Properties

Label 637.2.e.j
Level $637$
Weight $2$
Character orbit 637.e
Analytic conductor $5.086$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2696112.1
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{1} + \beta_{4} ) q^{5} + ( 2 + 2 \beta_{2} ) q^{6} + ( 1 + \beta_{3} ) q^{8} + ( -3 + 2 \beta_{1} + 3 \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{1} + \beta_{4} ) q^{5} + ( 2 + 2 \beta_{2} ) q^{6} + ( 1 + \beta_{3} ) q^{8} + ( -3 + 2 \beta_{1} + 3 \beta_{4} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{10} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 4 - 4 \beta_{4} ) q^{12} + q^{13} + ( -3 - 3 \beta_{2} - \beta_{3} ) q^{15} + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{16} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{18} + ( 1 + \beta_{1} - \beta_{4} ) q^{19} + 2 \beta_{2} q^{20} + ( -2 - 2 \beta_{2} ) q^{22} + ( -4 + 2 \beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{23} + 4 \beta_{4} q^{24} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} -\beta_{1} q^{26} + ( -4 - 4 \beta_{2} ) q^{27} + ( 8 + \beta_{3} ) q^{29} + ( -8 + 4 \beta_{1} + 8 \beta_{4} - 2 \beta_{5} ) q^{30} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{31} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{32} + ( 6 - 2 \beta_{1} - 6 \beta_{4} ) q^{33} + ( 4 - 2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 1 + 4 \beta_{2} + \beta_{3} ) q^{36} + ( 1 - 3 \beta_{1} - \beta_{4} - \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{38} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{39} + 2 \beta_{1} q^{40} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 4 + 2 \beta_{2} - 3 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{4} ) q^{44} + ( -5 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 2 \beta_{5} ) q^{45} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} - 3 \beta_{5} ) q^{46} + ( 3 - \beta_{1} - 3 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 8 + 4 \beta_{2} ) q^{48} + ( -5 - \beta_{3} ) q^{50} + ( 2 + 2 \beta_{1} - 2 \beta_{4} ) q^{51} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{53} + ( -12 + 4 \beta_{1} + 12 \beta_{4} - 4 \beta_{5} ) q^{54} + ( 3 + 3 \beta_{2} + \beta_{3} ) q^{55} + ( -1 - \beta_{2} + \beta_{3} ) q^{57} + ( -1 - 9 \beta_{1} + \beta_{4} - \beta_{5} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{59} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} ) q^{60} + ( 2 - 2 \beta_{4} ) q^{61} + ( -1 - \beta_{2} + \beta_{3} ) q^{62} + ( -5 + 2 \beta_{2} - \beta_{3} ) q^{64} + ( -1 + \beta_{1} + \beta_{4} ) q^{65} + ( -6 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{66} + ( -6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{67} + ( -6 - 4 \beta_{1} + 6 \beta_{4} - 2 \beta_{5} ) q^{68} + ( -5 - 9 \beta_{2} - 5 \beta_{3} ) q^{69} + ( -3 - 3 \beta_{2} - \beta_{3} ) q^{71} + ( -1 + 4 \beta_{1} + \beta_{4} - \beta_{5} ) q^{72} + ( -\beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -4 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} ) q^{74} + ( 6 - 2 \beta_{1} - 6 \beta_{4} - 2 \beta_{5} ) q^{75} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 + 2 \beta_{2} ) q^{78} + ( 6 - 4 \beta_{1} - 6 \beta_{4} + \beta_{5} ) q^{79} + ( 