Properties

Label 2415.2.a.q
Level $2415$
Weight $2$
Character orbit 2415.a
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2838720881\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.15751800.1
Defining polynomial: \(x^{6} - x^{5} - 8 x^{4} + 6 x^{3} + 16 x^{2} - 7 x - 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 8 x^{4} + 6 x^{3} + 16 x^{2} - 7 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + \nu^{2} + 4 \nu - 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} - \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 7 \nu^{3} + 11 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{3} + 5 \beta_{2} + \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 7 \beta_{2} + 17 \beta_{1} + 1\)

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
0.527344
−2.10679
−0.114413
2.07639
−1.66208
2.27955
−1.96273 −1.00000 1.85229 −1.00000 1.96273 1.00000 0.289915 1.00000 1.96273
1.2 −0.923950 −1.00000 −1.14632 −1.00000 0.923950 1.00000 2.90704 1.00000 0.923950
1.3 0.456154 −1.00000 −1.79192 −1.00000 −0.456154 1.00000 −1.72970 1.00000 −0.456154
1.4 0.646541 −1.00000 −1.58198 −1.00000 −0.646541 1.00000 −2.31590 1.00000 −0.646541
1.5 2.05680 −1.00000 2.23042 −1.00000 −2.05680 1.00000 0.473937 1.00000 −2.05680
1.6 2.72718 −1.00000 5.43751 −1.00000 −2.72718 1.00000 9.37471 1.00000 −2.72718
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 2415.2.a.q 6
3.b odd 2 1 7245.2.a.bj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.q 6 1.a even 1 1 trivial
7245.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{6} - 3 T_{2}^{5} - 4 T_{2}^{4} + 14 T_{2}^{3} - 9 T_{2} + 3 \)
\( T_{11}^{6} - 6 T_{11}^{5} - 15 T_{11}^{4} + 118 T_{11}^{3} - 45 T_{11}^{2} - 360 T_{11} + 270 \)
\( T_{13}^{6} + 6 T_{13}^{5} - 31 T_{13}^{4} - 174 T_{13}^{3} + 183 T_{13}^{2} + 1276 T_{13} + 698 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 9 T + 14 T^{3} - 4 T^{4} - 3 T^{5} + T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 270 - 360 T - 45 T^{2} + 118 T^{3} - 15 T^{4} - 6 T^{5} + T^{6} \)
$13$ \( 698 + 1276 T + 183 T^{2} - 174 T^{3} - 31 T^{4} + 6 T^{5} + T^{6} \)
$17$ \( 204 - 36 T - 267 T^{2} + 164 T^{3} - 11 T^{4} - 8 T^{5} + T^{6} \)
$19$ \( -916 + 1388 T + 191 T^{2} - 222 T^{3} - 25 T^{4} + 8 T^{5} + T^{6} \)
$23$ \( ( 1 + T )^{6} \)
$29$ \( 640 - 1600 T + 400 T^{2} + 248 T^{3} - 40 T^{4} - 8 T^{5} + T^{6} \)
$31$ \( 35744 + 8800 T - 2608 T^{2} - 792 T^{3} + 14 T^{4} + 16 T^{5} + T^{6} \)
$37$ \( -100638 - 13248 T + 6873 T^{2} + 718 T^{3} - 139 T^{4} - 8 T^{5} + T^{6} \)
$41$ \( 14884 - 2332 T - 3217 T^{2} + 1138 T^{3} - 89 T^{4} - 8 T^{5} + T^{6} \)
$43$ \( 28192 - 14160 T - 457 T^{2} + 1090 T^{3} - 117 T^{4} - 8 T^{5} + T^{6} \)
$47$ \( 10368 - 17280 T + 720 T^{2} + 992 T^{3} - 88 T^{4} - 10 T^{5} + T^{6} \)
$53$ \( 5664 + 9456 T - 13392 T^{2} + 1504 T^{3} + 154 T^{4} - 28 T^{5} + T^{6} \)
$59$ \( 31534 + 26568 T + 3381 T^{2} - 964 T^{3} - 153 T^{4} + 6 T^{5} + T^{6} \)
$61$ \( 57416 + 52600 T + 927 T^{2} - 2286 T^{3} - 169 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( 801072 - 206856 T - 5613 T^{2} + 4388 T^{3} - 175 T^{4} - 18 T^{5} + T^{6} \)
$71$ \( -84288 - 32832 T + 7536 T^{2} + 1240 T^{3} - 164 T^{4} - 10 T^{5} + T^{6} \)
$73$ \( -2510 - 260 T + 1555 T^{2} + 82 T^{3} - 103 T^{4} + 2 T^{5} + T^{6} \)
$79$ \( 38944 + 37696 T + 5536 T^{2} - 956 T^{3} - 190 T^{4} + 2 T^{5} + T^{6} \)
$83$ \( 1175996 - 342564 T - 12191 T^{2} + 6246 T^{3} - 117 T^{4} - 26 T^{5} + T^{6} \)
$89$ \( 36096 - 44544 T + 8496 T^{2} + 1048 T^{3} - 232 T^{4} - 8 T^{5} + T^{6} \)
$97$ \( 1501568 - 334080 T - 13040 T^{2} + 6432 T^{3} - 228 T^{4} - 20 T^{5} + T^{6} \)
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