Properties

Label 2415.2.a.p
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: 6.6.42978136.1
Defining polynomial: \(x^{6} - x^{5} - 8 x^{4} + 6 x^{3} + 16 x^{2} - 7 x - 9\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 8 x^{4} + 6 x^{3} + 16 x^{2} - 7 x - 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 7 \nu^{3} - 2 \nu^{2} + 8 \nu + 5 \)
\(\beta_{5}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 5 \nu^{2} - 9 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} + 7 \beta_{2} + \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(7 \beta_{5} + 8 \beta_{4} - 7 \beta_{3} + 9 \beta_{2} + 27 \beta_{1} + 1\)

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
−2.23082
−1.27660
−0.696045
1.21309
1.50731
2.48308
−2.23082 −1.00000 2.97658 −1.00000 2.23082 −1.00000 −2.17857 1.00000 2.23082
1.2 −1.27660 −1.00000 −0.370286 −1.00000 1.27660 −1.00000 3.02591 1.00000 1.27660
1.3 −0.696045 −1.00000 −1.51552 −1.00000 0.696045 −1.00000 2.44696 1.00000 0.696045
1.4 1.21309 −1.00000 −0.528418 −1.00000 −1.21309 −1.00000 −3.06719 1.00000 −1.21309
1.5 1.50731 −1.00000 0.271970 −1.00000 −1.50731 −1.00000 −2.60467 1.00000 −1.50731
1.6 2.48308 −1.00000 4.16568 −1.00000 −2.48308 −1.00000 5.37755 1.00000 −2.48308
\(n\): 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.p 6
3.b odd 2 1 7245.2.a.bk 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.p 6 1.a even 1 1 trivial
7245.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{6} - T_{2}^{5} - 8 T_{2}^{4} + 6 T_{2}^{3} + 16 T_{2}^{2} - 7 T_{2} - 9 \)
\( T_{11}^{6} + 6 T_{11}^{5} - 27 T_{11}^{4} - 182 T_{11}^{3} + 35 T_{11}^{2} + 936 T_{11} + 758 \)
\( T_{13}^{6} - 6 T_{13}^{5} - 31 T_{13}^{4} + 250 T_{13}^{3} - 117 T_{13}^{2} - 1628 T_{13} + 2394 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -9 - 7 T + 16 T^{2} + 6 T^{3} - 8 T^{4} - T^{5} + T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( 758 + 936 T + 35 T^{2} - 182 T^{3} - 27 T^{4} + 6 T^{5} + T^{6} \)
$13$ \( 2394 - 1628 T - 117 T^{2} + 250 T^{3} - 31 T^{4} - 6 T^{5} + T^{6} \)
$17$ \( 7484 - 52 T - 1815 T^{2} + 336 T^{3} + 49 T^{4} - 16 T^{5} + T^{6} \)
$19$ \( 972 + 2500 T + 1567 T^{2} - 42 T^{3} - 89 T^{4} + T^{6} \)
$23$ \( ( -1 + T )^{6} \)
$29$ \( -59776 + 4672 T + 6032 T^{2} - 312 T^{3} - 144 T^{4} + 4 T^{5} + T^{6} \)
$31$ \( -1248 + 1568 T + 2384 T^{2} - 8 T^{3} - 106 T^{4} + T^{6} \)
$37$ \( 5186 + 3944 T + 461 T^{2} - 254 T^{3} - 51 T^{4} + 4 T^{5} + T^{6} \)
$41$ \( -43548 - 9532 T + 8167 T^{2} + 386 T^{3} - 181 T^{4} - 4 T^{5} + T^{6} \)
$43$ \( 205888 + 48400 T - 8237 T^{2} - 2670 T^{3} - 97 T^{4} + 16 T^{5} + T^{6} \)
$47$ \( 20352 - 30080 T + 16240 T^{2} - 4224 T^{3} + 568 T^{4} - 38 T^{5} + T^{6} \)
$53$ \( 154848 - 33488 T - 8368 T^{2} + 1752 T^{3} + 66 T^{4} - 24 T^{5} + T^{6} \)
$59$ \( -246042 + 5224 T + 15477 T^{2} + 152 T^{3} - 229 T^{4} - 2 T^{5} + T^{6} \)
$61$ \( -214424 - 1816 T + 14999 T^{2} - 418 T^{3} - 225 T^{4} + 4 T^{5} + T^{6} \)
$67$ \( 17872 - 35608 T - 17257 T^{2} - 1348 T^{3} + 201 T^{4} + 30 T^{5} + T^{6} \)
$71$ \( -64 - 4480 T + 12688 T^{2} - 696 T^{3} - 252 T^{4} + 6 T^{5} + T^{6} \)
$73$ \( -899166 + 442364 T - 72009 T^{2} + 2910 T^{3} + 357 T^{4} - 38 T^{5} + T^{6} \)
$79$ \( -11552 + 5440 T + 1024 T^{2} - 548 T^{3} - 38 T^{4} + 14 T^{5} + T^{6} \)
$83$ \( 2141836 - 463772 T - 11083 T^{2} + 6934 T^{3} - 213 T^{4} - 22 T^{5} + T^{6} \)
$89$ \( 65792 - 9984 T - 10256 T^{2} + 2136 T^{3} - 8 T^{4} - 20 T^{5} + T^{6} \)
$97$ \( -131712 - 42496 T + 10192 T^{2} + 1760 T^{3} - 292 T^{4} - 4 T^{5} + T^{6} \)
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