Properties

Label 717.2.a.d
Level $717$
Weight $2$
Character orbit 717.a
Self dual yes
Analytic conductor $5.725$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 717 = 3 \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 717.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.72527382493\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - q^{3} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{4} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{5} -\beta_{5} q^{6} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -\beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} - q^{3} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{4} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{5} -\beta_{5} q^{6} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -\beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -2 - \beta_{3} + 2 \beta_{4} ) q^{10} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{11} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{12} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{14} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{15} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{16} + ( 2 - 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{17} + \beta_{5} q^{18} + ( -4 + \beta_{1} + \beta_{2} - \beta_{5} ) q^{19} + ( -2 - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{20} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{22} + ( -1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{23} + ( \beta_{2} - \beta_{3} ) q^{24} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{25} + ( 1 + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{26} - q^{27} + ( -4 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{28} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{29} + ( 2 + \beta_{3} - 2 \beta_{4} ) q^{30} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{31} + ( -1 + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{32} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{33} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{5} ) q^{34} + ( -2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{35} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{36} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{38} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{39} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{40} + ( 3 + \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{41} + ( \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{42} + ( \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{43} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{44} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{45} + ( -5 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{46} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{48} + ( 1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{49} + ( 3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{50} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{51} + ( -1 - 3 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} ) q^{52} + ( 1 + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{53} -\beta_{5} q^{54} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{55} + ( 1 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{56} + ( 4 - \beta_{1} - \beta_{2} + \beta_{5} ) q^{57} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{58} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{59} + ( 2 + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{60} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{61} + ( -7 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - \beta_{4} - 5 \beta_{5} ) q^{62} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} + ( -6 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{64} + ( -3 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{65} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{66} + ( -8 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{67} + ( 5 + 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} ) q^{68} + ( 1 + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{69} + ( 3 - 2 \beta_{1} - \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{70} + ( -4 - 2 \beta_{1} - 7 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{71} + ( -\beta_{2} + \beta_{3} ) q^{72} + ( -4 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{74} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{75} + ( -3 - 2 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{76} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{77} + ( -1 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{78} + ( -2 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{79} + ( 1 - \beta_{1} + 3 \beta_{3} + 2 \beta_{5} ) q^{80} + q^{81} + ( 6 - 2 \beta_{1} + \beta_{2} + 6 \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{82} + ( -3 - \beta_{1} - 3 \beta_{2} - 7 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{83} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{84} + ( -5 - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{85} + ( 9 + 2 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{86} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{87} + ( -5 + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{88} + ( 4 - 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{89} + ( -2 - \beta_{3} + 2 \beta_{4} ) q^{90} + ( -4 + 3 \beta_{2} - \beta_{4} ) q^{91} + ( -3 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{92} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{93} + ( 12 - \beta_{1} + 2 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{94} + ( 3 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{95} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{96} + ( -6 - 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{97} + ( -1 + 4 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{98} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 6q^{3} + 4q^{4} + 5q^{5} + 2q^{6} - 9q^{7} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - 2q^{2} - 6q^{3} + 4q^{4} + 5q^{5} + 2q^{6} - 9q^{7} - 3q^{8} + 6q^{9} - 11q^{10} - 13q^{11} - 4q^{12} - q^{13} - 5q^{15} - 4q^{16} + 11q^{17} - 2q^{18} - 22q^{19} - q^{20} + 9q^{21} - 2q^{22} - 12q^{23} + 3q^{24} - q^{25} + 12q^{26} - 6q^{27} - 16q^{28} + 11q^{30} - 18q^{31} + 7q^{32} + 13q^{33} - 3q^{34} - 9q^{35} + 4q^{36} - 8q^{37} - 5q^{38} + q^{39} - 11q^{40} + 10q^{41} - 14q^{43} - 4q^{44} + 5q^{45} - 18q^{46} - 9q^{47} + 4q^{48} + 5q^{49} + 4q^{50} - 11q^{51} - 16q^{52} - 8q^{53} + 2q^{54} - 20q^{55} + 11q^{56} + 22q^{57} - 15q^{58} - 10q^{59} + q^{60} - 12q^{61} - 13q^{62} - 9q^{63} - 31q^{64} - 11q^{65} + 2q^{66} - 36q^{67} + 22q^{68} + 12q^{69} + q^{70} - 3q^{71} - 3q^{72} - 32q^{73} + 9q^{74} + q^{75} - 4q^{76} + 6q^{77} - 12q^{78} - q^{79} - 7q^{80} + 6q^{81} + 7q^{82} - 7q^{83} + 16q^{84} - 14q^{85} + 45q^{86} - 15q^{88} + 17q^{89} - 11q^{90} - 23q^{91} - 12q^{92} + 18q^{93} + 50q^{94} - 7q^{96} - 28q^{97} + 13q^{98} - 13q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 4 \nu^{3} + 3 \nu^{2} - 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 3 \nu^{2} + 8 \nu - 1 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + \nu^{4} - 8 \nu^{3} - 5 \nu^{2} + 14 \nu + 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 4 \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 17 \beta_{1} + 2\)

