Properties

Label 637.2.q.d
Level $637$
Weight $2$
Character orbit 637.q
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-13})\)
Defining polynomial: \(x^{4} - 13 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{2} ) q^{2} + ( 1 - \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( 1 - 2 \beta_{2} ) q^{8} + ( 3 - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{2} ) q^{2} + ( 1 - \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( 1 - 2 \beta_{2} ) q^{8} + ( 3 - 3 \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{3} ) q^{10} + ( 4 - 2 \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{13} + 5 \beta_{2} q^{16} + ( \beta_{1} - 2 \beta_{3} ) q^{17} + ( -3 + 6 \beta_{2} ) q^{18} + 2 \beta_{1} q^{19} + \beta_{1} q^{20} + ( -6 + 6 \beta_{2} ) q^{22} -4 \beta_{2} q^{23} -8 q^{25} + ( -\beta_{1} + 2 \beta_{3} ) q^{26} -\beta_{2} q^{29} + ( -3 - 3 \beta_{2} ) q^{32} + 3 \beta_{3} q^{34} -3 \beta_{2} q^{36} + ( -2 + \beta_{2} ) q^{37} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{38} + ( 2 \beta_{1} - \beta_{3} ) q^{40} + ( -\beta_{1} + \beta_{3} ) q^{41} + ( -6 + 6 \beta_{2} ) q^{43} + ( 2 - 4 \beta_{2} ) q^{44} + 3 \beta_{1} q^{45} + ( 4 + 4 \beta_{2} ) q^{46} -2 \beta_{3} q^{47} + ( 16 - 8 \beta_{2} ) q^{50} -\beta_{3} q^{52} + 5 q^{53} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{55} + ( 1 + \beta_{2} ) q^{58} + 2 \beta_{1} q^{59} + ( -\beta_{1} + 2 \beta_{3} ) q^{61} - q^{64} + 13 \beta_{2} q^{65} + ( 16 - 8 \beta_{2} ) q^{67} + ( -\beta_{1} - \beta_{3} ) q^{68} + ( -6 - 6 \beta_{2} ) q^{71} + ( -3 - 3 \beta_{2} ) q^{72} + 3 \beta_{3} q^{73} + ( 3 - 3 \beta_{2} ) q^{74} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{76} -6 q^{79} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{80} -9 \beta_{2} q^{81} + ( \beta_{1} - 2 \beta_{3} ) q^{82} -2 \beta_{3} q^{83} + ( 13 + 13 \beta_{2} ) q^{85} + ( 6 - 12 \beta_{2} ) q^{86} -6 \beta_{2} q^{88} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{90} -4 q^{92} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{94} + ( -26 + 26 \beta_{2} ) q^{95} + 2 \beta_{1} q^{97} + ( 6 - 12 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{2} + 2q^{4} + 6q^{9} + O(q^{10}) \) \( 4q - 6q^{2} + 2q^{4} + 6q^{9} + 12q^{11} + 10q^{16} - 12q^{22} - 8q^{23} - 32q^{25} - 2q^{29} - 18q^{32} - 6q^{36} - 6q^{37} - 12q^{43} + 24q^{46} + 48q^{50} + 20q^{53} + 6q^{58} - 4q^{64} + 26q^{65} + 48q^{67} - 36q^{71} - 18q^{72} + 6q^{74} - 24q^{79} - 18q^{81} + 78q^{85} - 12q^{88} - 16q^{92} - 52q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 13 x^{2} + 169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/13\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(13 \beta_{2}\)
\(\nu^{3}\)\(=\)\(13 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
−3.12250 1.80278i
3.12250 + 1.80278i
3.12250 1.80278i
−3.12250 + 1.80278i
−1.50000 + 0.866025i 0 0.500000 0.866025i 3.60555i 0 0 1.73205i 1.50000 2.59808i 3.12250 + 5.40833i
491.2 −1.50000 + 0.866025i 0 0.500000 0.866025i 3.60555i 0 0 1.73205i 1.50000 2.59808i −3.12250 5.40833i
589.1 −1.50000 0.866025i 0 0.500000 + 0.866025i 3.60555i 0 0 1.73205i 1.50000 + 2.59808i −3.12250 + 5.40833i
589.2 −1.50000 0.866025i 0 0.500000 + 0.866025i 3.60555i 0 0 1.73205i 1.50000 + 2.59808i 3.12250 5.40833i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.e even 6 1 inner
91.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.q.d 4
7.b odd 2 1 inner 637.2.q.d 4
7.c even 3 1 637.2.k.e 4
7.c even 3 1 637.2.u.f 4
7.d odd 6 1 637.2.k.e 4
7.d odd 6 1 637.2.u.f 4
13.e even 6 1 inner 637.2.q.d 4
13.f odd 12 2 8281.2.a.br 4
91.k even 6 1 637.2.u.f 4
91.l odd 6 1 637.2.u.f 4
91.p odd 6 1 637.2.k.e 4
91.t odd 6 1 inner 637.2.q.d 4
91.u even 6 1 637.2.k.e 4
91.bc even 12 2 8281.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.e 4 7.c even 3 1
637.2.k.e 4 7.d odd 6 1
637.2.k.e 4 91.p odd 6 1
637.2.k.e 4 91.u even 6 1
637.2.q.d 4 1.a even 1 1 trivial
637.2.q.d 4 7.b odd 2 1 inner
637.2.q.d 4 13.e even 6 1 inner
637.2.q.d 4 91.t odd 6 1 inner
637.2.u.f 4 7.c even 3 1
637.2.u.f 4 7.d odd 6 1
637.2.u.f 4 91.k even 6 1
637.2.u.f 4 91.l odd 6 1
8281.2.a.br 4 13.f odd 12 2
8281.2.a.br 4 91.bc even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{2} + 3 T_{2} + 3 \)
\( T_{3} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 3 + 3 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 13 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 12 - 6 T + T^{2} )^{2} \)
$13$ \( 169 - 13 T^{2} + T^{4} \)
$17$ \( 1521 + 39 T^{2} + T^{4} \)
$19$ \( 2704 - 52 T^{2} + T^{4} \)
$23$ \( ( 16 + 4 T + T^{2} )^{2} \)
$29$ \( ( 1 + T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 3 + 3 T + T^{2} )^{2} \)
$41$ \( 169 - 13 T^{2} + T^{4} \)
$43$ \( ( 36 + 6 T + T^{2} )^{2} \)
$47$ \( ( 52 + T^{2} )^{2} \)
$53$ \( ( -5 + T )^{4} \)
$59$ \( 2704 - 52 T^{2} + T^{4} \)
$61$ \( 1521 + 39 T^{2} + T^{4} \)
$67$ \( ( 192 - 24 T + T^{2} )^{2} \)
$71$ \( ( 108 + 18 T + T^{2} )^{2} \)
$73$ \( ( 117 + T^{2} )^{2} \)
$79$ \( ( 6 + T )^{4} \)
$83$ \( ( 52 + T^{2} )^{2} \)
$89$ \( 2704 - 52 T^{2} + T^{4} \)
$97$ \( 2704 - 52 T^{2} + T^{4} \)
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