Properties

Label 5610.2.a.bz
Level 5610
Weight 2
Character orbit 5610.a
Self dual Yes
Analytic conductor 44.796
Analytic rank 1
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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
−1.48119
0.311108
2.17009
−1.00000 1.00000 1.00000 1.00000 −1.00000 −4.96239 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.37778 −1.00000 1.00000 −1.00000
1.3 −1.00000 1.00000 1.00000 1.00000 −1.00000 2.34017 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{3} + 4 T_{7}^{2} - 8 T_{7} - 16 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 12 T_{13} - 8 \)
\( T_{19}^{3} - 16 T_{19} + 16 \)
\( T_{23}^{3} - 16 T_{23} + 16 \)