Properties

Label 676.1.i.a
Level $676$
Weight $1$
Character orbit 676.i
Analytic conductor $0.337$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 676.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.337367948540\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{3} q^{5} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{3} q^{5} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} + \zeta_{12}^{4} q^{10} + \zeta_{12}^{4} q^{16} -\zeta_{12}^{2} q^{17} + \zeta_{12}^{3} q^{18} -\zeta_{12}^{5} q^{20} -\zeta_{12}^{4} q^{29} -\zeta_{12}^{5} q^{32} + \zeta_{12}^{3} q^{34} -\zeta_{12}^{4} q^{36} -\zeta_{12} q^{37} - q^{40} + \zeta_{12} q^{41} + \zeta_{12}^{5} q^{45} -\zeta_{12}^{4} q^{49} - q^{53} + \zeta_{12}^{5} q^{58} + \zeta_{12}^{2} q^{61} - q^{64} -\zeta_{12}^{4} q^{68} + \zeta_{12}^{5} q^{72} + \zeta_{12}^{3} q^{73} + \zeta_{12}^{2} q^{74} + \zeta_{12} q^{80} + \zeta_{12}^{4} q^{81} -\zeta_{12}^{2} q^{82} + \zeta_{12}^{5} q^{85} + 2 \zeta_{12} q^{89} + q^{90} -2 \zeta_{12}^{5} q^{97} + \zeta_{12}^{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 2q^{9} - 2q^{10} - 2q^{16} - 2q^{17} + 2q^{29} + 2q^{36} - 4q^{40} + 2q^{49} - 4q^{53} + 2q^{61} - 4q^{64} + 2q^{68} + 2q^{74} - 2q^{81} - 2q^{82} + 4q^{90} + O(q^{100}) \)

