Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.337367948540\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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: $C_6\times S_3$
Artin field: Galois closure of 12.0.3341233033216.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{5} - q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{5} - q^{8} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{10} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{17} - q^{18} -\zeta_{6} q^{20} -\zeta_{6}^{2} q^{29} + \zeta_{6} q^{32} + q^{34} + \zeta_{6}^{2} q^{36} + \zeta_{6}^{2} q^{37} - q^{40} + \zeta_{6}^{2} q^{41} -\zeta_{6} q^{45} + \zeta_{6}^{2} q^{49} - q^{53} -\zeta_{6} q^{58} + \zeta_{6} q^{61} + q^{64} -\zeta_{6}^{2} q^{68} + \zeta_{6} q^{72} + q^{73} + \zeta_{6} q^{74} + \zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} + \zeta_{6} q^{85} -2 \zeta_{6}^{2} q^{89} - q^{90} + 2 \zeta_{6} q^{97} + \zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{8} - q^{9} + O(q^{10}) \) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{8} - q^{9} + q^{10} - q^{16} + q^{17} - 2 q^{18} - q^{20} + q^{29} + q^{32} + 2 q^{34} - q^{36} - q^{37} - 2 q^{40} - q^{41} - q^{45} - q^{49} - 2 q^{53} - q^{58} + q^{61} + 2 q^{64} + q^{68} + q^{72} + 2 q^{73} + q^{74} - q^{80} - q^{81} + q^{82} + q^{85} + 2 q^{89} - 2 q^{90} + 2 q^{97} + q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 0 −1.00000 −0.500000 0.866025i 0.500000 0.866025i
315.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 0 −1.00000 −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.c even 3 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.j.a 2
4.b odd 2 1 CM 676.1.j.a 2
13.b even 2 1 52.1.j.a 2
13.c even 3 1 676.1.c.a 1
13.c even 3 1 inner 676.1.j.a 2
13.d odd 4 2 676.1.i.a 4
13.e even 6 1 52.1.j.a 2
13.e even 6 1 676.1.c.b 1
13.f odd 12 2 676.1.b.a 2
13.f odd 12 2 676.1.i.a 4
39.d odd 2 1 468.1.br.a 2
39.h odd 6 1 468.1.br.a 2
52.b odd 2 1 52.1.j.a 2
52.f even 4 2 676.1.i.a 4
52.i odd 6 1 52.1.j.a 2
52.i odd 6 1 676.1.c.b 1
52.j odd 6 1 676.1.c.a 1
52.j odd 6 1 inner 676.1.j.a 2
52.l even 12 2 676.1.b.a 2
52.l even 12 2 676.1.i.a 4
65.d even 2 1 1300.1.bc.a 2
65.h odd 4 2 1300.1.w.a 4
65.l even 6 1 1300.1.bc.a 2
65.r odd 12 2 1300.1.w.a 4
91.b odd 2 1 2548.1.bn.a 2
91.k even 6 1 2548.1.q.b 2
91.l odd 6 1 2548.1.q.a 2
91.p odd 6 1 2548.1.bi.a 2
91.r even 6 1 2548.1.q.b 2
91.r even 6 1 2548.1.bi.b 2
91.s odd 6 1 2548.1.q.a 2
91.s odd 6 1 2548.1.bi.a 2
91.t odd 6 1 2548.1.bn.a 2
91.u even 6 1 2548.1.bi.b 2
104.e even 2 1 832.1.bb.a 2
104.h odd 2 1 832.1.bb.a 2
104.p odd 6 1 832.1.bb.a 2
104.s even 6 1 832.1.bb.a 2
156.h even 2 1 468.1.br.a 2
156.r even 6 1 468.1.br.a 2
208.o odd 4 2 3328.1.v.b 4
208.p even 4 2 3328.1.v.b 4
208.bh even 12 2 3328.1.v.b 4
208.bi odd 12 2 3328.1.v.b 4
260.g odd 2 1 1300.1.bc.a 2
260.p even 4 2 1300.1.w.a 4
260.w odd 6 1 1300.1.bc.a 2
260.bg even 12 2 1300.1.w.a 4
364.h even 2 1 2548.1.bn.a 2
364.s odd 6 1 2548.1.bi.b 2
364.w even 6 1 2548.1.q.a 2
364.x even 6 1 2548.1.q.a 2
364.x even 6 1 2548.1.bi.a 2
364.bc even 6 1 2548.1.bn.a 2
364.bk odd 6 1 2548.1.q.b 2
364.bl odd 6 1 2548.1.q.b 2
364.bl odd 6 1 2548.1.bi.b 2
364.bp even 6 1 2548.1.bi.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 13.b even 2 1
52.1.j.a 2 13.e even 6 1
52.1.j.a 2 52.b odd 2 1
52.1.j.a 2 52.i odd 6 1
468.1.br.a 2 39.d odd 2 1
468.1.br.a 2 39.h odd 6 1
468.1.br.a 2 156.h even 2 1
468.1.br.a 2 156.r even 6 1
676.1.b.a 2 13.f odd 12 2
676.1.b.a 2 52.l even 12 2
676.1.c.a 1 13.c even 3 1
676.1.c.a 1 52.j odd 6 1
676.1.c.b 1 13.e even 6 1
676.1.c.b 1 52.i odd 6 1
676.1.i.a 4 13.d odd 4 2
676.1.i.a 4 13.f odd 12 2
676.1.i.a 4 52.f even 4 2
676.1.i.a 4 52.l even 12 2
676.1.j.a 2 1.a even 1 1 trivial
676.1.j.a 2 4.b odd 2 1 CM
676.1.j.a 2 13.c even 3 1 inner
676.1.j.a 2 52.j odd 6 1 inner
832.1.bb.a 2 104.e even 2 1
832.1.bb.a 2 104.h odd 2 1
832.1.bb.a 2 104.p odd 6 1
832.1.bb.a 2 104.s even 6 1
1300.1.w.a 4 65.h odd 4 2
1300.1.w.a 4 65.r odd 12 2
1300.1.w.a 4 260.p even 4 2
1300.1.w.a 4 260.bg even 12 2
1300.1.bc.a 2 65.d even 2 1
1300.1.bc.a 2 65.l even 6 1
1300.1.bc.a 2 260.g odd 2 1
1300.1.bc.a 2 260.w odd 6 1
2548.1.q.a 2 91.l odd 6 1
2548.1.q.a 2 91.s odd 6 1
2548.1.q.a 2 364.w even 6 1
2548.1.q.a 2 364.x even 6 1
2548.1.q.b 2 91.k even 6 1
2548.1.q.b 2 91.r even 6 1
2548.1.q.b 2 364.bk odd 6 1
2548.1.q.b 2 364.bl odd 6 1
2548.1.bi.a 2 91.p odd 6 1
2548.1.bi.a 2 91.s odd 6 1
2548.1.bi.a 2 364.x even 6 1
2548.1.bi.a 2 364.bp even 6 1
2548.1.bi.b 2 91.r even 6 1
2548.1.bi.b 2 91.u even 6 1
2548.1.bi.b 2 364.s odd 6 1
2548.1.bi.b 2 364.bl odd 6 1
2548.1.bn.a 2 91.b odd 2 1
2548.1.bn.a 2 91.t odd 6 1
2548.1.bn.a 2 364.h even 2 1
2548.1.bn.a 2 364.bc even 6 1
3328.1.v.b 4 208.o odd 4 2
3328.1.v.b 4 208.p even 4 2
3328.1.v.b 4 208.bh even 12 2
3328.1.v.b 4 208.bi odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

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