Properties

Label 676.2.e.b
Level $676$
Weight $2$
Character orbit 676.e
Analytic conductor $5.398$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.39788717664\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{5} - 2 \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - 2 \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - 6 \zeta_{6} q^{17} - 6 \zeta_{6} q^{19} + (8 \zeta_{6} - 8) q^{23} - q^{25} + (2 \zeta_{6} - 2) q^{29} - 10 q^{31} + 4 \zeta_{6} q^{35} + (6 \zeta_{6} - 6) q^{37} + (6 \zeta_{6} - 6) q^{41} - 4 \zeta_{6} q^{43} - 6 \zeta_{6} q^{45} + 2 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + 6 q^{53} + ( - 4 \zeta_{6} + 4) q^{55} - 10 \zeta_{6} q^{59} + 2 \zeta_{6} q^{61} + ( - 6 \zeta_{6} + 6) q^{63} + ( - 10 \zeta_{6} + 10) q^{67} + 10 \zeta_{6} q^{71} - 2 q^{73} + 4 q^{77} - 4 q^{79} + (9 \zeta_{6} - 9) q^{81} + 6 q^{83} + 12 \zeta_{6} q^{85} + (6 \zeta_{6} - 6) q^{89} + 12 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{17} - 6 q^{19} - 8 q^{23} - 2 q^{25} - 2 q^{29} - 20 q^{31} + 4 q^{35} - 6 q^{37} - 6 q^{41} - 4 q^{43} - 6 q^{45} + 4 q^{47} + 3 q^{49} + 12 q^{53} + 4 q^{55} - 10 q^{59} + 2 q^{61} + 6 q^{63} + 10 q^{67} + 10 q^{71} - 4 q^{73} + 8 q^{77} - 8 q^{79} - 9 q^{81} + 12 q^{83} + 12 q^{85} - 6 q^{89} + 12 q^{95} + 2 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.00000 0 −1.00000 1.73205i 0 1.50000 + 2.59808i 0
653.1 0 0 0 −2.00000 0 −1.00000 + 1.73205i 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.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(676, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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