Properties

Label 637.2.a.h
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 + \beta_{1} ) q^{5} + ( -3 \beta_{1} + \beta_{2} ) q^{6} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{8} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 + \beta_{1} ) q^{5} + ( -3 \beta_{1} + \beta_{2} ) q^{6} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{8} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{10} + ( -1 - \beta_{1} ) q^{11} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{12} + q^{13} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{15} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} + ( -1 - \beta_{2} ) q^{17} + ( 1 + 5 \beta_{1} ) q^{18} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( -6 + 5 \beta_{1} - 3 \beta_{2} ) q^{20} + ( -2 - \beta_{1} - \beta_{2} ) q^{22} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{23} + ( 10 - 4 \beta_{1} + 2 \beta_{2} ) q^{24} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{25} + ( -1 + \beta_{1} ) q^{26} + ( -6 - 4 \beta_{1} ) q^{27} + ( -3 + 2 \beta_{2} ) q^{29} + ( -8 + 5 \beta_{1} - 5 \beta_{2} ) q^{30} + ( \beta_{1} + 2 \beta_{2} ) q^{31} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{32} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{33} + ( -3 \beta_{1} + \beta_{2} ) q^{34} + ( 8 - \beta_{1} + 3 \beta_{2} ) q^{36} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( -3 - 3 \beta_{2} ) q^{38} + ( -1 - \beta_{2} ) q^{39} + ( 8 - 8 \beta_{1} + 6 \beta_{2} ) q^{40} + ( -2 - 4 \beta_{1} + 4 \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} -2 \beta_{1} q^{44} + ( -2 + 4 \beta_{1} - \beta_{2} ) q^{45} + ( -9 + 3 \beta_{1} - 4 \beta_{2} ) q^{46} + ( -6 + \beta_{2} ) q^{47} + ( -8 + 10 \beta_{1} - 2 \beta_{2} ) q^{48} + ( -10 + 4 \beta_{1} - 4 \beta_{2} ) q^{50} + ( 6 + \beta_{1} + \beta_{2} ) q^{51} + ( 2 - \beta_{1} + \beta_{2} ) q^{52} + ( 5 - \beta_{1} - \beta_{2} ) q^{53} + ( -6 - 6 \beta_{1} - 4 \beta_{2} ) q^{54} + ( -1 - \beta_{2} ) q^{55} + ( -1 + 5 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{58} + ( -8 + \beta_{1} + \beta_{2} ) q^{59} + ( 16 - 12 \beta_{1} + 6 \beta_{2} ) q^{60} + ( 8 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 5 + 4 \beta_{1} - \beta_{2} ) q^{62} + ( 8 - 4 \beta_{1} ) q^{64} + ( -2 + \beta_{1} ) q^{65} + ( 8 + 4 \beta_{1} + 2 \beta_{2} ) q^{66} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{68} + ( -3 + 8 \beta_{1} - \beta_{2} ) q^{69} + ( 1 + 3 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -10 + 4 \beta_{1} - 4 \beta_{2} ) q^{72} + ( -4 + 6 \beta_{1} + \beta_{2} ) q^{73} + ( 4 - 7 \beta_{1} + 3 \beta_{2} ) q^{74} + ( -4 + 8 \beta_{1} - 2 \beta_{2} ) q^{75} + ( 4 - 5 \beta_{1} + \beta_{2} ) q^{76} + ( -3 \beta_{1} + \beta_{2} ) q^{78} + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -14 + 10 \beta_{1} - 8 \beta_{2} ) q^{80} + ( 1 + 9 \beta_{1} + 3 \beta_{2} ) q^{81} + ( -6 + 6 \beta_{1} - 8 \beta_{2} ) q^{82} + ( -8 + 3 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{86} + ( -7 - 2 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -2 + 2 \beta_{1} ) q^{88} + ( 4 - 4 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 13 - 4 \beta_{1} + 5 \beta_{2} ) q^{90} + ( 12 - 11 \beta_{1} + 5 \beta_{2} ) q^{92} + ( -11 - 5 \beta_{1} ) q^{93} + ( 7 - 4 \beta_{1} - \beta_{2} ) q^{94} + ( -1 + 2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( 16 - 4 \beta_{1} + 8 \beta_{2} ) q^{96} + ( 2 - 4 \beta_{1} + \beta_{2} ) q^{97} + ( -7 - 7 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} - 4q^{3} + 6q^{4} - 5q^{5} - 2q^{6} - 6q^{8} + 11q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 4q^{3} + 6q^{4} - 5q^{5} - 2q^{6} - 6q^{8} + 11q^{9} + 14q^{10} - 4q^{11} - 18q^{12} + 3q^{13} + 2q^{15} + 4q^{16} - 4q^{17} + 8q^{18} - 7q^{19} - 16q^{20} - 8q^{22} + q^{23} + 28q^{24} + 4q^{25} - 2q^{26} - 22q^{27} - 7q^{29} - 24q^{30} + 3q^{31} - 24q^{32} + 10q^{33} - 2q^{34} + 26q^{36} - 10q^{37} - 12q^{38} - 4q^{39} + 22q^{40} - 6q^{41} + 9q^{43} - 2q^{44} - 3q^{45} - 28q^{46} - 17q^{47} - 16q^{48} - 30q^{50} + 20q^{51} + 6q^{52} + 13q^{53} - 28q^{54} - 4q^{55} + 4q^{57} + 14q^{58} - 22q^{59} + 42q^{60} + 24q^{61} + 18q^{62} + 20q^{64} - 5q^{65} + 30q^{66} - 14q^{67} - 18q^{68} - 2q^{69} + 4q^{71} - 30q^{72} - 5q^{73} + 8q^{74} - 6q^{75} + 8q^{76} - 2q^{78} + q^{79} - 40q^{80} + 15q^{81} - 20q^{82} - 23q^{83} + 2q^{85} + 6q^{86} - 20q^{87} - 4q^{88} + 11q^{89} + 40q^{90} + 30q^{92} - 38q^{93} + 16q^{94} - 5q^{95} + 52q^{96} + 3q^{97} - 30q^{99} + O(q^{100}) \)