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{80} + ( -6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{81} + ( -4 + 2 \beta_{1} + 4 \beta_{4} ) q^{82} + ( -1 + 9 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -3 + \beta_{2} - \beta_{3} ) q^{85} + ( 9 - \beta_{1} - 9 \beta_{4} + 5 \beta_{5} ) q^{86} + ( 7 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} + 11 \beta_{4} - 7 \beta_{5} ) q^{87} -4 \beta_{4} q^{88} + ( 1 - 5 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{89} + ( -13 - 11 \beta_{2} - 3 \beta_{3} ) q^{90} + ( -2 + 6 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 7 + \beta_{1} - 7 \beta_{4} - 3 \beta_{5} ) q^{93} + ( -7 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{94} + ( -\beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{95} + ( 4 - 8 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{96} + ( -3 + \beta_{2} ) q^{97} + ( 7 + 7 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} + 2q^{3} - 3q^{4} - 2q^{5} + 8q^{6} + 6q^{8} - 7q^{9} + O(q^{10}) \) \( 6q - q^{2} + 2q^{3} - 3q^{4} - 2q^{5} + 8q^{6} + 6q^{8} - 7q^{9} + 8q^{10} - 2q^{11} + 12q^{12} + 6q^{13} - 12q^{15} + q^{16} - 4q^{17} + 15q^{18} + 4q^{19} - 4q^{20} - 8q^{22} - 10q^{23} + 12q^{24} + 5q^{25} - q^{26} - 16q^{27} + 48q^{29} - 20q^{30} + 4q^{31} - 7q^{32} + 16q^{33} + 28q^{34} - 2q^{36} + 10q^{38} + 2q^{39} + 2q^{40} + 4q^{41} + 20q^{43} - 12q^{44} - 22q^{45} + 18q^{46} + 8q^{47} + 40q^{48} - 30q^{50} + 8q^{51} - 3q^{52} - 8q^{53} - 32q^{54} + 12q^{55} - 4q^{57} - 12q^{58} + 4q^{59} + 8q^{60} + 6q^{61} - 4q^{62} - 34q^{64} - 2q^{65} - 12q^{66} + 12q^{67} - 22q^{68} - 12q^{69} - 12q^{71} + q^{72} + 10q^{73} - 30q^{74} + 16q^{75} - 16q^{76} + 8q^{78} + 14q^{79} + 14q^{80} - 3q^{81} - 10q^{82} - 24q^{83} - 20q^{85} + 26q^{86} + 26q^{87} - 12q^{88} - 2q^{89} - 56q^{90} - 24q^{92} + 22q^{93} + 10q^{94} - 6q^{95} + 4q^{96} - 20q^{97} + 28q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} + 8 \nu - 40 \)\()/82\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 10 \nu^{4} + 9 \nu^{3} - 36 \nu^{2} + 16 \nu - 121 \)\()/41\)
\(\beta_{4}\)\(=\)\((\)\( -10 \nu^{5} + 9 \nu^{4} - 45 \nu^{3} - 25 \nu^{2} - 162 \nu + 72 \)\()/82\)
\(\beta_{5}\)\(=\)\((\)\( -26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 454 \nu - 26 \)\()/82\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 4 \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 13 \beta_{4} - 2 \beta_{1} - 13\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 18 \beta_{2} - 18 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.17146 2.02903i
0.235342 0.407624i
−0.906803 + 1.57063i
1.17146 + 2.02903i
0.235342 + 0.407624i
−0.906803 1.57063i
−1.17146 + 2.02903i 0.573183 + 0.992782i −1.74464 3.02181i 0.671462 1.16301i −2.68585 0 3.48929 0.842923 1.45999i 1.57318 + 2.72483i
79.2 −0.235342 + 0.407624i −1.12457 1.94781i 0.889229 + 1.54019i −0.264658 + 0.458402i 1.05863 0 −1.77846 −1.02932 + 1.78283i −0.124570 0.215762i
79.3 0.906803 1.57063i 1.55139 + 2.68708i −0.644584 1.11645i −1.40680 + 2.43665i 5.62721 0 1.28917 −3.31361 + 5.73933i 2.55139 + 4.41913i
508.1 −1.17146 2.02903i 0.573183 0.992782i −1.74464 + 3.02181i 0.671462 + 1.16301i −2.68585 0 3.48929 0.842923 + 1.45999i 1.57318 2.72483i
