Properties

Label 5610.2.a.ci
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: 5.5.18569692.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 23 x^{3} - 32 x^{2} + 26 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} + 20 \nu^{2} + 18 \nu - 12 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} - \nu^{3} + 26 \nu^{2} + 46 \nu - 30 \)\()/2\)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 22 \nu^{2} + 34 \nu - 12 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + 22 \nu^{2} + 36 \nu - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_{1} + 9\)
\(\nu^{3}\)\(=\)\(10 \beta_{4} - 13 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 18\)
\(\nu^{4}\)\(=\)\(39 \beta_{4} - 62 \beta_{3} + 22 \beta_{2} + 22 \beta_{1} + 186\)

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
0.380617
−2.65676
−3.25711
0.228960
5.30430
1.00000 −1.00000 1.00000 1.00000 −1.00000 −4.10711 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −3.13372 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 1.89495 1.00000 1.00000 1.00000
1.4 1.00000 −1.00000 1.00000 1.00000 −1.00000 3.06482 1.00000 1.00000 1.00000
1.5 1.00000 −1.00000 1.00000 1.00000 −1.00000 4.28107 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{5} - 2 T_{7}^{4} - 27 T_{7}^{3} + 52 T_{7}^{2} + 168 T_{7} - 320 \)
\( T_{13}^{5} + T_{13}^{4} - 32 T_{13}^{3} + 36 T_{13}^{2} + 112 T_{13} - 128 \)
\( T_{19}^{5} + T_{19}^{4} - 32 T_{19}^{3} + 36 T_{19}^{2} + 112 T_{19} - 128 \)
\( T_{23}^{5} - 97 T_{23}^{3} + 76 T_{23}^{2} + 2236 T_{23} - 4720 \)