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

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

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
−1.65544
−0.210756
2.86620
−2.65544 −2.39593 5.05137 −3.65544 6.36226 0 −8.10275 2.74049 9.70682
1.2 −1.21076 1.74483 −0.534070 −2.21076 −2.11256 0 3.06814 0.0444180 2.67669
1.3 1.86620 −3.34889 1.48270 0.866198 −6.24970 0 −0.965392 8.21509 1.61650
\(n\): 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.h 3
3.b odd 2 1 5733.2.a.be 3
7.b odd 2 1 637.2.a.i yes 3
7.c even 3 2 637.2.e.l 6
7.d odd 6 2 637.2.e.k 6
13.b even 2 1 8281.2.a.bh 3
21.c even 2 1 5733.2.a.bd 3
91.b odd 2 1 8281.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 1.a even 1 1 trivial
637.2.a.i yes 3 7.b odd 2 1
637.2.e.k 6 7.d odd 6 2
637.2.e.l 6 7.c even 3 2
5733.2.a.bd 3 21.c even 2 1
5733.2.a.be 3 3.b odd 2 1
8281.2.a.bh 3 13.b even 2 1
8281.2.a.bk 3 91.b odd 2 1

Hecke kernels

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

\( T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} - 6 \)
\( T_{3}^{3} + 4 T_{3}^{2} - 2 T_{3} - 14 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 2 T_{17} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 - 4 T + 2 T^{2} + T^{3} \)
$3$ \( -14 - 2 T + 4 T^{2} + T^{3} \)
$5$ \( -7 + 3 T + 5 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -2 + 4 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( -14 - 2 T + 4 T^{2} + T^{3} \)
$19$ \( -63 - 3 T + 7 T^{2} + T^{3} \)
$23$ \( -43 - 41 T - T^{2} + T^{3} \)
$29$ \( -3 - 13 T + 7 T^{2} + T^{3} \)
$31$ \( 49 - 41 T - 3 T^{2} + T^{3} \)
$37$ \( -82 + 8 T + 10 T^{2} + T^{3} \)
$41$ \( -504 - 116 T + 6 T^{2} + T^{3} \)
$43$ \( 101 - 5 T - 9 T^{2} + T^{3} \)
$47$ \( 147 + 89 T + 17 T^{2} + T^{3} \)
$53$ \( 9 + 39 T - 13 T^{2} + T^{3} \)
$59$ \( 252 + 144 T + 22 T^{2} + T^{3} \)
$61$ \( -224 + 160 T - 24 T^{2} + T^{3} \)
$67$ \( -648 - 36 T + 14 T^{2} + T^{3} \)
$71$ \( 194 - 44 T - 4 T^{2} + T^{3} \)
$73$ \( -1561 - 219 T + 5 T^{2} + T^{3} \)
$79$ \( -99 - 69 T - T^{2} + T^{3} \)
$83$ \( 203 + 127 T + 23 T^{2} + T^{3} \)
$89$ \( -21 - 55 T - 11 T^{2} + T^{3} \)
$97$ \( 7 - 71 T - 3 T^{2} + T^{3} \)
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