508.2 −0.235342 0.407624i −1.12457 + 1.94781i 0.889229 1.54019i −0.264658 0.458402i 1.05863 0 −1.77846 −1.02932 1.78283i −0.124570 + 0.215762i
508.3 0.906803 + 1.57063i 1.55139 2.68708i −0.644584 + 1.11645i −1.40680 2.43665i 5.62721 0 1.28917 −3.31361 5.73933i 2.55139 4.41913i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 508.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.j 6
7.b odd 2 1 637.2.e.i 6
7.c even 3 1 91.2.a.d 3
7.c even 3 1 inner 637.2.e.j 6
7.d odd 6 1 637.2.a.j 3
7.d odd 6 1 637.2.e.i 6
21.g even 6 1 5733.2.a.x 3
21.h odd 6 1 819.2.a.i 3
28.g odd 6 1 1456.2.a.t 3
35.j even 6 1 2275.2.a.m 3
56.k odd 6 1 5824.2.a.bs 3
56.p even 6 1 5824.2.a.by 3
91.r even 6 1 1183.2.a.i 3
91.s odd 6 1 8281.2.a.bg 3
91.z odd 12 2 1183.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 7.c even 3 1
637.2.a.j 3 7.d odd 6 1
637.2.e.i 6 7.b odd 2 1
637.2.e.i 6 7.d odd 6 1
637.2.e.j 6 1.a even 1 1 trivial
637.2.e.j 6 7.c even 3 1 inner
819.2.a.i 3 21.h odd 6 1
1183.2.a.i 3 91.r even 6 1
1183.2.c.f 6 91.z odd 12 2
1456.2.a.t 3 28.g odd 6 1
2275.2.a.m 3 35.j even 6 1
5733.2.a.x 3 21.g even 6 1
5824.2.a.bs 3 56.k odd 6 1
5824.2.a.by 3 56.p even 6 1
8281.2.a.bg 3 91.s odd 6 1

Hecke kernels

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

\( T_{2}^{6} + T_{2}^{5} + 5 T_{2}^{4} + 18 T_{2}^{2} + 8 T_{2} + 4 \)
\( T_{3}^{6} - 2 T_{3}^{5} + 10 T_{3}^{4} - 4 T_{3}^{3} + 52 T_{3}^{2} - 48 T_{3} + 64 \)
\( T_{5}^{6} + 2 T_{5}^{5} + 7 T_{5}^{4} - 2 T_{5}^{3} + 13 T_{5}^{2} + 6 T_{5} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 8 T + 18 T^{2} + 5 T^{4} + T^{5} + T^{6} \)
$3$ \( 64 - 48 T + 52 T^{2} - 4 T^{3} + 10 T^{4} - 2 T^{5} + T^{6} \)
$5$ \( 4 + 6 T + 13 T^{2} - 2 T^{3} + 7 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 64 + 48 T + 52 T^{2} + 4 T^{3} + 10 T^{4} + 2 T^{5} + T^{6} \)
$13$ \( ( -1 + T )^{6} \)
$17$ \( 16 - 40 T + 84 T^{2} - 48 T^{3} + 26 T^{4} + 4 T^{5} + T^{6} \)
$19$ \( 16 + 4 T + 17 T^{2} - 12 T^{3} + 15 T^{4} - 4 T^{5} + T^{6} \)
$23$ \( 18496 - 136 T + 1361 T^{2} + 282 T^{3} + 99 T^{4} + 10 T^{5} + T^{6} \)
$29$ \( ( -454 + 185 T - 24 T^{2} + T^{3} )^{2} \)
$31$ \( 256 + 304 T + 297 T^{2} + 108 T^{3} + 35 T^{4} - 4 T^{5} + T^{6} \)
$37$ \( 15376 - 7192 T + 3364 T^{2} - 248 T^{3} + 58 T^{4} + T^{6} \)
$41$ \( ( -8 - 28 T - 2 T^{2} + T^{3} )^{2} \)
$43$ \( ( 628 - 71 T - 10 T^{2} + T^{3} )^{2} \)
$47$ \( 295936 - 42976 T + 10593 T^{2} - 456 T^{3} + 143 T^{4} - 8 T^{5} + T^{6} \)
$53$ \( 484 - 770 T + 1049 T^{2} - 324 T^{3} + 99 T^{4} + 8 T^{5} + T^{6} \)
$59$ \( 473344 - 107328 T + 27088 T^{2} - 752 T^{3} + 172 T^{4} - 4 T^{5} + T^{6} \)
$61$ \( ( 4 - 2 T + T^{2} )^{3} \)
$67$ \( 952576 - 121024 T + 27088 T^{2} - 464 T^{3} + 268 T^{4} - 12 T^{5} + T^{6} \)
$71$ \( ( 16 - 22 T + 6 T^{2} + T^{3} )^{2} \)
$73$ \( 75076 - 27126 T + 12541 T^{2} + 442 T^{3} + 199 T^{4} - 10 T^{5} + T^{6} \)
$79$ \( 256 + 80 T + 249 T^{2} - 102 T^{3} + 191 T^{4} - 14 T^{5} + T^{6} \)
$83$ \( ( -3268 - 271 T + 12 T^{2} + T^{3} )^{2} \)
$89$ \( 178084 + 40090 T + 9869 T^{2} + 654 T^{3} + 99 T^{4} + 2 T^{5} + T^{6} \)
$97$ \( ( 22 + 29 T + 10 T^{2} + T^{3} )^{2} \)
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