Properties

Label 2415.2.a.w
Level $2415$
Weight $2$
Character orbit 2415.a
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 16 x^{8} + 30 x^{7} + 87 x^{6} - 143 x^{5} - 196 x^{4} + 244 x^{3} + 160 x^{2} - 89 x - 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 16 x^{8} + 30 x^{7} + 87 x^{6} - 143 x^{5} - 196 x^{4} + 244 x^{3} + 160 x^{2} - 89 x - 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 7 \nu^{2} + 6 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} - \nu^{8} - 15 \nu^{7} + 11 \nu^{6} + 72 \nu^{5} - 25 \nu^{4} - 123 \nu^{3} - 5 \nu^{2} + 37 \nu - 2 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} - \nu^{8} - 15 \nu^{7} + 11 \nu^{6} + 72 \nu^{5} - 25 \nu^{4} - 125 \nu^{3} - 5 \nu^{2} + 47 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\( \nu^{9} - 16 \nu^{7} - 2 \nu^{6} + 84 \nu^{5} + 24 \nu^{4} - 158 \nu^{3} - 62 \nu^{2} + 57 \nu + 7 \)
\(\beta_{7}\)\(=\)\( -\nu^{9} + 16 \nu^{7} + 3 \nu^{6} - 83 \nu^{5} - 35 \nu^{4} + 149 \nu^{3} + 91 \nu^{2} - 39 \nu - 14 \)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{9} - \nu^{8} - 47 \nu^{7} + 7 \nu^{6} + 238 \nu^{5} + 23 \nu^{4} - 421 \nu^{3} - 131 \nu^{2} + 119 \nu + 16 \)\()/2\)
\(\beta_{9}\)\(=\)\( -2 \nu^{9} + 33 \nu^{7} + 4 \nu^{6} - 178 \nu^{5} - 47 \nu^{4} + 335 \nu^{3} + 121 \nu^{2} - 103 \nu - 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 7 \beta_{2} + 22\)
\(\nu^{5}\)\(=\)\(-\beta_{8} + \beta_{6} - 9 \beta_{5} + 10 \beta_{4} - \beta_{2} + 29 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(\beta_{8} + \beta_{7} - \beta_{4} + 11 \beta_{3} + 49 \beta_{2} - 2 \beta_{1} + 135\)
\(\nu^{7}\)\(=\)\(\beta_{9} - 10 \beta_{8} + 12 \beta_{6} - 71 \beta_{5} + 81 \beta_{4} - \beta_{3} - 14 \beta_{2} + 184 \beta_{1} + 38\)
\(\nu^{8}\)\(=\)\(\beta_{9} + 15 \beta_{8} + 13 \beta_{7} + \beta_{6} + 2 \beta_{5} - 19 \beta_{4} + 93 \beta_{3} + 349 \beta_{2} - 35 \beta_{1} + 885\)
\(\nu^{9}\)\(=\)\(16 \beta_{9} - 74 \beta_{8} + 2 \beta_{7} + 109 \beta_{6} - 538 \beta_{5} + 612 \beta_{4} - 18 \beta_{3} - 148 \beta_{2} + 1237 \beta_{1} + 161\)

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
−2.75158
−1.90103
−1.78717
−0.910389
−0.115446
0.509075
1.88703
1.91628
2.51086
2.64238
−2.75158 1.00000 5.57120 1.00000 −2.75158 1.00000 −9.82644 1.00000 −2.75158
1.2 −1.90103 1.00000 1.61393 1.00000 −1.90103 1.00000 0.733935 1.00000 −1.90103
1.3 −1.78717 1.00000 1.19399 1.00000 −1.78717 1.00000 1.44048 1.00000 −1.78717
1.4 −0.910389 1.00000 −1.17119 1.00000 −0.910389 1.00000 2.88702 1.00000 −0.910389
1.5 −0.115446 1.00000 −1.98667 1.00000 −0.115446 1.00000 0.460244 1.00000 −0.115446
1.6 0.509075 1.00000 −1.74084 1.00000 0.509075 1.00000 −1.90437 1.00000 0.509075
1.7 1.88703 1.00000 1.56089 1.00000 1.88703 1.00000 −0.828610 1.00000 1.88703
1.8 1.91628 1.00000 1.67212 1.00000 1.91628 1.00000 −0.628312 1.00000 1.91628
1.9 2.51086 1.00000 4.30443 1.00000 2.51086 1.00000 5.78609 1.00000 2.51086
1.10 2.64238 1.00000 4.98215 1.00000 2.64238 1.00000 7.87996 1.00000 2.64238
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.w 10
3.b odd 2 1 7245.2.a.bv 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.w 10 1.a even 1 1 trivial
7245.2.a.bv 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):