Character values

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

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 1.00000i 0 0 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
23.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 0 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
147.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
147.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
\(n\): 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 \(\Q(\sqrt{-1}) \)
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
52.b odd 2 1 inner
52.i odd 6 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.1.i.a 4
4.b odd 2 1 CM 676.1.i.a 4
13.b even 2 1 inner 676.1.i.a 4
13.c even 3 1 676.1.b.a 2
13.c even 3 1 inner 676.1.i.a 4
13.d odd 4 1 52.1.j.a 2
13.d odd 4 1 676.1.j.a 2
13.e even 6 1 676.1.b.a 2
13.e even 6 1 inner 676.1.i.a 4
13.f odd 12 1 52.1.j.a 2
13.f odd 12 1 676.1.c.a 1
13.f odd 12 1 676.1.c.b 1
13.f odd 12 1 676.1.j.a 2
39.f even 4 1 468.1.br.a 2
39.k even 12 1 468.1.br.a 2
52.b odd 2 1 inner 676.1.i.a 4
52.f even 4 1 52.1.j.a 2
52.f even 4 1 676.1.j.a 2
52.i odd 6 1 676.1.b.a 2
52.i odd 6 1 inner 676.1.i.a 4
52.j odd 6 1 676.1.b.a 2
52.j odd 6 1 inner 676.1.i.a 4
52.l even 12 1 52.1.j.a 2
52.l even 12 1 676.1.c.a 1
52.l even 12 1 676.1.c.b 1
52.l even 12 1 676.1.j.a 2
65.f even 4 1 1300.1.w.a 4
65.g odd 4 1 1300.1.bc.a 2
65.k even 4 1 1300.1.w.a 4
65.o even 12 1 1300.1.w.a 4
65.s odd 12 1 1300.1.bc.a 2
65.t even 12 1 1300.1.w.a 4
91.i even 4 1 2548.1.bn.a 2
91.w even 12 1 2548.1.bi.a 2
91.x odd 12 1 2548.1.q.b 2
91.z odd 12 1 2548.1.q.b 2
91.z odd 12 1 2548.1.bi.b 2
91.ba even 12 1 2548.1.q.a 2
91.bb even 12 1 2548.1.q.a 2
91.bb even 12 1 2548.1.bi.a 2
91.bc even 12 1 2548.1.bn.a 2
91.bd odd 12 1 2548.1.bi.b 2
104.j odd 4 1 832.1.bb.a 2
104.m even 4 1 832.1.bb.a 2
104.u even 12 1 832.1.bb.a 2
104.x odd 12 1 832.1.bb.a 2
156.l odd 4 1 468.1.br.a 2
156.v odd 12 1 468.1.br.a 2
208.l even 4 1 3328.1.v.b 4
208.m odd 4 1 3328.1.v.b 4
208.r odd 4 1 3328.1.v.b 4
208.s even 4 1 3328.1.v.b 4
208.be odd 12 1 3328.1.v.b 4
208.bf even 12 1 3328.1.v.b 4
208.bk even 12 1 3328.1.v.b 4
208.bl odd 12 1 3328.1.v.b 4
260.l odd 4 1 1300.1.w.a 4
260.s odd 4 1 1300.1.w.a 4
260.u even 4 1 1300.1.bc.a 2
260.bc even 12 1 1300.1.bc.a 2
260.be odd 12 1 1300.1.w.a 4
260.bl odd 12 1 1300.1.w.a 4
364.p odd 4 1 2548.1.bn.a 2
364.bt even 12 1 2548.1.bi.b 2
364.bv odd 12 1 2548.1.bn.a 2
364.bw odd 12 1 2548.1.q.a 2
364.bw odd 12 1 2548.1.bi.a 2
364.bz odd 12 1 2548.1.q.a 2
364.ca even 12 1 2548.1.q.b 2
364.ce even 12 1 2548.1.q.b 2
364.ce even 12 1 2548.1.bi.b 2
364.cg odd 12 1 2548.1.bi.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 13.d odd 4 1
52.1.j.a 2 13.f odd 12 1
52.1.j.a 2 52.f even 4 1
52.1.j.a 2 52.l even 12 1
468.1.br.a 2 39.f even 4 1
468.1.br.a 2 39.k even 12 1
468.1.br.a 2 156.l odd 4 1
468.1.br.a 2 156.v odd 12 1
676.1.b.a 2 13.c even 3 1
676.1.b.a 2 13.e even 6 1
676.1.b.a 2 52.i odd 6 1
676.1.b.a 2 52.j odd 6 1
676.1.c.a 1 13.f odd 12 1
676.1.c.a 1 52.l even 12 1
676.1.c.b 1 13.f odd 12 1
676.1.c.b 1 52.l even 12 1
676.1.i.a 4 1.a even 1 1 trivial
676.1.i.a 4 4.b odd 2 1 CM
676.1.i.a 4 13.b even 2 1 inner
676.1.i.a 4 13.c even 3 1 inner
676.1.i.a 4 13.e even 6 1 inner
676.1.i.a 4 52.b odd 2 1 inner
676.1.i.a 4 52.i odd 6 1 inner
676.1.i.a 4 52.j odd 6 1 inner
676.1.j.a 2 13.d odd 4 1
676.1.j.a 2 13.f odd 12 1
676.1.j.a 2 52.f even 4 1
676.1.j.a 2 52.l even 12 1
832.1.bb.a 2 104.j odd 4 1
832.1.bb.a 2 104.m even 4 1
832.1.bb.a 2 104.u even 12 1
832.1.bb.a 2 104.x odd 12 1
1300.1.w.a 4 65.f even 4 1
1300.1.w.a 4 65.k even 4 1
1300.1.w.a 4 65.o even 12 1
1300.1.w.a 4 65.t even 12 1
1300.1.w.a 4 260.l odd 4 1
1300.1.w.a 4 260.s odd 4 1
1300.1.w.a 4 260.be odd 12 1
1300.1.w.a 4 260.bl odd 12 1
1300.1.bc.a 2 65.g odd 4 1
1300.1.bc.a 2 65.s odd 12 1
1300.1.bc.a 2 260.u even 4 1
1300.1.bc.a 2 260.bc even 12 1
2548.1.q.a 2 91.ba even 12 1
2548.1.q.a 2 91.bb even 12 1
2548.1.q.a 2 364.bw odd 12 1
2548.1.q.a 2 364.bz odd 12 1
2548.1.q.b 2 91.x odd 12 1
2548.1.q.b 2 91.z odd 12 1
2548.1.q.b 2 364.ca even 12 1
2548.1.q.b 2 364.ce even 12 1
2548.1.bi.a 2 91.w even 12 1
2548.1.bi.a 2 91.bb even 12 1
2548.1.bi.a 2 364.bw odd 12 1
2548.1.bi.a 2 364.cg odd 12 1
2548.1.bi.b 2 91.z odd 12 1
2548.1.bi.b 2 91.bd odd 12 1
2548.1.bi.b 2 364.bt even 12 1
2548.1.bi.b 2 364.ce even 12 1
2548.1.bn.a 2 91.i even 4 1
2548.1.bn.a 2 91.bc even 12 1
2548.1.bn.a 2 364.p odd 4 1
2548.1.bn.a 2 364.bv odd 12 1
3328.1.v.b 4 208.l even 4 1
3328.1.v.b 4 208.m odd 4 1
3328.1.v.b 4 208.r odd 4 1
3328.1.v.b 4 208.s even 4 1
3328.1.v.b 4 208.be odd 12 1
3328.1.v.b 4 208.bf even 12 1
3328.1.v.b 4 208.bk even 12 1
3328.1.v.b 4 208.bl odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(676, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 1 + T + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 1 - T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 1 - T^{2} + T^{4} \)
$41$ \( 1 - T^{2} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 1 + T )^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 16 - 4 T^{2} + T^{4} \)
$97$ \( 16 - 4 T^{2} + T^{4} \)
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