Properties

Label 5610.2.a.cj
Level 5610
Weight 2
Character orbit 5610.a
Self dual Yes
Analytic conductor 44.796
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 5610.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(44.7960755339\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1284160.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 2 \nu + 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 4 \nu^{2} - 6 \nu + 5 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 6 \nu - 6 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 10 \nu^{2} - 4 \nu + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} - \beta_{3} + \beta_{2} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + \beta_{2} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{4} - 3 \beta_{3} + 9 \beta_{2} + 4 \beta_{1} + 17\)\()/4\)
\(\nu^{4}\)\(=\)\(-5 \beta_{4} - \beta_{3} + 6 \beta_{2} + 5 \beta_{1} + 19\)

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
1.30524
−1.20302
0.480230
3.11852
−1.70097
1.00000 1.00000 1.00000 1.00000 1.00000 −4.74909 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 −1.61835 1.00000 1.00000 1.00000
1.3 1.00000 1.00000 1.00000 1.00000 1.00000 0.869660 1.00000 1.00000 1.00000
1.4 1.00000 1.00000 1.00000 1.00000 1.00000 2.25889 1.00000 1.00000 1.00000
1.5 1.00000 1.00000 1.00000 1.00000 1.00000 4.23889 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(17\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{5} - T_{7}^{4} - 24 T_{7}^{3} + 32 T_{7}^{2} + 64 T_{7} - 64 \)
\( T_{13}^{5} - 9 T_{13}^{4} + 8 T_{13}^{3} + 56 T_{13}^{2} - 48 T_{13} - 16 \)
\( T_{19}^{5} + 5 T_{19}^{4} - 64 T_{19}^{3} - 256 T_{19}^{2} + 640 T_{19} + 2240 \)
\( T_{23}^{5} - 11 T_{23}^{4} - 56 T_{23}^{3} + 704 T_{23}^{2} + 832 T_{23} - 10816 \)