Properties

Label 637.2.a.a
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 2q^{4} + 3q^{5} - 3q^{9} + O(q^{10}) \) \( q - 2q^{2} + 2q^{4} + 3q^{5} - 3q^{9} - 6q^{10} - 6q^{11} + q^{13} - 4q^{16} - 4q^{17} + 6q^{18} - 5q^{19} + 6q^{20} + 12q^{22} + 3q^{23} + 4q^{25} - 2q^{26} - 5q^{29} + 3q^{31} + 8q^{32} + 8q^{34} - 6q^{36} - 4q^{37} + 10q^{38} + 6q^{41} - q^{43} - 12q^{44} - 9q^{45} - 6q^{46} - 7q^{47} - 8q^{50} + 2q^{52} - 9q^{53} - 18q^{55} + 10q^{58} - 8q^{59} + 10q^{61} - 6q^{62} - 8q^{64} + 3q^{65} - 6q^{67} - 8q^{68} - 8q^{71} + 13q^{73} + 8q^{74} - 10q^{76} + 3q^{79} - 12q^{80} + 9q^{81} - 12q^{82} - 15q^{83} - 12q^{85} + 2q^{86} - 3q^{89} + 18q^{90} + 6q^{92} + 14q^{94} - 15q^{95} - 7q^{97} + 18q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 0 2.00000 3.00000 0 0 0 −3.00000 −6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.a.a 1
3.b odd 2 1 5733.2.a.l 1
7.b odd 2 1 91.2.a.a 1
7.c even 3 2 637.2.e.d 2
7.d odd 6 2 637.2.e.e 2
13.b even 2 1 8281.2.a.l 1
21.c even 2 1 819.2.a.f 1
28.d even 2 1 1456.2.a.g 1
35.c odd 2 1 2275.2.a.h 1
56.e even 2 1 5824.2.a.t 1
56.h odd 2 1 5824.2.a.s 1
91.b odd 2 1 1183.2.a.b 1
91.i even 4 2 1183.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 7.b odd 2 1
637.2.a.a 1 1.a even 1 1 trivial
637.2.e.d 2 7.c even 3 2
637.2.e.e 2 7.d odd 6 2
819.2.a.f 1 21.c even 2 1
1183.2.a.b 1 91.b odd 2 1
1183.2.c.b 2 91.i even 4 2
1456.2.a.g 1 28.d even 2 1
2275.2.a.h 1 35.c odd 2 1
5733.2.a.l 1 3.b odd 2 1
5824.2.a.s 1 56.h odd 2 1
5824.2.a.t 1 56.e even 2 1
8281.2.a.l 1 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2} + 2 \)
\( T_{3} \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( 6 + T \)
$13$ \( -1 + T \)
$17$ \( 4 + T \)
$19$ \( 5 + T \)
$23$ \( -3 + T \)
$29$ \( 5 + T \)
$31$ \( -3 + T \)
$37$ \( 4 + T \)
$41$ \( -6 + T \)
$43$ \( 1 + T \)
$47$ \( 7 + T \)
$53$ \( 9 + T \)
$59$ \( 8 + T \)
$61$ \( -10 + T \)
$67$ \( 6 + T \)
$71$ \( 8 + T \)
$73$ \( -13 + T \)
$79$ \( -3 + T \)
$83$ \( 15 + T \)
$89$ \( 3 + T \)
$97$ \( 7 + T \)
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