Properties

Label 637.2.r.b
Level $637$
Weight $2$
Character orbit 637.r
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM discriminant -91
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (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: \(x^{4} - 13 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 \beta_{2} q^{4} -\beta_{1} q^{5} + ( 3 - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q -2 \beta_{2} q^{4} -\beta_{1} q^{5} + ( 3 - 3 \beta_{2} ) q^{9} + \beta_{3} q^{13} + ( -4 + 4 \beta_{2} ) q^{16} -\beta_{1} q^{19} + 2 \beta_{3} q^{20} + ( -1 + \beta_{2} ) q^{23} + 8 \beta_{2} q^{25} -5 q^{29} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{31} -6 q^{36} + 2 \beta_{3} q^{41} + 9 q^{43} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{45} -\beta_{1} q^{47} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{52} -11 \beta_{2} q^{53} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{59} + 8 q^{64} + ( 13 - 13 \beta_{2} ) q^{65} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{73} + 2 \beta_{3} q^{76} + ( -15 + 15 \beta_{2} ) q^{79} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{80} -9 \beta_{2} q^{81} -5 \beta_{3} q^{83} -\beta_{1} q^{89} + 2 q^{92} + 13 \beta_{2} q^{95} -5 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 6q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 6q^{9} - 8q^{16} - 2q^{23} + 16q^{25} - 20q^{29} - 24q^{36} + 36q^{43} - 22q^{53} + 32q^{64} + 26q^{65} - 30q^{79} - 18q^{81} + 8q^{92} + 26q^{95} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
3.12250 1.80278i
−3.12250 + 1.80278i
3.12250 + 1.80278i
−3.12250 1.80278i
0 0 −1.00000 + 1.73205i −3.12250 + 1.80278i 0 0 0 1.50000 + 2.59808i 0
116.2 0 0 −1.00000 + 1.73205i 3.12250 1.80278i 0 0 0 1.50000 + 2.59808i 0
324.1 0 0 −1.00000 1.73205i −3.12250 1.80278i 0 0 0 1.50000 2.59808i 0
324.2 0 0 −1.00000 1.73205i 3.12250 + 1.80278i 0 0 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
91.b odd 2 1 CM by \(\Q(\sqrt{-91}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
13.b even 2 1 inner
91.r even 6 1 inner
91.s 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.r.b 4
7.b odd 2 1 inner 637.2.r.b 4
7.c even 3 1 637.2.c.b 2
7.c even 3 1 inner 637.2.r.b 4
7.d odd 6 1 637.2.c.b 2
7.d odd 6 1 inner 637.2.r.b 4
13.b even 2 1 inner 637.2.r.b 4
91.b odd 2 1 CM 637.2.r.b 4
91.r even 6 1 637.2.c.b 2
91.r even 6 1 inner 637.2.r.b 4
91.s odd 6 1 637.2.c.b 2
91.s odd 6 1 inner 637.2.r.b 4
91.z odd 12 2 8281.2.a.u 2
91.bb even 12 2 8281.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.b 2 7.c even 3 1
637.2.c.b 2 7.d odd 6 1
637.2.c.b 2 91.r even 6 1
637.2.c.b 2 91.s odd 6 1
637.2.r.b 4 1.a even 1 1 trivial
637.2.r.b 4 7.b odd 2 1 inner
637.2.r.b 4 7.c even 3 1 inner
637.2.r.b 4 7.d odd 6 1 inner
637.2.r.b 4 13.b even 2 1 inner
637.2.r.b 4 91.b odd 2 1 CM
637.2.r.b 4 91.r even 6 1 inner
637.2.r.b 4 91.s odd 6 1 inner
8281.2.a.u 2 91.z odd 12 2
8281.2.a.u 2 91.bb even 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 169 - 13 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 13 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( 169 - 13 T^{2} + T^{4} \)
$23$ \( ( 1 + T + T^{2} )^{2} \)
$29$ \( ( 5 + T )^{4} \)
$31$ \( 13689 - 117 T^{2} + T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 52 + T^{2} )^{2} \)
$43$ \( ( -9 + T )^{4} \)
$47$ \( 169 - 13 T^{2} + T^{4} \)
$53$ \( ( 121 + 11 T + T^{2} )^{2} \)
$59$ \( 43264 - 208 T^{2} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 13689 - 117 T^{2} + T^{4} \)
$79$ \( ( 225 + 15 T + T^{2} )^{2} \)
$83$ \( ( 325 + T^{2} )^{2} \)
$89$ \( 169 - 13 T^{2} + T^{4} \)
$97$ \( ( 325 + T^{2} )^{2} \)
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