Properties

Label 5610.2.a.bx
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.

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

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

\( T_{7}^{2} - T_{7} - 8 \)
\( T_{13}^{2} + 2 T_{13} - 32 \)
\( T_{19} - 4 \)
\( T_{23}^{2} + T_{23} - 8 \)