Properties

Label 637.2.a.j
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{5} + ( -2 + 2 \beta_{1} ) q^{6} + ( 1 + \beta_{2} ) q^{8} + ( 3 - 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{5} + ( -2 + 2 \beta_{1} ) q^{6} + ( 1 + \beta_{2} ) q^{8} + ( 3 - 2 \beta_{1} ) q^{9} + ( 3 - \beta_{1} + \beta_{2} ) q^{10} + ( 1 - \beta_{1} + \beta_{2} ) q^{11} + 4 q^{12} - q^{13} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{15} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( -1 - \beta_{1} - \beta_{2} ) q^{17} + ( -6 + 3 \beta_{1} - 2 \beta_{2} ) q^{18} + ( 1 + \beta_{1} ) q^{19} + 2 \beta_{1} q^{20} + ( -2 + 2 \beta_{1} ) q^{22} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{23} + 4 q^{24} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{25} -\beta_{1} q^{26} + ( 4 - 4 \beta_{1} ) q^{27} + ( 8 + \beta_{2} ) q^{29} + ( 8 - 4 \beta_{1} + 2 \beta_{2} ) q^{30} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{32} + ( 6 - 2 \beta_{1} ) q^{33} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{34} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{36} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( 3 + \beta_{1} + \beta_{2} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} ) q^{39} + 2 \beta_{1} q^{40} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} - 3 \beta_{2} ) q^{43} + 4 q^{44} + ( -9 + 5 \beta_{1} - 2 \beta_{2} ) q^{45} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{46} + ( 3 - \beta_{1} + 4 \beta_{2} ) q^{47} + ( -8 + 4 \beta_{1} ) q^{48} + ( -5 - \beta_{2} ) q^{50} + ( -2 - 2 \beta_{1} ) q^{51} + ( -1 - \beta_{2} ) q^{52} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( -12 + 4 \beta_{1} - 4 \beta_{2} ) q^{54} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} ) q^{57} + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{58} + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -4 + 4 \beta_{1} ) q^{60} + 2 q^{61} + ( 1 - \beta_{1} - \beta_{2} ) q^{62} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( 1 - \beta_{1} ) q^{65} + ( -6 + 6 \beta_{1} - 2 \beta_{2} ) q^{66} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{67} + ( -6 - 4 \beta_{1} - 2 \beta_{2} ) q^{68} + ( 5 - 9 \beta_{1} + 5 \beta_{2} ) q^{69} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{72} + ( 3 + \beta_{1} + 4 \beta_{2} ) q^{73} + ( 10 + 4 \beta_{2} ) q^{74} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 2 - 2 \beta_{1} ) q^{78} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{79} + ( 6 - 4 \beta_{1} + 2 \beta_{2} ) q^{80} + ( 3 - 6 \beta_{1} + 4 \beta_{2} ) q^{81} + ( -4 + 2 \beta_{1} ) q^{82} + ( 1 + 9 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} ) q^{85} + ( -9 + \beta_{1} - 5 \beta_{2} ) q^{86} + ( 11 - 7 \beta_{1} + 7 \beta_{2} ) q^{87} + 4 q^{88} + ( 1 - 5 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 13 - 11 \beta_{1} + 3 \beta_{2} ) q^{90} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -7 - \beta_{1} + 3 \beta_{2} ) q^{93} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{94} + ( 2 + \beta_{2} ) q^{95} + ( 4 - 8 \beta_{1} + 4 \beta_{2} ) q^{96} + ( 3 + \beta_{1} ) q^{97} + ( 7 - 7 