AD in Constantinopolis
Obverse:

Reverse:

Diameter: 38 mm
Die Orientation: -
Weight: 17.25 g
No notes for this coin
No references provided for this coin
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An AE Nummus struck 491-498 AD in Constantinopolis
Obverse: draped, diademed bust right; (illegible letters)

Reverse: Monogram of Anastasius

Diameter: 9 mm
Die Orientation: -
Weight: 1.06 g
No notes for this coin
BMC 32:128, Wroth 26
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An AR Siliqua struck 530-580 in Constantinopolis?
Obverse: Helmeted, draped, cuirassed bust right,

Reverse: K

Diameter: 14 mm
Die Orientation: -
Weight: 1.2 g
Anonymous Issue. Ex Cng coin
Vagi 3051; Bendall Type 8c
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An AU Solidus struck Jan. 566 AD in Constantinopolis
Obverse: D N I - VST - INVS PP AVI, Helmeted bust facing, holding Victory on globe and shield.

Reverse: VICTORI - A AVGGG ? (officina 4) around, CONOB in exergue, Constantinipolis seated facing, head r., holding spear and cross on globe.

Diameter: -
Die Orientation: -
Weight: 4.31 g
No notes for this coin
Berk-59, Hahn-4, Sear-344.
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An AE unit struck 539-540 AD in Constantinopolis
Obverse: DN IVSTINIANVS PP AVG, Bust of Justinian wearing helmet and cuirass, facing holding cross on globe and shield; cross to right

Reverse: Large M, ANNO to left, cross above, regnal year (XIII) to right, officina letter (ϵ) below; CON in exergue

Diameter: 42 mm
Die Orientation: 6 H
Weight: 21.4 g

Byzantine Empire
AE Follis

Minted at the fifth officina in Constantinople during regnal year 13 which corresponds to 539/540 AD (Reigns typically didn't start on January 1st, so there is overlap)

SB 163
(6) Justin | Justinian | Justin II
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An AE 12 nummi struck 518-578 AD in Alexandria
Obverse: Uncertain inscription, Bust facing right wearing diadem, cuirass, and drapery

Reverse: Cross between I and B; AΛEΞ in exergue

Diameter: 13 mm
Die Orientation: 5 H
Weight: 2.61 g
No notes for this coin
SB 112?
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An AU Solidus struck 590-602 AD in Constantinopolis
Obverse: DN mAVRC TIb PP AVG, Bust of Maurice facing, wearing plumed helmet, and cuirass, holding cross on globe

Reverse: VICTORIA AVGGI, Angel standing facing, holding long P-headed cross, and cross on globe; CONOB in exergue

Diameter: 21 mm
Die Orientation: 6 H
Weight: 4.24 g
VICTORIA AVGG="to the Victory of the two Emperors" (referring to himself and his son co-emperor Theodosius)
The I at the end denotes the 10th officina.

Given that Theodosius was not raised as co-emperor until 590, it makes sense to date the coin to his reign in my opinion.

CONOB="Constantinople, 1/72 pound pure gold"
SB 478
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An AE Half-Follis struck 593 AD in Constantinopolis
Obverse: D N MAVR TIBER PP AV. Helmeted and cuirassed bust facing, holding globus cruciger and shield.

Reverse: Large mark of value K; ANNO to left, cross above, X/I to right, A in exergue.

Diameter: 22 mm
Die Orientation: -
Weight: 5.15 g
1st officina. Dated RY 11.
SBV 497; DOC 56a