Properties

Label 3136.1.bc.a
Level $3136$
Weight $1$
Character orbit 3136.bc
Analytic conductor $1.565$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3136.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.14336.1
Artin image: $C_6\times C_4{\rm wrC}_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{9} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{11} -\zeta_{12} q^{25} + ( -1 + \zeta_{12}^{3} ) q^{29} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{37} + ( -1 - \zeta_{12}^{3} ) q^{43} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{53} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{67} + 2 \zeta_{12}^{3} q^{71} -\zeta_{12}^{4} q^{81} + ( 1 - \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 2q^{11} - 4q^{29} + 2q^{37} - 4q^{43} - 2q^{53} + 2q^{67} + 2q^{81} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
913.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 0 0 0 0 0 0 0.866025 0.500000i 0
1489.1 0 0 0 0 0 0 0 −0.866025 0.500000i 0
2481.1 0 0 0 0 0 0 0 −0.866025 + 0.500000i 0
3057.1 0 0 0 0 0 0 0 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner
112.w even 12 1 inner
112.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.1.bc.a 4
4.b odd 2 1 784.1.y.a 4
7.b odd 2 1 CM 3136.1.bc.a 4
7.c even 3 1 448.1.l.a 2
7.c even 3 1 inner 3136.1.bc.a 4
7.d odd 6 1 448.1.l.a 2
7.d odd 6 1 inner 3136.1.bc.a 4
16.e even 4 1 inner 3136.1.bc.a 4
16.f odd 4 1 784.1.y.a 4
28.d even 2 1 784.1.y.a 4
28.f even 6 1 112.1.l.a 2
28.f even 6 1 784.1.y.a 4
28.g odd 6 1 112.1.l.a 2
28.g odd 6 1 784.1.y.a 4
56.j odd 6 1 896.1.l.a 2
56.k odd 6 1 896.1.l.b 2
56.m even 6 1 896.1.l.b 2
56.p even 6 1 896.1.l.a 2
84.j odd 6 1 1008.1.u.b 2
84.n even 6 1 1008.1.u.b 2
112.j even 4 1 784.1.y.a 4
112.l odd 4 1 inner 3136.1.bc.a 4
112.u odd 12 1 112.1.l.a 2
112.u odd 12 1 784.1.y.a 4
112.u odd 12 1 896.1.l.b 2
112.v even 12 1 112.1.l.a 2
112.v even 12 1 784.1.y.a 4
112.v even 12 1 896.1.l.b 2
112.w even 12 1 448.1.l.a 2
112.w even 12 1 896.1.l.a 2
112.w even 12 1 inner 3136.1.bc.a 4
112.x odd 12 1 448.1.l.a 2
112.x odd 12 1 896.1.l.a 2
112.x odd 12 1 inner 3136.1.bc.a 4
140.p odd 6 1 2800.1.z.a 2
140.s even 6 1 2800.1.z.a 2
140.w even 12 1 2800.1.bf.a 2
140.w even 12 1 2800.1.bf.b 2
140.x odd 12 1 2800.1.bf.a 2
140.x odd 12 1 2800.1.bf.b 2
336.br odd 12 1 1008.1.u.b 2
336.bu even 12 1 1008.1.u.b 2
560.ce odd 12 1 2800.1.bf.b 2
560.cf even 12 1 2800.1.bf.a 2
560.co even 12 1 2800.1.z.a 2
560.cs odd 12 1 2800.1.z.a 2
560.da odd 12 1 2800.1.bf.a 2
560.db even 12 1 2800.1.bf.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a 2 28.f even 6 1
112.1.l.a 2 28.g odd 6 1
112.1.l.a 2 112.u odd 12 1
112.1.l.a 2 112.v even 12 1
448.1.l.a 2 7.c even 3 1
448.1.l.a 2 7.d odd 6 1
448.1.l.a 2 112.w even 12 1
448.1.l.a 2 112.x odd 12 1
784.1.y.a 4 4.b odd 2 1
784.1.y.a 4 16.f odd 4 1
784.1.y.a 4 28.d even 2 1
784.1.y.a 4 28.f even 6 1
784.1.y.a 4 28.g odd 6 1
784.1.y.a 4 112.j even 4 1
784.1.y.a 4 112.u odd 12 1
784.1.y.a 4 112.v even 12 1
896.1.l.a 2 56.j odd 6 1
896.1.l.a 2 56.p even 6 1
896.1.l.a 2 112.w even 12 1
896.1.l.a 2 112.x odd 12 1
896.1.l.b 2 56.k odd 6 1
896.1.l.b 2 56.m even 6 1
896.1.l.b 2 112.u odd 12 1
896.1.l.b 2 112.v even 12 1
1008.1.u.b 2 84.j odd 6 1
1008.1.u.b 2 84.n even 6 1
1008.1.u.b 2 336.br odd 12 1
1008.1.u.b 2 336.bu even 12 1
2800.1.z.a 2 140.p odd 6 1
2800.1.z.a 2 140.s even 6 1
2800.1.z.a 2 560.co even 12 1
2800.1.z.a 2 560.cs odd 12 1
2800.1.bf.a 2 140.w even 12 1
2800.1.bf.a 2 140.x odd 12 1
2800.1.bf.a 2 560.cf even 12 1
2800.1.bf.a 2 560.da odd 12 1
2800.1.bf.b 2 140.w even 12 1
2800.1.bf.b 2 140.x odd 12 1
2800.1.bf.b 2 560.ce odd 12 1
2800.1.bf.b 2 560.db even 12 1
3136.1.bc.a 4 1.a even 1 1 trivial
3136.1.bc.a 4 7.b odd 2 1 CM
3136.1.bc.a 4 7.c even 3 1 inner
3136.1.bc.a 4 7.d odd 6 1 inner
3136.1.bc.a 4 16.e even 4 1 inner
3136.1.bc.a 4 112.l odd 4 1 inner
3136.1.bc.a 4 112.w even 12 1 inner
3136.1.bc.a 4 112.x odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 2 + 2 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 2 + 2 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( 4 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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