Properties

Label 2415.2.a.r
Level $2415$
Weight $2$
Character orbit 2415.a
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 9 x^{5} + 16 x^{4} + 20 x^{3} - 29 x^{2} - 12 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 9 x^{5} + 16 x^{4} + 20 x^{3} - 29 x^{2} - 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{5} + 9 \nu^{4} - 13 \nu^{3} - 20 \nu^{2} + 11 \nu + 9 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{6} - \nu^{5} - 18 \nu^{4} + 2 \nu^{3} + 34 \nu^{2} + 8 \nu - 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 9 \nu^{4} + 16 \nu^{3} + 20 \nu^{2} - 26 \nu - 12 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 12 \nu^{4} - 11 \nu^{3} + 35 \nu^{2} + 22 \nu - 15 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{6} + \beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(8 \beta_{5} + \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 30 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(-9 \beta_{6} + 3 \beta_{5} + 11 \beta_{4} + 13 \beta_{3} + 47 \beta_{2} + 15 \beta_{1} + 91\)

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
2.76794
1.88240
1.48389
0.451228
−0.748534
−1.43461
−2.40232
−2.76794 1.00000 5.66150 −1.00000 −2.76794 −1.00000 −10.1348 1.00000 2.76794
1.2 −1.88240 1.00000 1.54343 −1.00000 −1.88240 −1.00000 0.859442 1.00000 1.88240
1.3 −1.48389 1.00000 0.201931 −1.00000 −1.48389 −1.00000 2.66814 1.00000 1.48389
1.4 −0.451228 1.00000 −1.79639 −1.00000 −0.451228 −1.00000 1.71304 1.00000 0.451228
1.5 0.748534 1.00000 −1.43970 −1.00000 0.748534 −1.00000 −2.57473 1.00000 −0.748534
1.6 1.43461 1.00000 0.0581079 −1.00000 1.43461 −1.00000 −2.78586 1.00000 −1.43461
1.7 2.40232 1.00000 3.77112 −1.00000 2.40232 −1.00000 4.25479 1.00000 −2.40232
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.r 7
3.b odd 2 1 7245.2.a.bn 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.r 7 1.a even 1 1 trivial
7245.2.a.bn 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{7} + 2 T_{2}^{6} - 9 T_{2}^{5} - 16 T_{2}^{4} + 20 T_{2}^{3} + 29 T_{2}^{2} - 12 T_{2} - 9 \)
\( T_{11}^{7} + 2 T_{11}^{6} - 55 T_{11}^{5} - 64 T_{11}^{4} + 831 T_{11}^{3} + 674 T_{11}^{2} - 3222 T_{11} - 870 \)
\( T_{13}^{7} - 6 T_{13}^{6} - 31 T_{13}^{5} + 220 T_{13}^{4} + 111 T_{13}^{3} - 2006 T_{13}^{2} + 2002 T_{13} + 1086 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -9 - 12 T + 29 T^{2} + 20 T^{3} - 16 T^{4} - 9 T^{5} + 2 T^{6} + T^{7} \)
$3$ \( ( -1 + T )^{7} \)
$5$ \( ( 1 + T )^{7} \)
$7$ \( ( 1 + T )^{7} \)
$11$ \( -870 - 3222 T + 674 T^{2} + 831 T^{3} - 64 T^{4} - 55 T^{5} + 2 T^{6} + T^{7} \)
$13$ \( 1086 + 2002 T - 2006 T^{2} + 111 T^{3} + 220 T^{4} - 31 T^{5} - 6 T^{6} + T^{7} \)
$17$ \( -16200 + 14796 T - 1902 T^{2} - 1651 T^{3} + 462 T^{4} + 17 T^{5} - 14 T^{6} + T^{7} \)
$19$ \( 3180 - 1248 T - 1558 T^{2} + 479 T^{3} + 212 T^{4} - 41 T^{5} - 6 T^{6} + T^{7} \)
$23$ \( ( 1 + T )^{7} \)
$29$ \( -192 - 928 T + 960 T^{2} + 400 T^{3} - 252 T^{4} - 60 T^{5} + 4 T^{6} + T^{7} \)
$31$ \( 351968 - 81728 T - 31680 T^{2} + 7048 T^{3} + 722 T^{4} - 162 T^{5} - 4 T^{6} + T^{7} \)
$37$ \( -43686 + 68856 T - 25128 T^{2} - 3113 T^{3} + 1426 T^{4} - 13 T^{5} - 18 T^{6} + T^{7} \)
$41$ \( -141480 - 139956 T - 22422 T^{2} + 7931 T^{3} + 1088 T^{4} - 149 T^{5} - 10 T^{6} + T^{7} \)
$43$ \( 137736 - 182604 T + 89266 T^{2} - 18897 T^{3} + 1176 T^{4} + 159 T^{5} - 26 T^{6} + T^{7} \)
$47$ \( 6528 - 154816 T + 31584 T^{2} + 10496 T^{3} - 832 T^{4} - 200 T^{5} + 4 T^{6} + T^{7} \)
$53$ \( -96 - 176 T + 192 T^{2} + 312 T^{3} - 58 T^{4} - 86 T^{5} - 2 T^{6} + T^{7} \)
$59$ \( -37146 + 119768 T - 80506 T^{2} + 11191 T^{3} + 1882 T^{4} - 271 T^{5} - 6 T^{6} + T^{7} \)
$61$ \( -540 - 932 T + 656 T^{2} + 1051 T^{3} - 282 T^{4} - 81 T^{5} + 4 T^{6} + T^{7} \)
$67$ \( 25088 + 15680 T - 4136 T^{2} - 2589 T^{3} + 412 T^{4} + 105 T^{5} - 22 T^{6} + T^{7} \)
$71$ \( 359232 - 320384 T + 54688 T^{2} + 12120 T^{3} - 1980 T^{4} - 248 T^{5} + 8 T^{6} + T^{7} \)
$73$ \( 43110 + 108642 T - 32292 T^{2} - 10825 T^{3} + 3002 T^{4} - 11 T^{5} - 24 T^{6} + T^{7} \)
$79$ \( 1228320 + 21696 T - 171056 T^{2} + 23156 T^{3} + 1714 T^{4} - 336 T^{5} - 2 T^{6} + T^{7} \)
$83$ \( 2916 + 648 T - 6372 T^{2} + 237 T^{3} + 1442 T^{4} + 67 T^{5} - 26 T^{6} + T^{7} \)
$89$ \( -4800 - 123104 T + 107008 T^{2} - 32544 T^{3} + 3804 T^{4} - 32 T^{5} - 22 T^{6} + T^{7} \)
$97$ \( -4672 + 18688 T - 27248 T^{2} + 17664 T^{3} - 5116 T^{4} + 700 T^{5} - 44 T^{6} + T^{7} \)
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