Properties

Label 5610.2.a.bu
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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.23607 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 3.23607 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{2} - 2 T_{7} - 4 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)
\( T_{19}^{2} + 2 T_{19} - 44 \)
\( T_{23}^{2} + 6 T_{23} + 4 \)