# Properties

 Label 128.4.g.a Level $128$ Weight $4$ Character orbit 128.g Analytic conductor $7.552$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.55224448073$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$11$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 4q^{3} - 4q^{5} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 4q^{3} - 4q^{5} + 4q^{7} - 4q^{9} + 4q^{11} - 4q^{13} + 4q^{19} - 4q^{21} - 324q^{23} - 4q^{25} + 268q^{27} - 4q^{29} + 752q^{31} - 8q^{33} + 460q^{35} - 4q^{37} - 596q^{39} - 4q^{41} - 804q^{43} + 104q^{45} + 1384q^{51} + 748q^{53} + 292q^{55} - 4q^{57} - 1372q^{59} - 1828q^{61} - 2512q^{63} - 8q^{65} - 2036q^{67} - 1060q^{69} - 220q^{71} - 4q^{73} + 1712q^{75} + 1900q^{77} - 2436q^{83} + 496q^{85} + 1292q^{87} - 4q^{89} + 3604q^{91} - 112q^{93} + 6088q^{95} - 8q^{97} + 5424q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −3.54796 + 8.56554i 0 −7.55322 + 3.12865i 0 −7.16166 + 7.16166i 0 −41.6886 41.6886i 0
17.2 0 −2.92731 + 7.06715i 0 13.5234 5.60159i 0 23.0737 23.0737i 0 −22.2836 22.2836i 0
17.3 0 −1.94138 + 4.68690i 0 4.93338 2.04347i 0 −14.0755 + 14.0755i 0 0.893826 + 0.893826i 0
17.4 0 −1.64064 + 3.96085i 0 −11.8087 + 4.89132i 0 5.11236 5.11236i 0 6.09524 + 6.09524i 0
17.5 0 −0.729459 + 1.76107i 0 4.29822 1.78038i 0 −1.47807 + 1.47807i 0 16.5226 + 16.5226i 0
17.6 0 0.477813 1.15354i 0 −16.3468 + 6.77105i 0 18.0222 18.0222i 0 17.9895 + 17.9895i 0
17.7 0 0.998206 2.40988i 0 17.4005 7.20752i 0 −4.37099 + 4.37099i 0 14.2808 + 14.2808i 0
17.8 0 1.20185 2.90153i 0 −3.98512 + 1.65069i 0 −22.4050 + 22.4050i 0 12.1174 + 12.1174i 0
17.9 0 1.90169 4.59109i 0 0.188811 0.0782080i 0 11.4103 11.4103i 0 1.63019 + 1.63019i 0
17.10 0 3.21198 7.75440i 0 −13.6472 + 5.65283i 0 −9.07689 + 9.07689i 0 −30.7220 30.7220i 0
17.11 0 3.28810 7.93817i 0 11.2895 4.67626i 0 11.8490 11.8490i 0 −33.1111 33.1111i 0
49.1 0 −7.57022 + 3.13569i 0 −8.03744 + 19.4041i 0 1.85119 + 1.85119i 0 28.3838 28.3838i 0
49.2 0 −6.97998 + 2.89120i 0 1.57150 3.79394i 0 −15.6607 15.6607i 0 21.2692 21.2692i 0
49.3 0 −5.53310 + 2.29188i 0 4.22177 10.1923i 0 11.6451 + 11.6451i 0 6.27055 6.27055i 0
49.4 0 −4.56924 + 1.89264i 0 1.37033 3.30826i 0 6.14642 + 6.14642i 0 −1.79604 + 1.79604i 0
49.5 0 −0.143768 + 0.0595506i 0 −0.767542 + 1.85301i 0 5.47741 + 5.47741i 0 −19.0748 + 19.0748i 0
49.6 0 1.36212 0.564209i 0 6.58151 15.8892i 0 −14.5517 14.5517i 0 −17.5548 + 17.5548i 0
49.7 0 1.65706 0.686375i 0 −4.13953 + 9.99370i 0 −24.2273 24.2273i 0 −16.8172 + 16.8172i 0
49.8 0 2.66269 1.10292i 0 −5.50788 + 13.2972i 0 6.48055 + 6.48055i 0 −13.2184 + 13.2184i 0
49.9 0 5.56908 2.30679i 0 6.28381 15.1704i 0 16.6573 + 16.6573i 0 6.60151 6.60151i 0
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.11 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.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.4.g.a 44
4.b odd 2 1 32.4.g.a 44
8.b even 2 1 256.4.g.a 44
8.d odd 2 1 256.4.g.b 44
32.g even 8 1 inner 128.4.g.a 44
32.g even 8 1 256.4.g.a 44
32.h odd 8 1 32.4.g.a 44
32.h odd 8 1 256.4.g.b 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.g.a 44 4.b odd 2 1
32.4.g.a 44 32.h odd 8 1
128.4.g.a 44 1.a even 1 1 trivial
128.4.g.a 44 32.g even 8 1 inner
256.4.g.a 44 8.b even 2 1
256.4.g.a 44 32.g even 8 1
256.4.g.b 44 8.d odd 2 1
256.4.g.b 44 32.h odd 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database