Properties

Label 637.2.u.f
Level $637$
Weight $2$
Character orbit 637.u
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.u (of order \(6\), degree \(2\), not 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: \( x^{4} - 13x^{2} + 169 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2) q^{2} + ( - \beta_{2} + 1) q^{4} + \beta_1 q^{5} + (2 \beta_{2} - 1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 2) q^{2} + ( - \beta_{2} + 1) q^{4} + \beta_1 q^{5} + (2 \beta_{2} - 1) q^{8} - 3 q^{9} + ( - \beta_{3} + 2 \beta_1) q^{10} + ( - 4 \beta_{2} + 2) q^{11} + \beta_1 q^{13} + 5 \beta_{2} q^{16} + ( - 2 \beta_{3} + \beta_1) q^{17} + (3 \beta_{2} - 6) q^{18} + 2 \beta_{3} q^{19} + ( - \beta_{3} + \beta_1) q^{20} - 6 \beta_{2} q^{22} - 4 \beta_{2} q^{23} + 8 \beta_{2} q^{25} + ( - \beta_{3} + 2 \beta_1) q^{26} + (\beta_{2} - 1) q^{29} + (3 \beta_{2} + 3) q^{32} - 3 \beta_{3} q^{34} + (3 \beta_{2} - 3) q^{36} + ( - \beta_{2} + 2) q^{37} + (2 \beta_{3} + 2 \beta_1) q^{38} + (2 \beta_{3} - \beta_1) q^{40} - \beta_1 q^{41} - 6 \beta_{2} q^{43} + ( - 2 \beta_{2} - 2) q^{44} - 3 \beta_1 q^{45} + ( - 4 \beta_{2} - 4) q^{46} - 2 \beta_1 q^{47} + (8 \beta_{2} + 8) q^{50} + ( - \beta_{3} + \beta_1) q^{52} - 5 \beta_{2} q^{53} + ( - 4 \beta_{3} + 2 \beta_1) q^{55} + (2 \beta_{2} - 1) q^{58} - 2 \beta_1 q^{59} + ( - \beta_{3} + 2 \beta_1) q^{61} - q^{64} + 13 \beta_{2} q^{65} + ( - 16 \beta_{2} + 8) q^{67} + ( - \beta_{3} - \beta_1) q^{68} + (6 \beta_{2} - 12) q^{71} + ( - 6 \beta_{2} + 3) q^{72} + (3 \beta_{3} - 3 \beta_1) q^{73} + ( - 3 \beta_{2} + 3) q^{74} + 2 \beta_1 q^{76} + ( - 6 \beta_{2} + 6) q^{79} + 5 \beta_{3} q^{80} + 9 q^{81} + (\beta_{3} - 2 \beta_1) q^{82} + 2 \beta_{3} q^{83} + ( - 13 \beta_{2} + 26) q^{85} + ( - 6 \beta_{2} - 6) q^{86} + 6 q^{88} + ( - 2 \beta_{3} + 2 \beta_1) q^{89} + (3 \beta_{3} - 6 \beta_1) q^{90} - 4 q^{92} + (2 \beta_{3} - 4 \beta_1) q^{94} + (26 \beta_{2} - 26) q^{95} + ( - 2 \beta_{3} + 2 \beta_1) q^{97} + (12 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 2 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 2 q^{4} - 12 q^{9} + 10 q^{16} - 18 q^{18} - 12 q^{22} - 8 q^{23} + 16 q^{25} - 2 q^{29} + 18 q^{32} - 6 q^{36} + 6 q^{37} - 12 q^{43} - 12 q^{44} - 24 q^{46} + 48 q^{50} - 10 q^{53} - 4 q^{64} + 26 q^{65} - 36 q^{71} + 6 q^{74} + 12 q^{79} + 36 q^{81} + 78 q^{85} - 36 q^{86} + 24 q^{88} - 16 q^{92} - 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
−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 −3.12250 1.80278i 0 0 1.73205i −3.00000 −6.24500
30.2 1.50000 0.866025i 0 0.500000 0.866025i 3.12250 + 1.80278i 0 0 1.73205i −3.00000 6.24500
361.1 1.50000 + 0.866025i 0 0.500000 + 0.866025i −3.12250 + 1.80278i 0 0 1.73205i −3.00000 −6.24500
361.2 1.50000 + 0.866025i 0 0.500000 + 0.866025i 3.12250 1.80278i 0 0 1.73205i −3.00000 6.24500
\(n\): 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
91.p odd 6 1 inner
91.u even 6 1 inner

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 39T^{2} + 1521 \) Copy content Toggle raw display
$19$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
$61$ \( (T^{2} - 39)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 117 T^{2} + 13689 \) Copy content Toggle raw display
$79$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
$97$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
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