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
−1.65963
−0.360520
1.39213
−1.94590
2.30642
0.267500
−2.22058 −1.00000 2.93097 −0.0570810 2.22058 −3.64374 −2.06730 1.00000 0.126753
1.2 −2.15574 −1.00000 2.64721 3.41325 2.15574 0.201367 −1.39522 1.00000 −7.35809
1.3 −0.899709 −1.00000 −1.19052 1.67380 0.899709 −1.75765 2.87054 1.00000 −1.50593
1.4 0.104133 −1.00000 −1.98916 −0.431998 −0.104133 2.76657 −0.415402 1.00000 −0.0449852
1.5 1.05161 −1.00000 −0.894124 2.87285 −1.05161 −4.11383 −3.04348 1.00000 3.02110
1.6 2.12029 −1.00000 2.49562 −2.47082 −2.12029 −2.45271 1.05086 1.00000 −5.23885
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(239\) \(1\)

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 717.2.a.d 6
3.b odd 2 1 2151.2.a.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.d 6 1.a even 1 1 trivial
2151.2.a.e 6 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(717))\):

\( T_{2}^{6} + 2 T_{2}^{5} - 6 T_{2}^{4} - 11 T_{2}^{3} + 7 T_{2}^{2} + 9 T_{2} - 1 \)
\( T_{5}^{6} - 5 T_{5}^{5} - 2 T_{5}^{4} + 34 T_{5}^{3} - 24 T_{5}^{2} - 19 T_{5} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 9 T + 7 T^{2} - 11 T^{3} - 6 T^{4} + 2 T^{5} + T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( -1 - 19 T - 24 T^{2} + 34 T^{3} - 2 T^{4} - 5 T^{5} + T^{6} \)
$7$ \( 36 - 138 T - 193 T^{2} - 51 T^{3} + 17 T^{4} + 9 T^{5} + T^{6} \)
$11$ \( -521 - 1110 T - 721 T^{2} - 99 T^{3} + 41 T^{4} + 13 T^{5} + T^{6} \)
$13$ \( -44 + 78 T + 167 T^{2} - 59 T^{3} - 43 T^{4} + T^{5} + T^{6} \)
$17$ \( 1459 - 655 T - 408 T^{2} + 176 T^{3} + 14 T^{4} - 11 T^{5} + T^{6} \)
$19$ \( 436 + 1328 T + 1453 T^{2} + 735 T^{3} + 184 T^{4} + 22 T^{5} + T^{6} \)
$23$ \( 5584 - 140 T - 2291 T^{2} - 701 T^{3} - 24 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( -1 + 6 T + 36 T^{2} + 24 T^{3} - 22 T^{4} + T^{6} \)
$31$ \( -5869 - 8533 T - 3931 T^{2} - 505 T^{3} + 63 T^{4} + 18 T^{5} + T^{6} \)
$37$ \( 36 + 60 T - 13 T^{2} - 47 T^{3} - 4 T^{4} + 8 T^{5} + T^{6} \)
$41$ \( -484 - 902 T + 501 T^{2} + 338 T^{3} - 43 T^{4} - 10 T^{5} + T^{6} \)
$43$ \( -10924 - 13648 T - 5963 T^{2} - 1001 T^{3} - 8 T^{4} + 14 T^{5} + T^{6} \)
$47$ \( -2396 - 5644 T - 4157 T^{2} - 1135 T^{3} - 79 T^{4} + 9 T^{5} + T^{6} \)
$53$ \( 4 + 98 T + 335 T^{2} - 175 T^{3} - 60 T^{4} + 8 T^{5} + T^{6} \)
$59$ \( 2000 - 2225 T^{2} - 1000 T^{3} - 75 T^{4} + 10 T^{5} + T^{6} \)
$61$ \( 2719 + 2075 T - 571 T^{2} - 441 T^{3} - 19 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( -61424 - 36664 T - 3440 T^{2} + 1700 T^{3} + 435 T^{4} + 36 T^{5} + T^{6} \)
$71$ \( -27584 + 69304 T + 19451 T^{2} - 903 T^{3} - 283 T^{4} + 3 T^{5} + T^{6} \)
$73$ \( 8236 - 4086 T - 3943 T^{2} + 449 T^{3} + 302 T^{4} + 32 T^{5} + T^{6} \)
$79$ \( -37484 - 81018 T + 26057 T^{2} + 148 T^{3} - 334 T^{4} + T^{5} + T^{6} \)
$83$ \( 140741 + 349864 T + 40967 T^{2} - 3091 T^{3} - 413 T^{4} + 7 T^{5} + T^{6} \)
$89$ \( 5744 - 17980 T + 10199 T^{2} + 2406 T^{3} - 174 T^{4} - 17 T^{5} + T^{6} \)
$97$ \( 1192624 + 151580 T - 32531 T^{2} - 5364 T^{3} + T^{4} + 28 T^{5} + T^{6} \)
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