Properties

Label 637.2.a.n
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8\)

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
−1.66745
2.35100
−1.20475
1.90903
−0.146243
0.758419
−2.44785 −0.667452 3.99195 0.910286 1.63382 0 −4.87599 −2.55451 −2.22824
1.2 −1.17619 3.35100 −0.616586 3.14862 −3.94140 0 3.07759 8.22917 −3.70337
1.3 −0.656184 −0.204753 −1.56942 −1.35996 0.134356 0 2.34220 −2.95808 0.892385
1.4 0.264627 2.90903 −1.92997 −1.43515 0.769807 0 −1.03998 5.46247 −0.379780
1.5 1.83237 0.853757 1.35758 2.62555 1.56440 0 −1.17715 −2.27110 4.81098
1.6 2.18322 1.75842 2.76645 2.11065 3.83901 0 1.67333 0.0920365 4.60802
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.n yes 6
3.b odd 2 1 5733.2.a.br 6
7.b odd 2 1 637.2.a.m 6
7.c even 3 2 637.2.e.n 12
7.d odd 6 2 637.2.e.o 12
13.b even 2 1 8281.2.a.cd 6
21.c even 2 1 5733.2.a.bu 6
91.b odd 2 1 8281.2.a.cc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 7.b odd 2 1
637.2.a.n yes 6 1.a even 1 1 trivial
637.2.e.n 12 7.c even 3 2
637.2.e.o 12 7.d odd 6 2
5733.2.a.br 6 3.b odd 2 1
5733.2.a.bu 6 21.c even 2 1
8281.2.a.cc 6 91.b odd 2 1
8281.2.a.cd 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{6} - 8 T_{2}^{4} + 14 T_{2}^{2} + 4 T_{2} - 2 \)
\( T_{3}^{6} - 8 T_{3}^{5} + 20 T_{3}^{4} - 12 T_{3}^{3} - 12 T_{3}^{2} + 8 T_{3} + 2 \)
\( T_{17}^{6} - 16 T_{17}^{5} + 56 T_{17}^{4} + 324 T_{17}^{3} - 2792 T_{17}^{2} + 6792 T_{17} - 5294 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + 4 T + 14 T^{2} - 8 T^{4} + T^{6} \)
$3$ \( 2 + 8 T - 12 T^{2} - 12 T^{3} + 20 T^{4} - 8 T^{5} + T^{6} \)
$5$ \( 31 - 26 T - 31 T^{2} + 24 T^{3} + 5 T^{4} - 6 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( -562 - 692 T + 186 T^{2} + 156 T^{3} - 38 T^{4} - 4 T^{5} + T^{6} \)
$13$ \( ( -1 + T )^{6} \)
$17$ \( -5294 + 6792 T - 2792 T^{2} + 324 T^{3} + 56 T^{4} - 16 T^{5} + T^{6} \)
$19$ \( -73 - 6 T + 83 T^{2} + 16 T^{3} - 17 T^{4} - 2 T^{5} + T^{6} \)
$23$ \( 529 + 874 T - 169 T^{2} - 272 T^{3} - 37 T^{4} + 6 T^{5} + T^{6} \)
$29$ \( 529 + 230 T - 401 T^{2} - 268 T^{3} - 33 T^{4} + 6 T^{5} + T^{6} \)
$31$ \( -44249 - 10046 T + 4181 T^{2} + 508 T^{3} - 115 T^{4} - 6 T^{5} + T^{6} \)
$37$ \( 254 - 716 T + 378 T^{2} + 236 T^{3} - 82 T^{4} + T^{6} \)
$41$ \( 28784 + 19744 T + 1720 T^{2} - 968 T^{3} - 120 T^{4} + 8 T^{5} + T^{6} \)
$43$ \( 35153 + 29214 T + 6575 T^{2} - 188 T^{3} - 161 T^{4} - 2 T^{5} + T^{6} \)
$47$ \( -135617 + 107350 T - 25621 T^{2} + 1480 T^{3} + 215 T^{4} - 30 T^{5} + T^{6} \)
$53$ \( -1319 + 3386 T + 1199 T^{2} - 680 T^{3} - 37 T^{4} + 14 T^{5} + T^{6} \)
$59$ \( 1532 + 760 T - 1236 T^{2} + 44 T^{3} + 146 T^{4} - 24 T^{5} + T^{6} \)
$61$ \( -216584 - 20768 T + 16252 T^{2} + 112 T^{3} - 246 T^{4} + T^{6} \)
$67$ \( 6112 - 3712 T - 1984 T^{2} + 800 T^{3} - 16 T^{5} + T^{6} \)
$71$ \( -1206162 - 104028 T + 33274 T^{2} + 1856 T^{3} - 316 T^{4} - 8 T^{5} + T^{6} \)
$73$ \( -142657 + 22234 T + 12011 T^{2} - 808 T^{3} - 241 T^{4} + 6 T^{5} + T^{6} \)
$79$ \( 7913 + 9486 T - 11129 T^{2} - 3172 T^{3} - 65 T^{4} + 22 T^{5} + T^{6} \)
$83$ \( -167041 + 13034 T + 26145 T^{2} - 7992 T^{3} + 941 T^{4} - 50 T^{5} + T^{6} \)
$89$ \( 9959 - 26 T - 9053 T^{2} + 1188 T^{3} + 131 T^{4} - 26 T^{5} + T^{6} \)
$97$ \( 217287 + 193902 T + 13171 T^{2} - 3620 T^{3} - 261 T^{4} + 14 T^{5} + T^{6} \)
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