\(T_{2}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)
\(T_{13}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -12 - 89 T + 160 T^{2} + 244 T^{3} - 196 T^{4} - 143 T^{5} + 87 T^{6} + 30 T^{7} - 16 T^{8} - 2 T^{9} + T^{10} \)
$3$ \( ( -1 + T )^{10} \)
$5$ \( ( -1 + T )^{10} \)
$7$ \( ( -1 + T )^{10} \)
$11$ \( 13056 - 2888 T - 22130 T^{2} + 12018 T^{3} + 5694 T^{4} - 4185 T^{5} - 223 T^{6} + 415 T^{7} - 31 T^{8} - 9 T^{9} + T^{10} \)
$13$ \( 25024 + 9688 T - 46448 T^{2} - 9078 T^{3} + 21592 T^{4} - 3002 T^{5} - 1811 T^{6} + 448 T^{7} + 23 T^{8} - 14 T^{9} + T^{10} \)
$17$ \( -214464 - 137408 T + 189488 T^{2} + 121792 T^{3} - 24292 T^{4} - 16356 T^{5} + 1977 T^{6} + 720 T^{7} - 87 T^{8} - 8 T^{9} + T^{10} \)
$19$ \( -7808 + 46096 T - 35148 T^{2} - 41692 T^{3} + 39686 T^{4} - 4895 T^{5} - 3033 T^{6} + 835 T^{7} - 15 T^{8} - 13 T^{9} + T^{10} \)
$23$ \( ( 1 + T )^{10} \)
$29$ \( 24611328 - 6211072 T - 6715136 T^{2} + 2536704 T^{3} + 104192 T^{4} - 128048 T^{5} + 7192 T^{6} + 2028 T^{7} - 176 T^{8} - 10 T^{9} + T^{10} \)
$31$ \( 188416 - 1298944 T + 1661696 T^{2} + 774752 T^{3} - 184288 T^{4} - 58240 T^{5} + 8792 T^{6} + 1282 T^{7} - 168 T^{8} - 8 T^{9} + T^{10} \)
$37$ \( 4336304 + 147224 T - 2694756 T^{2} + 532266 T^{3} + 381198 T^{4} - 144146 T^{5} + 8417 T^{6} + 2140 T^{7} - 219 T^{8} - 8 T^{9} + T^{10} \)
$41$ \( 2321376 - 4804624 T + 1519728 T^{2} + 686776 T^{3} - 285854 T^{4} - 20645 T^{5} + 14455 T^{6} - 299 T^{7} - 207 T^{8} + 5 T^{9} + T^{10} \)
$43$ \( -55440128 - 21844224 T + 18807536 T^{2} + 2312768 T^{3} - 1192248 T^{4} - 76424 T^{5} + 28951 T^{6} + 950 T^{7} - 289 T^{8} - 4 T^{9} + T^{10} \)
$47$ \( -16662528 + 9049088 T + 7575680 T^{2} - 1471296 T^{3} - 1034272 T^{4} + 6752 T^{5} + 32112 T^{6} + 152 T^{7} - 320 T^{8} - T^{9} + T^{10} \)
$53$ \( 1388928 + 1692224 T - 1912096 T^{2} + 92528 T^{3} + 323480 T^{4} - 93384 T^{5} + 2046 T^{6} + 2086 T^{7} - 182 T^{8} - 9 T^{9} + T^{10} \)
$59$ \( 15264 + 277600 T - 527122 T^{2} - 828112 T^{3} + 115100 T^{4} + 95877 T^{5} - 1741 T^{6} - 2977 T^{7} - 131 T^{8} + 17 T^{9} + T^{10} \)
$61$ \( -41801456 - 47216528 T - 9648204 T^{2} + 4072804 T^{3} + 859308 T^{4} - 200861 T^{5} - 17251 T^{6} + 5045 T^{7} - 153 T^{8} - 19 T^{9} + T^{10} \)
$67$ \( 2011136 - 963584 T - 1569536 T^{2} + 1175008 T^{3} - 101024 T^{4} - 81230 T^{5} + 15859 T^{6} + 622 T^{7} - 239 T^{8} + T^{10} \)
$71$ \( -49152 + 69632 T + 1044224 T^{2} - 171200 T^{3} - 202112 T^{4} + 23296 T^{5} + 11224 T^{6} - 628 T^{7} - 232 T^{8} + T^{10} \)
$73$ \( 3179632 + 7888344 T + 4504660 T^{2} - 1283518 T^{3} - 1076520 T^{4} - 30522 T^{5} + 32569 T^{6} + 1124 T^{7} - 321 T^{8} - 6 T^{9} + T^{10} \)
$79$ \( 1067433984 + 18851328 T - 116668608 T^{2} + 4699808 T^{3} + 4266112 T^{4} - 364792 T^{5} - 50048 T^{6} + 6338 T^{7} + 82 T^{8} - 32 T^{9} + T^{10} \)
$83$ \( -1263122688 - 125945152 T + 192825296 T^{2} + 3208404 T^{3} - 6322980 T^{4} - 4876 T^{5} + 83741 T^{6} - 350 T^{7} - 485 T^{8} + 2 T^{9} + T^{10} \)
$89$ \( 771072 - 809599744 T + 124389760 T^{2} + 34117952 T^{3} - 4983584 T^{4} - 540848 T^{5} + 72288 T^{6} + 3812 T^{7} - 448 T^{8} - 10 T^{9} + T^{10} \)
$97$ \( 329532928 - 812644096 T + 6616832 T^{2} + 69063552 T^{3} - 1891968 T^{4} - 1555232 T^{5} + 74408 T^{6} + 9716 T^{7} - 520 T^{8} - 18 T^{9} + T^{10} \)
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