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Properties

Label 961.2.a.k
Level $961$
Weight $2$
Character orbit 961.a
Self dual yes
Analytic conductor $7.674$
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] = [961,2,Mod(1,961)] 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("961.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,16,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.67362363425\)
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: \( x^{12} - 28x^{10} + 302x^{8} - 1560x^{6} + 3844x^{4} - 3648x^{2} + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_1 q^{3} + (\beta_{11} + 1) q^{4} + ( - \beta_{11} + \beta_{8} - \beta_{3} + 1) q^{5} + (\beta_{9} + \beta_{7} - \beta_{2}) q^{6} + ( - \beta_{11} + \beta_{8} + 1) q^{7} + (\beta_{11} - \beta_{6} + 1) q^{8}+ \cdots + ( - 2 \beta_{10} - \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} + 8 q^{5} + 8 q^{7} + 12 q^{8} + 20 q^{9} + 20 q^{10} + 28 q^{14} - 16 q^{16} - 8 q^{18} + 16 q^{19} - 20 q^{20} + 12 q^{25} - 20 q^{28} + 24 q^{32} - 40 q^{33} + 56 q^{35} + 40 q^{36} + 36 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 28x^{10} + 302x^{8} - 1560x^{6} + 3844x^{4} - 3648x^{2} + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 4\nu^{9} - 314\nu^{7} + 2280\nu^{5} - 4540\nu^{3} + 864\nu ) / 496 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 4\nu^{8} - 314\nu^{6} + 2280\nu^{4} - 4540\nu^{2} + 864 ) / 496 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 4\nu^{8} - 314\nu^{6} + 2280\nu^{4} - 4044\nu^{2} - 1616 ) / 496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{11} + 104\nu^{9} - 662\nu^{7} + 752\nu^{5} + 4844\nu^{3} - 9776\nu ) / 992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{10} - 104\nu^{8} + 786\nu^{6} - 2612\nu^{4} + 3340\nu^{2} - 144 ) / 248 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{11} + 158\nu^{9} - 1274\nu^{7} + 4252\nu^{5} - 4428\nu^{3} - 1832\nu ) / 496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7\nu^{10} - 158\nu^{8} + 1274\nu^{6} - 4376\nu^{4} + 5668\nu^{2} - 648 ) / 248 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} + 81\nu^{9} - 794\nu^{7} + 3390\nu^{5} - 5724\nu^{3} + 1996\nu ) / 248 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -23\nu^{11} + 528\nu^{9} - 4434\nu^{7} + 16752\nu^{5} - 28508\nu^{3} + 18320\nu ) / 992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7\nu^{10} - 158\nu^{8} + 1274\nu^{6} - 4314\nu^{4} + 5172\nu^{2} - 152 ) / 124 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} - \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 4\beta_{8} + 8\beta_{4} - 8\beta_{3} + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{10} + 12\beta_{9} + 8\beta_{7} - 10\beta_{5} - 12\beta_{2} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{11} - 48\beta_{8} + 6\beta_{6} + 62\beta_{4} - 66\beta_{3} + 222 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -84\beta_{10} + 116\beta_{9} + 48\beta_{7} - 72\beta_{5} - 114\beta_{2} + 284\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 192\beta_{11} - 460\beta_{8} + 112\beta_{6} + 484\beta_{4} - 540\beta_{3} + 1616 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -676\beta_{10} + 1056\beta_{9} + 224\beta_{7} - 452\beta_{5} - 1000\beta_{2} + 2100\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1580\beta_{11} - 4112\beta_{8} + 1436\beta_{6} + 3832\beta_{4} - 4368\beta_{3} + 12120 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -5412\beta_{10} + 9380\beta_{9} + 476\beta_{7} - 2540\beta_{5} - 8480\beta_{2} + 15952\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.90485
−2.90485
0.190967
−0.190967
2.67863
−2.67863
1.89866
−1.89866
1.62154
−1.62154
2.47305
−2.47305
−2.13825 −2.90485 2.57209 −1.29612 6.21128 −2.71034 −1.22328 5.43815 2.77143
