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Properties

Label 4312.2.a.bj
Level $4312$
Weight $2$
Character orbit 4312.a
Self dual yes
Analytic conductor $34.431$
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] = [4312,2,Mod(1,4312)] 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("4312.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4312 = 2^{3} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4312.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,32,0,-12,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.4314933516\)
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: \( x^{12} - 34x^{10} + 421x^{8} - 2246x^{6} + 4540x^{4} - 1832x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{5} + (\beta_{2} + 3) q^{9} - q^{11} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{13} + (\beta_{8} - \beta_{7} + \cdots + \beta_{2}) q^{15} + \beta_{10} q^{17} + (\beta_{11} - \beta_{9}) q^{19}+ \cdots + ( - \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 32 q^{9} - 12 q^{11} + 16 q^{23} + 40 q^{25} + 12 q^{29} + 28 q^{37} - 16 q^{39} - 12 q^{43} + 24 q^{51} + 36 q^{53} + 16 q^{57} + 72 q^{65} - 20 q^{67} + 20 q^{71} - 16 q^{79} + 124 q^{81} - 32 q^{85}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 34x^{10} + 421x^{8} - 2246x^{6} + 4540x^{4} - 1832x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{10} + 192\nu^{8} - 1357\nu^{6} + 4364\nu^{4} - 8340\nu^{2} - 800 ) / 4016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{11} + 868\nu^{9} - 10909\nu^{7} + 58864\nu^{5} - 122228\nu^{3} + 62480\nu ) / 8032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33\nu^{10} - 704\nu^{8} + 3637\nu^{6} + 2740\nu^{4} - 17612\nu^{2} - 11792 ) / 4016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 83\nu^{11} - 2440\nu^{9} + 25455\nu^{7} - 110972\nu^{5} + 178652\nu^{3} - 60448\nu ) / 8032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\nu^{10} - 704\nu^{8} + 3637\nu^{6} + 4748\nu^{4} - 39700\nu^{2} + 6280 ) / 2008 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -83\nu^{10} + 2440\nu^{8} - 25455\nu^{6} + 110972\nu^{4} - 174636\nu^{2} + 28320 ) / 4016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 133\nu^{11} - 4176\nu^{9} + 47273\nu^{7} - 228700\nu^{5} + 415076\nu^{3} - 113120\nu ) / 8032 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -167\nu^{11} + 6240\nu^{9} - 83635\nu^{7} + 469636\nu^{5} - 931180\nu^{3} + 259136\nu ) / 8032 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 193\nu^{11} - 6460\nu^{9} + 79077\nu^{7} - 420776\nu^{5} + 864244\nu^{3} - 355440\nu ) / 8032 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{6} - 2\beta_{4} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 2\beta_{5} + 11\beta_{2} + 57 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} - 14\beta_{9} + 15\beta_{6} - 18\beta_{4} + 88\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{7} - 31\beta_{5} - 11\beta_{3} + 118\beta_{2} + 571 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{11} - 17\beta_{10} - 177\beta_{9} + 188\beta_{6} - 119\beta_{4} + 883\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6\beta_{8} + 165\beta_{7} - 388\beta_{5} - 268\beta_{3} + 1265\beta_{2} + 5839 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 268\beta_{11} - 223\beta_{10} - 2150\beta_{9} + 2245\beta_{6} - 426\beta_{4} + 8984\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 128\beta_{8} + 1894\beta_{7} - 4573\beta_{5} - 4505\beta_{3} + 13602\beta_{2} + 60461 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4505\beta_{11} - 2679\beta_{10} - 25487\beta_{9} + 26340\beta_{6} + 4211\beta_{4} + 92317\beta_1 \) 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
−3.34898
−3.10061
−3.01818
−1.88304
−0.690446
−0.196334
0.196334
0.690446
1.88304
3.01818
3.10061
3.34898
0 −3.34898 0 −2.92555 0 0 0 8.21568 0
1.2 0 −3.10061 0 −2.32216 0 0 0 6.61381 0
1.3 0 −3.01818 0 4.29141 0 0 0 6.10943 0
1.4 0 −1.88304 0 1.20884 0 0 0 0.545823 0
1.5 0 −0.690446 0 3.23978 0 0 0 −2.52328 0
1.6 0 −0.196334 0 −2.38224 0 0 0 −2.96145 0
1.7 0 0.196334 0 2.38224 0 0 0 −2.96145 0
1.8 0 0.690446 0 −3.23978 0 0 0 −2.52328 0
1.9 0 1.88304 0 −1.20884 0 0 0 0.545823 0
1.10 0 3.01818 0 −4.29141 0 0 0 6.10943 0
1.11 0 3.10061 0 2.32216 0 0 0 6.61381 0
1.12 0 3.34898 0 2.92555 0 0 0 8.21568 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(11\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4312.2.a.bj ✓ 12
4.b odd 2 1 8624.2.a.dg 12
7.b odd 2 1 inner 4312.2.a.bj ✓ 12
28.d even 2 1 8624.2.a.dg 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4312.2.a.bj ✓ 12 1.a even 1 1 trivial
4312.2.a.bj ✓ 12 7.b odd 2 1 inner
8624.2.a.dg 12 4.b odd 2 1
8624.2.a.dg 12 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4312))\):

\( T_{3}^{12} - 34T_{3}^{10} + 421T_{3}^{8} - 2246T_{3}^{6} + 4540T_{3}^{4} - 1832T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{12} - 50T_{5}^{10} + 957T_{5}^{8} - 8974T_{5}^{6} + 43020T_{5}^{4} - 97096T_{5}^{2} + 73984 \) Copy content Toggle raw display
\( T_{13}^{12} - 146T_{13}^{10} + 8288T_{13}^{8} - 232704T_{13}^{6} + 3367168T_{13}^{4} - 23244800T_{13}^{2} + 57032704 \) Copy content Toggle raw display
\( T_{17}^{12} - 200T_{17}^{10} + 15568T_{17}^{8} - 589824T_{17}^{6} + 10971136T_{17}^{4} - 84338688T_{17}^{2} + 102252544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 34 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 50 T^{10} + \cdots + 73984 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T + 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 146 T^{10} + \cdots + 57032704 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 102252544 \) Copy content Toggle raw display
$19$ \( T^{12} - 146 T^{10} + \cdots + 589824 \) Copy content Toggle raw display
$23$ \( (T^{6} - 8 T^{5} + \cdots + 656)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots + 22656)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 188 T^{10} + \cdots + 99856 \) Copy content Toggle raw display
$37$ \( (T^{6} - 14 T^{5} + \cdots + 18352)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 232 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$43$ \( (T^{6} + 6 T^{5} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 146506816 \) Copy content Toggle raw display
$53$ \( (T^{6} - 18 T^{5} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14086841344 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 123588808704 \) Copy content Toggle raw display
$67$ \( (T^{6} + 10 T^{5} + \cdots + 155072)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 10 T^{5} + \cdots - 808512)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1605124096 \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17994612736 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 306388818576 \) Copy content Toggle raw display
$97$ \( T^{12} - 292 T^{10} + \cdots + 75272976 \) Copy content Toggle raw display
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