Properties

Label 637.4.a.a
Level $637$
Weight $4$
Character orbit 637.a
Self dual yes
Analytic conductor $37.584$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5842166737\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{2} + 7 q^{3} + 17 q^{4} + 7 q^{5} - 35 q^{6} - 45 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 7 q^{3} + 17 q^{4} + 7 q^{5} - 35 q^{6} - 45 q^{8} + 22 q^{9} - 35 q^{10} - 26 q^{11} + 119 q^{12} - 13 q^{13} + 49 q^{15} + 89 q^{16} - 77 q^{17} - 110 q^{18} + 126 q^{19} + 119 q^{20} + 130 q^{22} - 96 q^{23} - 315 q^{24} - 76 q^{25} + 65 q^{26} - 35 q^{27} - 82 q^{29} - 245 q^{30} - 196 q^{31} - 85 q^{32} - 182 q^{33} + 385 q^{34} + 374 q^{36} - 131 q^{37} - 630 q^{38} - 91 q^{39} - 315 q^{40} - 336 q^{41} - 201 q^{43} - 442 q^{44} + 154 q^{45} + 480 q^{46} + 105 q^{47} + 623 q^{48} + 380 q^{50} - 539 q^{51} - 221 q^{52} - 432 q^{53} + 175 q^{54} - 182 q^{55} + 882 q^{57} + 410 q^{58} + 294 q^{59} + 833 q^{60} + 56 q^{61} + 980 q^{62} - 287 q^{64} - 91 q^{65} + 910 q^{66} + 478 q^{67} - 1309 q^{68} - 672 q^{69} + 9 q^{71} - 990 q^{72} - 98 q^{73} + 655 q^{74} - 532 q^{75} + 2142 q^{76} + 455 q^{78} + 1304 q^{79} + 623 q^{80} - 839 q^{81} + 1680 q^{82} + 308 q^{83} - 539 q^{85} + 1005 q^{86} - 574 q^{87} + 1170 q^{88} + 1190 q^{89} - 770 q^{90} - 1632 q^{92} - 1372 q^{93} - 525 q^{94} + 882 q^{95} - 595 q^{96} - 70 q^{97} - 572 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−5.00000 7.00000 17.0000 7.00000 −35.0000 0 −45.0000 22.0000 −35.0000
\(n\): 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 637.4.a.a 1
7.b odd 2 1 13.4.a.a 1
21.c even 2 1 117.4.a.b 1
28.d even 2 1 208.4.a.g 1
35.c odd 2 1 325.4.a.d 1
35.f even 4 2 325.4.b.b 2
56.e even 2 1 832.4.a.a 1
56.h odd 2 1 832.4.a.r 1
77.b even 2 1 1573.4.a.a 1
84.h odd 2 1 1872.4.a.k 1
91.b odd 2 1 169.4.a.e 1
91.i even 4 2 169.4.b.a 2
91.n odd 6 2 169.4.c.e 2
91.t odd 6 2 169.4.c.a 2
91.bc even 12 4 169.4.e.e 4
273.g even 2 1 1521.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 7.b odd 2 1
117.4.a.b 1 21.c even 2 1
169.4.a.e 1 91.b odd 2 1
169.4.b.a 2 91.i even 4 2
169.4.c.a 2 91.t odd 6 2
169.4.c.e 2 91.n odd 6 2
169.4.e.e 4 91.bc even 12 4
208.4.a.g 1 28.d even 2 1
325.4.a.d 1 35.c odd 2 1
325.4.b.b 2 35.f even 4 2
637.4.a.a 1 1.a even 1 1 trivial
832.4.a.a 1 56.e even 2 1
832.4.a.r 1 56.h odd 2 1
1521.4.a.a 1 273.g even 2 1
1573.4.a.a 1 77.b even 2 1
1872.4.a.k 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2} + 5 \) Copy content Toggle raw display
\( T_{3} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T - 7 \) Copy content Toggle raw display
$5$ \( T - 7 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 26 \) Copy content Toggle raw display
$13$ \( T + 13 \) Copy content Toggle raw display
$17$ \( T + 77 \) Copy content Toggle raw display
$19$ \( T - 126 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T + 82 \) Copy content Toggle raw display
$31$ \( T + 196 \) Copy content Toggle raw display
$37$ \( T + 131 \) Copy content Toggle raw display
$41$ \( T + 336 \) Copy content Toggle raw display
$43$ \( T + 201 \) Copy content Toggle raw display
$47$ \( T - 105 \) Copy content Toggle raw display
$53$ \( T + 432 \) Copy content Toggle raw display
$59$ \( T - 294 \) Copy content Toggle raw display
$61$ \( T - 56 \) Copy content Toggle raw display
$67$ \( T - 478 \) Copy content Toggle raw display
$71$ \( T - 9 \) Copy content Toggle raw display
$73$ \( T + 98 \) Copy content Toggle raw display
$79$ \( T - 1304 \) Copy content Toggle raw display
$83$ \( T - 308 \) Copy content Toggle raw display
$89$ \( T - 1190 \) Copy content Toggle raw display
$97$ \( T + 70 \) Copy content Toggle raw display
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