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{5} - 4q^{6} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{5} - 4q^{6} + 3q^{8} + 7q^{9} + 8q^{10} + 2q^{11} + 12q^{12} - 3q^{13} - 6q^{15} - q^{16} - 4q^{17} - 15q^{18} + 4q^{19} + 2q^{20} - 4q^{22} + 10q^{23} + 12q^{24} - 5q^{25} - q^{26} + 8q^{27} + 24q^{29} + 20q^{30} + 4q^{31} + 7q^{32} + 16q^{33} - 14q^{34} - q^{36} + 10q^{38} - 2q^{39} + 2q^{40} - 2q^{41} + 10q^{43} + 12q^{44} - 22q^{45} - 18q^{46} + 8q^{47} - 20q^{48} - 15q^{50} - 8q^{51} - 3q^{52} + 8q^{53} - 32q^{54} - 6q^{55} - 2q^{57} + 12q^{58} + 4q^{59} - 8q^{60} + 6q^{61} + 2q^{62} - 17q^{64} + 2q^{65} - 12q^{66} - 12q^{67} - 22q^{68} + 6q^{69} - 6q^{71} - q^{72} + 10q^{73} + 30q^{74} + 16q^{75} + 8q^{76} + 4q^{78} - 14q^{79} + 14q^{80} + 3q^{81} - 10q^{82} + 12q^{83} - 10q^{85} - 26q^{86} + 26q^{87} + 12q^{88} - 2q^{89} + 28q^{90} - 12q^{92} - 22q^{93} + 10q^{94} + 6q^{95} + 4q^{96} + 10q^{97} + 14q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.81361
0.470683
2.34292
−1.81361 3.10278 1.28917 −2.81361 −5.62721 0 1.28917 6.62721 5.10278
1.2 0.470683 −2.24914 −1.77846 −0.529317 −1.05863 0 −1.77846 2.05863 −0.249141
1.3 2.34292 1.14637 3.48929 1.34292 2.68585 0 3.48929 −1.68585 3.14637
\(n\): 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.j 3
3.b odd 2 1 5733.2.a.x 3
7.b odd 2 1 91.2.a.d 3
7.c even 3 2 637.2.e.i 6
7.d odd 6 2 637.2.e.j 6
13.b even 2 1 8281.2.a.bg 3
21.c even 2 1 819.2.a.i 3
28.d even 2 1 1456.2.a.t 3
35.c odd 2 1 2275.2.a.m 3
56.e even 2 1 5824.2.a.bs 3
56.h odd 2 1 5824.2.a.by 3
91.b odd 2 1 1183.2.a.i 3
91.i even 4 2 1183.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 7.b odd 2 1
637.2.a.j 3 1.a even 1 1 trivial
637.2.e.i 6 7.c even 3 2
637.2.e.j 6 7.d odd 6 2
819.2.a.i 3 21.c even 2 1
1183.2.a.i 3 91.b odd 2 1
1183.2.c.f 6 91.i even 4 2
1456.2.a.t 3 28.d even 2 1
2275.2.a.m 3 35.c odd 2 1
5733.2.a.x 3 3.b odd 2 1
5824.2.a.bs 3 56.e even 2 1
5824.2.a.by 3 56.h odd 2 1
8281.2.a.bg 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 2 \)
\( T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 10 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 - 4 T - T^{2} + T^{3} \)
$3$ \( 8 - 6 T - 2 T^{2} + T^{3} \)
$5$ \( -2 - 3 T + 2 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 8 - 6 T - 2 T^{2} + T^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( 4 - 10 T + 4 T^{2} + T^{3} \)
$19$ \( 4 + T - 4 T^{2} + T^{3} \)
$23$ \( 136 + T - 10 T^{2} + T^{3} \)
$29$ \( -454 + 185 T - 24 T^{2} + T^{3} \)
$31$ \( -16 - 19 T - 4 T^{2} + T^{3} \)
$37$ \( -124 - 58 T + T^{3} \)
$41$ \( 8 - 28 T + 2 T^{2} + T^{3} \)
$43$ \( 628 - 71 T - 10 T^{2} + T^{3} \)
$47$ \( 544 - 79 T - 8 T^{2} + T^{3} \)
$53$ \( -22 - 35 T - 8 T^{2} + T^{3} \)
$59$ \( 688 - 156 T - 4 T^{2} + T^{3} \)
$61$ \( ( -2 + T )^{3} \)
$67$ \( -976 - 124 T + 12 T^{2} + T^{3} \)
$71$ \( 16 - 22 T + 6 T^{2} + T^{3} \)
$73$ \( 274 - 99 T - 10 T^{2} + T^{3} \)
$79$ \( -16 + 5 T + 14 T^{2} + T^{3} \)
$83$ \( 3268 - 271 T - 12 T^{2} + T^{3} \)
$89$ \( -422 - 95 T + 2 T^{2} + T^{3} \)
$97$ \( -22 + 29 T - 10 T^{2} + T^{3} \)
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