Properties

Label 5610.2.a.bt
Level 5610
Weight 2
Character orbit 5610.a
Self dual Yes
Analytic conductor 44.796
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7} \)
\( T_{13} + 2 \)
\( T_{19} \)
\( T_{23}^{2} - 32 \)