Properties

Label 5610.2.a.cd
Level 5610
Weight 2
Character orbit 5610.a
Self dual Yes
Analytic conductor 44.796
Analytic rank 0
Dimension 3
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: \(3\)
Coefficient field: 3.3.2089.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 13 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{2} + \beta_{1} + 16\)\()/2\)

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
−0.309984
3.75054
−3.44055
1.00000 1.00000 1.00000 −1.00000 1.00000 −3.79696 1.00000 1.00000 −1.00000
1.2 1.00000 1.00000 1.00000 −1.00000 1.00000 1.15799 1.00000 1.00000 −1.00000
1.3 1.00000 1.00000 1.00000 −1.00000 1.00000 3.63897 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{3} - T_{7}^{2} - 14 T_{7} + 16 \)
\( T_{13}^{3} - 5 T_{13}^{2} - 6 T_{13} + 8 \)
\( T_{19}^{3} - 13 T_{19}^{2} + 42 T_{19} - 8 \)
\( T_{23}^{3} + 11 T_{23}^{2} + 26 T_{23} + 8 \)