Properties

Label 676.2.h.c
Level $676$
Weight $2$
Character orbit 676.h
Analytic conductor $5.398$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} - \beta_1 q^{7} + ( - 3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} - \beta_1 q^{7} + ( - 3 \beta_{2} + 3) q^{9} + ( - \beta_{3} + \beta_1) q^{11} + ( - 6 \beta_{2} + 6) q^{17} + 3 \beta_1 q^{19} + 8 \beta_{2} q^{23} + q^{25} - 2 \beta_{2} q^{29} + 5 \beta_{3} q^{31} + ( - 4 \beta_{2} + 4) q^{35} + ( - 3 \beta_{3} + 3 \beta_1) q^{37} + (3 \beta_{3} - 3 \beta_1) q^{41} + ( - 4 \beta_{2} + 4) q^{43} + 3 \beta_1 q^{45} + \beta_{3} q^{47} - 3 \beta_{2} q^{49} + 6 q^{53} + 4 \beta_{2} q^{55} - 5 \beta_1 q^{59} + ( - 2 \beta_{2} + 2) q^{61} + (3 \beta_{3} - 3 \beta_1) q^{63} + ( - 5 \beta_{3} + 5 \beta_1) q^{67} - 5 \beta_1 q^{71} - \beta_{3} q^{73} - 4 q^{77} - 4 q^{79} - 9 \beta_{2} q^{81} - 3 \beta_{3} q^{83} + 6 \beta_1 q^{85} + ( - 3 \beta_{3} + 3 \beta_1) q^{89} + (12 \beta_{2} - 12) q^{95} - \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 12 q^{17} + 16 q^{23} + 4 q^{25} - 4 q^{29} + 8 q^{35} + 8 q^{43} - 6 q^{49} + 24 q^{53} + 8 q^{55} + 4 q^{61} - 16 q^{77} - 16 q^{79} - 18 q^{81} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 2.00000i 0 1.73205 + 1.00000i 0 1.50000 2.59808i 0
361.2 0 0 0 2.00000i 0 −1.73205 1.00000i 0 1.50000 2.59808i 0
485.1 0 0 0 2.00000i 0 −1.73205 + 1.00000i 0 1.50000 + 2.59808i 0
485.2 0 0 0 2.00000i 0 1.73205 1.00000i 0 1.50000 + 2.59808i 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
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.2.h.c 4
13.b even 2 1 inner 676.2.h.c 4
13.c even 3 1 676.2.d.c 2
13.c even 3 1 inner 676.2.h.c 4
13.d odd 4 1 676.2.e.b 2
13.d odd 4 1 676.2.e.c 2
13.e even 6 1 676.2.d.c 2
13.e even 6 1 inner 676.2.h.c 4
13.f odd 12 1 52.2.a.a 1
13.f odd 12 1 676.2.a.c 1
13.f odd 12 1 676.2.e.b 2
13.f odd 12 1 676.2.e.c 2
39.h odd 6 1 6084.2.b.m 2
39.i odd 6 1 6084.2.b.m 2
39.k even 12 1 468.2.a.b 1
39.k even 12 1 6084.2.a.m 1
52.i odd 6 1 2704.2.f.f 2
52.j odd 6 1 2704.2.f.f 2
52.l even 12 1 208.2.a.c 1
52.l even 12 1 2704.2.a.g 1
65.o even 12 1 1300.2.c.c 2
65.s odd 12 1 1300.2.a.d 1
65.t even 12 1 1300.2.c.c 2
91.w even 12 1 2548.2.j.f 2
91.x odd 12 1 2548.2.j.e 2
91.ba even 12 1 2548.2.j.f 2
91.bc even 12 1 2548.2.a.e 1
91.bd odd 12 1 2548.2.j.e 2
104.u even 12 1 832.2.a.f 1
104.x odd 12 1 832.2.a.e 1
117.w odd 12 1 4212.2.i.d 2
117.x even 12 1 4212.2.i.i 2
117.bb odd 12 1 4212.2.i.d 2
117.bc even 12 1 4212.2.i.i 2
143.o even 12 1 6292.2.a.g 1
156.v odd 12 1 1872.2.a.f 1
208.be odd 12 1 3328.2.b.q 2
208.bf even 12 1 3328.2.b.e 2
208.bk even 12 1 3328.2.b.e 2
208.bl odd 12 1 3328.2.b.q 2
260.bc even 12 1 5200.2.a.q 1
312.bo even 12 1 7488.2.a.bn 1
312.bq odd 12 1 7488.2.a.bw 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 13.f odd 12 1
208.2.a.c 1 52.l even 12 1
468.2.a.b 1 39.k even 12 1
676.2.a.c 1 13.f odd 12 1
676.2.d.c 2 13.c even 3 1
676.2.d.c 2 13.e even 6 1
676.2.e.b 2 13.d odd 4 1
676.2.e.b 2 13.f odd 12 1
676.2.e.c 2 13.d odd 4 1
676.2.e.c 2 13.f odd 12 1
676.2.h.c 4 1.a even 1 1 trivial
676.2.h.c 4 13.b even 2 1 inner
676.2.h.c 4 13.c even 3 1 inner
676.2.h.c 4 13.e even 6 1 inner
832.2.a.e 1 104.x odd 12 1
832.2.a.f 1 104.u even 12 1
1300.2.a.d 1 65.s odd 12 1
1300.2.c.c 2 65.o even 12 1
1300.2.c.c 2 65.t even 12 1
1872.2.a.f 1 156.v odd 12 1
2548.2.a.e 1 91.bc even 12 1
2548.2.j.e 2 91.x odd 12 1
2548.2.j.e 2 91.bd odd 12 1
2548.2.j.f 2 91.w even 12 1
2548.2.j.f 2 91.ba even 12 1
2704.2.a.g 1 52.l even 12 1
2704.2.f.f 2 52.i odd 6 1
2704.2.f.f 2 52.j odd 6 1
3328.2.b.e 2 208.bf even 12 1
3328.2.b.e 2 208.bk even 12 1
3328.2.b.q 2 208.be odd 12 1
3328.2.b.q 2 208.bl odd 12 1
4212.2.i.d 2 117.w odd 12 1
4212.2.i.d 2 117.bb odd 12 1
4212.2.i.i 2 117.x even 12 1
4212.2.i.i 2 117.bc even 12 1
5200.2.a.q 1 260.bc even 12 1
6084.2.a.m 1 39.k even 12 1
6084.2.b.m 2 39.h odd 6 1
6084.2.b.m 2 39.i odd 6 1
6292.2.a.g 1 143.o even 12 1
7488.2.a.bn 1 312.bo even 12 1
7488.2.a.bw 1 312.bq odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(676, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$71$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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