Properties

Label 5610.2.a.bv
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: \(\Q(\sqrt{41}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.70156
−2.70156
1.00000 −1.00000 1.00000 1.00000 −1.00000 −4.70156 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 1.70156 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{2} + 3 T_{7} - 8 \)
\( T_{13}^{2} + T_{13} - 10 \)
\( T_{19}^{2} - 5 T_{19} - 4 \)
\( T_{23}^{2} + 3 T_{23} - 8 \)