# Properties

 Label 256.3.h.a Level $256$ Weight $3$ Character orbit 256.h Analytic conductor $6.975$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 256.h (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.97549476762$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{3} + 4q^{5} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{3} + 4q^{5} + 4q^{7} - 4q^{9} - 4q^{11} + 4q^{13} + 8q^{15} - 4q^{19} + 4q^{21} + 68q^{23} - 4q^{25} - 100q^{27} + 4q^{29} - 8q^{33} + 92q^{35} + 4q^{37} - 188q^{39} - 4q^{41} + 92q^{43} + 40q^{45} + 8q^{47} + 224q^{51} + 164q^{53} - 252q^{55} - 4q^{57} + 124q^{59} + 68q^{61} - 8q^{65} - 164q^{67} - 188q^{69} + 260q^{71} - 4q^{73} - 488q^{75} - 220q^{77} + 520q^{79} - 484q^{83} - 96q^{85} + 452q^{87} - 4q^{89} - 196q^{91} - 32q^{93} - 8q^{97} + 216q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 0 −4.68670 + 1.94129i 0 4.51028 + 1.86822i 0 3.85317 3.85317i 0 11.8326 11.8326i 0
31.2 0 −3.70255 + 1.53365i 0 −7.20074 2.98264i 0 −4.26150 + 4.26150i 0 4.99283 4.99283i 0
31.3 0 −1.37292 + 0.568682i 0 −2.28872 0.948019i 0 6.37744 6.37744i 0 −4.80245 + 4.80245i 0
31.4 0 −0.374985 + 0.155324i 0 7.60625 + 3.15061i 0 −6.84161 + 6.84161i 0 −6.24747 + 6.24747i 0
31.5 0 1.58190 0.655246i 0 −4.18866 1.73500i 0 −3.93197 + 3.93197i 0 −4.29089 + 4.29089i 0
31.6 0 2.49683 1.03422i 0 0.452310 + 0.187353i 0 −0.429965 + 0.429965i 0 −1.19943 + 1.19943i 0
31.7 0 4.35131 1.80237i 0 2.81639 + 1.16659i 0 6.23443 6.23443i 0 9.32143 9.32143i 0
95.1 0 −1.73217 + 4.18183i 0 1.85856 + 4.48696i 0 5.27676 + 5.27676i 0 −8.12333 8.12333i 0
95.2 0 −1.31872 + 3.18367i 0 0.659338 + 1.59178i 0 −9.54718 9.54718i 0 −2.03276 2.03276i 0
95.3 0 −1.10785 + 2.67458i 0 −2.95565 7.13556i 0 4.18452 + 4.18452i 0 0.437918 + 0.437918i 0
95.4 0 0.299792 0.723762i 0 −1.34740 3.25291i 0 −0.583225 0.583225i 0 5.93000 + 5.93000i 0
95.5 0 0.527719 1.27403i 0 0.642823 + 1.55191i 0 4.95044 + 4.95044i 0 5.01930 + 5.01930i 0
95.6 0 0.936461 2.26082i 0 3.18221 + 7.68254i 0 −3.67370 3.67370i 0 2.12963 + 2.12963i 0
95.7 0 2.10187 5.07436i 0 −1.74699 4.21761i 0 0.392379 + 0.392379i 0 −14.9674 14.9674i 0
159.1 0 −1.73217 4.18183i 0 1.85856 4.48696i 0 5.27676 5.27676i 0 −8.12333 + 8.12333i 0
159.2 0 −1.31872 3.18367i 0 0.659338 1.59178i 0 −9.54718 + 9.54718i 0 −2.03276 + 2.03276i 0
159.3 0 −1.10785 2.67458i 0 −2.95565 + 7.13556i 0 4.18452 4.18452i 0 0.437918 0.437918i 0
159.4 0 0.299792 + 0.723762i 0 −1.34740 + 3.25291i 0 −0.583225 + 0.583225i 0 5.93000 5.93000i 0
159.5 0 0.527719 + 1.27403i 0 0.642823 1.55191i 0 4.95044 4.95044i 0 5.01930 5.01930i 0
159.6 0 0.936461 + 2.26082i 0 3.18221 7.68254i 0 −3.67370 + 3.67370i 0 2.12963 2.12963i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.7 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 256.3.h.a 28
4.b odd 2 1 256.3.h.b 28
8.b even 2 1 128.3.h.a 28
8.d odd 2 1 32.3.h.a 28
24.f even 2 1 288.3.u.a 28
32.g even 8 1 32.3.h.a 28
32.g even 8 1 256.3.h.b 28
32.h odd 8 1 128.3.h.a 28
32.h odd 8 1 inner 256.3.h.a 28
96.p odd 8 1 288.3.u.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.h.a 28 8.d odd 2 1
32.3.h.a 28 32.g even 8 1
128.3.h.a 28 8.b even 2 1
128.3.h.a 28 32.h odd 8 1
256.3.h.a 28 1.a even 1 1 trivial
256.3.h.a 28 32.h odd 8 1 inner
256.3.h.b 28 4.b odd 2 1
256.3.h.b 28 32.g even 8 1
288.3.u.a 28 24.f even 2 1
288.3.u.a 28 96.p odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{28} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database