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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,3,Mod(31,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 256.h (of order \(8\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 31.7
Significant digits:
Format:

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 \( T_{3}^{28} + 4 T_{3}^{27} + 10 T_{3}^{26} + 52 T_{3}^{25} + 162 T_{3}^{24} - 848 T_{3}^{23} + \cdots + 30700437632 \) acting on \(S_{3}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display