1.2 −2.13825 2.90485 2.57209 −1.29612 −6.21128 −2.71034 −1.22328 5.43815 2.77143
1.3 −1.80264 −0.190967 1.24951 −2.46636 0.344245 −1.05215 1.35286 −2.96353 4.44597
1.4 −1.80264 0.190967 1.24951 −2.46636 −0.344245 −1.05215 1.35286 −2.96353 4.44597
1.5 −1.24512 −2.67863 −0.449667 2.61876 3.33523 1.20454 3.05014 4.17508 −3.26068
1.6 −1.24512 2.67863 −0.449667 2.61876 −3.33523 1.20454 3.05014 4.17508 −3.26068
1.7 0.720616 −1.89866 −1.48071 2.78711 −1.36820 4.20133 −2.50826 0.604891 2.00844
1.8 0.720616 1.89866 −1.48071 2.78711 1.36820 4.20133 −2.50826 0.604891 2.00844
1.9 1.96916 −1.62154 1.87757 3.50580 −3.19307 2.09158 −0.241077 −0.370593 6.90346
1.10 1.96916 1.62154 1.87757 3.50580 3.19307 2.09158 −0.241077 −0.370593 6.90346
1.11 2.49624 −2.47305 4.23120 −1.14918 −6.17333 0.265036 5.56961 3.11600 −2.86862
1.12 2.49624 2.47305 4.23120 −1.14918 6.17333 0.265036 5.56961 3.11600 −2.86862
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(31\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.a.k ✓ 12
3.b odd 2 1 8649.2.a.bp 12
31.b odd 2 1 inner 961.2.a.k ✓ 12
31.c even 3 2 961.2.c.k 24
31.d even 5 4 961.2.d.r 48
31.e odd 6 2 961.2.c.k 24
31.f odd 10 4 961.2.d.r 48
31.g even 15 8 961.2.g.v 96
31.h odd 30 8 961.2.g.v 96
93.c even 2 1 8649.2.a.bp 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.k ✓ 12 1.a even 1 1 trivial
961.2.a.k ✓ 12 31.b odd 2 1 inner
961.2.c.k 24 31.c even 3 2
961.2.c.k 24 31.e odd 6 2
961.2.d.r 48 31.d even 5 4
961.2.d.r 48 31.f odd 10 4
961.2.g.v 96 31.g even 15 8
961.2.g.v 96 31.h odd 30 8
8649.2.a.bp 12 3.b odd 2 1
8649.2.a.bp 12 93.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{6} - 10T_{2}^{4} - 2T_{2}^{3} + 28T_{2}^{2} + 8T_{2} - 17 \) Copy content Toggle raw display
\( T_{3}^{12} - 28T_{3}^{10} + 302T_{3}^{8} - 1560T_{3}^{6} + 3844T_{3}^{4} - 3648T_{3}^{2} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 10 T^{4} + \cdots - 17)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 28 T^{10} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( (T^{6} - 4 T^{5} - 10 T^{4} + \cdots - 94)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} - 8 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 56 T^{10} + \cdots + 128 \) Copy content Toggle raw display
$13$ \( T^{12} - 64 T^{10} + \cdots + 84872 \) Copy content Toggle raw display
$17$ \( T^{12} - 128 T^{10} + \cdots + 61693832 \) Copy content Toggle raw display
$19$ \( (T^{6} - 8 T^{5} - 24 T^{4} + \cdots - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 84 T^{10} + \cdots + 2367488 \) Copy content Toggle raw display
$29$ \( T^{12} - 128 T^{10} + \cdots + 1486088 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 144296072 \) Copy content Toggle raw display
$41$ \( (T^{6} - 16 T^{5} + \cdots + 33248)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 197130368 \) Copy content Toggle raw display
$47$ \( (T^{6} - 24 T^{5} + \cdots - 4384)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 84 T^{10} + \cdots + 2048 \) Copy content Toggle raw display
$59$ \( (T^{6} - 20 T^{5} + \cdots + 6392)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} - 208 T^{10} + \cdots + 7388168 \) Copy content Toggle raw display
$67$ \( (T^{6} - 8 T^{5} + \cdots - 359872)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 36 T^{5} + \cdots - 343408)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 6416632328 \) Copy content Toggle raw display
$79$ \( T^{12} - 212 T^{10} + \cdots + 1968128 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 311101568 \) Copy content Toggle raw display
$89$ \( T^{12} - 552 T^{10} + \cdots + 86119688 \) Copy content Toggle raw display
$97$ \( (T^{6} + 4 T^{5} + \cdots + 7550)^{2} \) Copy content Toggle raw display
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