# Properties

 Label 256.4.g.b Level $256$ Weight $4$ Character orbit 256.g Analytic conductor $15.104$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 256.g (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1044889615$$ 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1 0 −3.54796 + 8.56554i 0 7.55322 3.12865i 0 7.16166 7.16166i 0 −41.6886 41.6886i 0
33.2 0 −2.92731 + 7.06715i 0 −13.5234 + 5.60159i 0 −23.0737 + 23.0737i 0 −22.2836 22.2836i 0
33.3 0 −1.94138 + 4.68690i 0 −4.93338 + 2.04347i 0 14.0755 14.0755i 0 0.893826 + 0.893826i 0
33.4 0 −1.64064 + 3.96085i 0 11.8087 4.89132i 0 −5.11236 + 5.11236i 0 6.09524 + 6.09524i 0
33.5 0 −0.729459 + 1.76107i 0 −4.29822 + 1.78038i 0 1.47807 1.47807i 0 16.5226 + 16.5226i 0
33.6 0 0.477813 1.15354i 0 16.3468 6.77105i 0 −18.0222 + 18.0222i 0 17.9895 + 17.9895i 0
33.7 0 0.998206 2.40988i 0 −17.4005 + 7.20752i 0 4.37099 4.37099i 0 14.2808 + 14.2808i 0
33.8 0 1.20185 2.90153i 0 3.98512 1.65069i 0 22.4050 22.4050i 0 12.1174 + 12.1174i 0
33.9 0 1.90169 4.59109i 0 −0.188811 + 0.0782080i 0 −11.4103 + 11.4103i 0 1.63019 + 1.63019i 0
33.10 0 3.21198 7.75440i 0 13.6472 5.65283i 0 9.07689 9.07689i 0 −30.7220 30.7220i 0
33.11 0 3.28810 7.93817i 0 −11.2895 + 4.67626i 0 −11.8490 + 11.8490i 0 −33.1111 33.1111i 0
97.1 0 −7.57022 + 3.13569i 0 8.03744 19.4041i 0 −1.85119 1.85119i 0 28.3838 28.3838i 0
97.2 0 −6.97998 + 2.89120i 0 −1.57150 + 3.79394i 0 15.6607 + 15.6607i 0 21.2692 21.2692i 0
97.3 0 −5.53310 + 2.29188i 0 −4.22177 + 10.1923i 0 −11.6451 11.6451i 0 6.27055 6.27055i 0
97.4 0 −4.56924 + 1.89264i 0 −1.37033 + 3.30826i 0 −6.14642 6.14642i 0 −1.79604 + 1.79604i 0
97.5 0 −0.143768 + 0.0595506i 0 0.767542 1.85301i 0 −5.47741 5.47741i 0 −19.0748 + 19.0748i 0
97.6 0 1.36212 0.564209i 0 −6.58151 + 15.8892i 0 14.5517 + 14.5517i 0 −17.5548 + 17.5548i 0
97.7 0 1.65706 0.686375i 0 4.13953 9.99370i 0 24.2273 + 24.2273i 0 −16.8172 + 16.8172i 0
97.8 0 2.66269 1.10292i 0 5.50788 13.2972i 0 −6.48055 6.48055i 0 −13.2184 + 13.2184i 0
97.9 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 225.11 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 256.4.g.b 44
4.b odd 2 1 256.4.g.a 44
8.b even 2 1 32.4.g.a 44
8.d odd 2 1 128.4.g.a 44
32.g even 8 1 32.4.g.a 44
32.g even 8 1 inner 256.4.g.b 44
32.h odd 8 1 128.4.g.a 44
32.h odd 8 1 256.4.g.a 44

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

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database