# Properties

 Label 128.5.h.a Level 128 Weight 5 Character orbit 128.h Analytic conductor 13.231 Analytic rank 0 Dimension 60 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 128.h (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2313552747$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$15$$ over $$\Q(\zeta_{8})$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$60q + 4q^{3} - 4q^{5} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q + 4q^{3} - 4q^{5} + 4q^{7} - 4q^{9} + 4q^{11} - 4q^{13} + 8q^{15} + 4q^{19} - 4q^{21} + 1156q^{23} - 4q^{25} - 3644q^{27} - 4q^{29} - 8q^{33} + 5188q^{35} - 4q^{37} + 2692q^{39} - 4q^{41} - 5564q^{43} - 328q^{45} + 8q^{47} - 8384q^{51} + 956q^{53} + 11780q^{55} - 4q^{57} + 13060q^{59} + 7548q^{61} - 8q^{65} - 18876q^{67} - 19588q^{69} - 19964q^{71} - 4q^{73} + 200q^{75} + 9404q^{77} + 50184q^{79} - 10556q^{83} + 2496q^{85} - 49276q^{87} - 4q^{89} - 31868q^{91} + 320q^{93} - 8q^{97} + 46920q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1 0 −15.8039 + 6.54620i 0 21.3091 + 8.82651i 0 18.2180 18.2180i 0 149.636 149.636i 0
15.2 0 −12.3594 + 5.11942i 0 −32.9803 13.6609i 0 −59.2707 + 59.2707i 0 69.2702 69.2702i 0
15.3 0 −10.5427 + 4.36694i 0 −27.2781 11.2990i 0 51.6577 51.6577i 0 34.8033 34.8033i 0
15.4 0 −8.76479 + 3.63049i 0 8.51003 + 3.52497i 0 −21.5633 + 21.5633i 0 6.36534 6.36534i 0
15.5 0 −7.43335 + 3.07899i 0 8.82167 + 3.65406i 0 −10.5220 + 10.5220i 0 −11.5012 + 11.5012i 0
15.6 0 −4.81945 + 1.99628i 0 42.1637 + 17.4648i 0 27.8831 27.8831i 0 −38.0337 + 38.0337i 0
15.7 0 −0.421454 + 0.174572i 0 −33.4127 13.8400i 0 −1.19092 + 1.19092i 0 −57.1285 + 57.1285i 0
15.8 0 0.239783 0.0993213i 0 −10.4583 4.33197i 0 51.7271 51.7271i 0 −57.2280 + 57.2280i 0
15.9 0 3.21611 1.33215i 0 23.1494 + 9.58878i 0 −30.5337 + 30.5337i 0 −48.7069 + 48.7069i 0
15.10 0 4.25478 1.76239i 0 −19.2280 7.96449i 0 −0.0963230 + 0.0963230i 0 −42.2785 + 42.2785i 0
15.11 0 5.05753 2.09490i 0 14.0253 + 5.80946i 0 −61.2653 + 61.2653i 0 −36.0856 + 36.0856i 0
15.12 0 10.9321 4.52821i 0 −0.302610 0.125345i 0 32.5244 32.5244i 0 41.7299 41.7299i 0
15.13 0 10.9680 4.54309i 0 35.3175 + 14.6290i 0 47.3057 47.3057i 0 42.3816 42.3816i 0
15.14 0 12.3801 5.12803i 0 −36.8127 15.2483i 0 −16.3058 + 16.3058i 0 69.6958 69.6958i 0
15.15 0 14.8038 6.13192i 0 5.46900 + 2.26533i 0 −27.5680 + 27.5680i 0 124.275 124.275i 0
47.1 0 −6.02892 + 14.5551i 0 1.24804 + 3.01304i 0 −48.9953 48.9953i 0 −118.227 118.227i 0
47.2 0 −5.09563 + 12.3019i 0 −6.07195 14.6590i 0 −3.01401 3.01401i 0 −68.0967 68.0967i 0
47.3 0 −4.50708 + 10.8811i 0 −0.960017 2.31769i 0 43.4088 + 43.4088i 0 −40.8081 40.8081i 0
47.4 0 −3.05748 + 7.38140i 0 −13.2656 32.0260i 0 14.4988 + 14.4988i 0 12.1387 + 12.1387i 0
47.5 0 −3.03024 + 7.31565i 0 12.4171 + 29.9775i 0 51.5981 + 51.5981i 0 12.9392 + 12.9392i 0
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 111.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.5.h.a 60
4.b odd 2 1 32.5.h.a 60
32.g even 8 1 32.5.h.a 60
32.h odd 8 1 inner 128.5.h.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.5.h.a 60 4.b odd 2 1
32.5.h.a 60 32.g even 8 1
128.5.h.a 60 1.a even 1 1 trivial
128.5.h.a 60 32.h odd 8 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database