Properties

Label 5610.2.a.bq
Level 5610
Weight 2
Character orbit 5610.a
Self dual Yes
Analytic conductor 44.796
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.37228
−2.37228
−1.00000 −1.00000 1.00000 1.00000 1.00000 −3.37228 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 2.37228 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{2} + T_{7} - 8 \)
\( T_{13} + 4 \)
\( T_{19} - 4 \)
\( T_{23}^{2} + 9 T_{23} + 12 \)