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Properties

Label 256.12.b.p
Level $256$
Weight $12$
Character orbit 256.b
Analytic conductor $196.696$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,12,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-98736,0,-627628] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(196.695854223\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 2789 x^{10} - 107880 x^{9} + 4082152 x^{8} + 294993082 x^{7} - 2405623951 x^{6} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{110}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{6} + 4 \beta_{2} - 9 \beta_1) q^{5} + ( - \beta_{3} - 8228) q^{7} + (\beta_{8} + 4 \beta_{4} + \cdots - 52304) q^{9} + (\beta_{11} - \beta_{10} + \cdots - 3612 \beta_1) q^{11}+ \cdots + (35428 \beta_{11} + \cdots - 1241901196 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 98736 q^{7} - 627628 q^{9} + 9656528 q^{15} + 8255272 q^{17} - 19842640 q^{23} - 103266004 q^{25} - 492266240 q^{31} - 675982576 q^{33} - 1296172112 q^{39} - 973831672 q^{41} + 1207950176 q^{47}+ \cdots + 437793250984 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 2789 x^{10} - 107880 x^{9} + 4082152 x^{8} + 294993082 x^{7} - 2405623951 x^{6} + \cdots + 10\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!72 \nu^{11} + \cdots - 30\!\cdots\!00 ) / 22\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18\!\cdots\!96 \nu^{11} + \cdots + 44\!\cdots\!50 ) / 74\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\!\cdots\!16 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 13\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!28 \nu^{11} + \cdots + 17\!\cdots\!15 ) / 27\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!48 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 93\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 62\!\cdots\!44 \nu^{11} + \cdots - 15\!\cdots\!50 ) / 10\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!68 \nu^{11} + \cdots - 36\!\cdots\!65 ) / 27\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 59\!\cdots\!84 \nu^{11} + \cdots + 11\!\cdots\!95 ) / 27\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24\!\cdots\!60 \nu^{11} + \cdots + 59\!\cdots\!50 ) / 82\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 44\!\cdots\!32 \nu^{11} + \cdots - 10\!\cdots\!00 ) / 82\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51\!\cdots\!08 \nu^{11} + \cdots - 33\!\cdots\!00 ) / 82\!\cdots\!95 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 16 \beta_{10} - 8 \beta_{8} + 4 \beta_{7} - 1808 \beta_{6} + 116 \beta_{5} + 937 \beta_{4} + \cdots + 130761 ) / 786432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 292 \beta_{11} - 688 \beta_{10} + 145 \beta_{9} - 488 \beta_{8} - 52 \beta_{7} + 25036 \beta_{6} + \cdots + 182912303 ) / 393216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9816 \beta_{11} + 592 \beta_{10} + 21414 \beta_{9} + 74616 \beta_{8} + 10804 \beta_{7} + \cdots + 22306961165 ) / 786432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 118872 \beta_{11} - 13488 \beta_{10} + 2190822 \beta_{9} - 2331752 \beta_{8} - 210844 \beta_{7} + \cdots + 8919188633 ) / 786432 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 64453200 \beta_{11} - 115544272 \beta_{10} + 20433204 \beta_{9} - 60178440 \beta_{8} + \cdots + 4499207819393 ) / 786432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 105531620 \beta_{11} - 129552020 \beta_{10} + 432090857 \beta_{9} + 900464728 \beta_{8} + \cdots + 231786546024725 ) / 98304 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3662975496 \beta_{11} + 77457101072 \beta_{10} + 151252671954 \beta_{9} - 340972355864 \beta_{8} + \cdots - 48\!\cdots\!81 ) / 786432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 505833104540 \beta_{11} - 1148469255200 \beta_{10} + 83606279775 \beta_{9} - 193277535168 \beta_{8} + \cdots - 49\!\cdots\!14 ) / 65536 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 329186938896 \beta_{11} + 27707104258936 \beta_{10} + 59785658405820 \beta_{9} + \cdots + 41\!\cdots\!57 ) / 196608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 865747227043052 \beta_{11} + \cdots - 42\!\cdots\!78 ) / 393216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13\!\cdots\!48 \beta_{11} + \cdots - 40\!\cdots\!85 ) / 196608 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−31.7571 + 11.4470i
6.00626 − 18.8120i
35.5542 + 3.32595i
−21.8554 + 2.67314i
39.2519 + 8.89394i
−26.2000 + 40.5002i
−26.2000 − 40.5002i
39.2519 − 8.89394i
−21.8554 − 2.67314i
35.5542 − 3.32595i
6.00626 + 18.8120i
−31.7571 − 11.4470i
0 − 795.850i 0 3925.84i 0 −13521.5 0 −456230. 0
129.2 0 − 597.717i 0 9917.53i 0 −77908.4 0 −180119. 0
129.3 0 − 551.275i 0 − 3654.34i 0 40487.8 0 −126757. 0
129.4 0 − 219.634i 0 − 4589.61i 0 −25262.2 0 128908. 0
129.5 0 − 181.938i 0 − 4531.11i 0 62513.8 0 144046. 0
129.6 0 − 28.4295i 0 − 13262.1i 0 −35677.5 0 176339. 0
129.7 0 28.4295i 0 13262.1i 0 −35677.5 0 176339. 0
129.8 0 181.938i 0 4531.11i 0 62513.8 0 144046. 0
129.9 0 219.634i 0 4589.61i 0 −25262.2 0 128908. 0
129.10 0 551.275i 0 3654.34i 0 40487.8 0 −126757. 0
129.11 0 597.717i 0 − 9917.53i 0 −77908.4 0 −180119. 0
129.12 0 795.850i 0 − 3925.84i 0 −13521.5 0 −456230. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.12.b.p 12
4.b odd 2 1 256.12.b.q 12
8.b even 2 1 inner 256.12.b.p 12
8.d odd 2 1 256.12.b.q 12
16.e even 4 1 128.12.a.e ✓ 6
16.e even 4 1 128.12.a.h yes 6
16.f odd 4 1 128.12.a.f yes 6
16.f odd 4 1 128.12.a.g yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.12.a.e ✓ 6 16.e even 4 1
128.12.a.f yes 6 16.f odd 4 1
128.12.a.g yes 6 16.f odd 4 1
128.12.a.h yes 6 16.e even 4 1
256.12.b.p 12 1.a even 1 1 trivial
256.12.b.p 12 8.b even 2 1 inner
256.12.b.q 12 4.b odd 2 1
256.12.b.q 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{12} + 1376696 T_{3}^{10} + 635352457072 T_{3}^{8} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{6} + 49368 T_{7}^{5} - 5636820048 T_{7}^{4} - 242979558680320 T_{7}^{3} + \cdots + 24\!\cdots\!40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 49\!\cdots\!40)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 21\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 75\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 29\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 60\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
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