Properties

Label 676.1.c.a
Level $676$
Weight $1$
Character orbit 676.c
Self dual yes
Analytic conductor $0.337$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 676.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.337367948540\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.5940688.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{5} - q^{8} + q^{9} - q^{10} + q^{16} - q^{17} - q^{18} + q^{20} - q^{29} - q^{32} + q^{34} + q^{36} + q^{37} - q^{40} + q^{41} + q^{45} + q^{49} - q^{53} + q^{58} - q^{61} + q^{64} - q^{68} - q^{72} + q^{73} - q^{74} + q^{80} + q^{81} - q^{82} - q^{85} - 2q^{89} - q^{90} - 2q^{97} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
0
−1.00000 0 1.00000 1.00000 0 0 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.1.c.a 1
4.b odd 2 1 CM 676.1.c.a 1
13.b even 2 1 676.1.c.b 1
13.c even 3 2 676.1.j.a 2
13.d odd 4 2 676.1.b.a 2
13.e even 6 2 52.1.j.a 2
13.f odd 12 4 676.1.i.a 4
39.h odd 6 2 468.1.br.a 2
52.b odd 2 1 676.1.c.b 1
52.f even 4 2 676.1.b.a 2
52.i odd 6 2 52.1.j.a 2
52.j odd 6 2 676.1.j.a 2
52.l even 12 4 676.1.i.a 4
65.l even 6 2 1300.1.bc.a 2
65.r odd 12 4 1300.1.w.a 4
91.k even 6 2 2548.1.bi.b 2
91.l odd 6 2 2548.1.bi.a 2
91.p odd 6 2 2548.1.q.a 2
91.t odd 6 2 2548.1.bn.a 2
91.u even 6 2 2548.1.q.b 2
104.p odd 6 2 832.1.bb.a 2
104.s even 6 2 832.1.bb.a 2
156.r even 6 2 468.1.br.a 2
208.bh even 12 4 3328.1.v.b 4
208.bi odd 12 4 3328.1.v.b 4
260.w odd 6 2 1300.1.bc.a 2
260.bg even 12 4 1300.1.w.a 4
364.s odd 6 2 2548.1.q.b 2
364.w even 6 2 2548.1.bi.a 2
364.bc even 6 2 2548.1.bn.a 2
364.bk odd 6 2 2548.1.bi.b 2
364.bp even 6 2 2548.1.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 13.e even 6 2
52.1.j.a 2 52.i odd 6 2
468.1.br.a 2 39.h odd 6 2
468.1.br.a 2 156.r even 6 2
676.1.b.a 2 13.d odd 4 2
676.1.b.a 2 52.f even 4 2
676.1.c.a 1 1.a even 1 1 trivial
676.1.c.a 1 4.b odd 2 1 CM
676.1.c.b 1 13.b even 2 1
676.1.c.b 1 52.b odd 2 1
676.1.i.a 4 13.f odd 12 4
676.1.i.a 4 52.l even 12 4
676.1.j.a 2 13.c even 3 2
676.1.j.a 2 52.j odd 6 2
832.1.bb.a 2 104.p odd 6 2
832.1.bb.a 2 104.s even 6 2
1300.1.w.a 4 65.r odd 12 4
1300.1.w.a 4 260.bg even 12 4
1300.1.bc.a 2 65.l even 6 2
1300.1.bc.a 2 260.w odd 6 2
2548.1.q.a 2 91.p odd 6 2
2548.1.q.a 2 364.bp even 6 2
2548.1.q.b 2 91.u even 6 2
2548.1.q.b 2 364.s odd 6 2
2548.1.bi.a 2 91.l odd 6 2
2548.1.bi.a 2 364.w even 6 2
2548.1.bi.b 2 91.k even 6 2
2548.1.bi.b 2 364.bk odd 6 2
2548.1.bn.a 2 91.t odd 6 2
2548.1.bn.a 2 364.bc even 6 2
3328.1.v.b 4 208.bh even 12 4
3328.1.v.b 4 208.bi odd 12 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1 \) acting on \(S_{1}^{\mathrm{new}}(676, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( 1 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 1 + T \)
$31$ \( T \)
$37$ \( -1 + T \)
$41$ \( -1 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( 1 + T \)
$59$ \( T \)
$61$ \( 1 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -1 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( 2 + T \)
$97$ \( 2 + T \)
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