Properties

Label 5610.2.a.cf
Level 5610
Weight 2
Character orbit 5610.a
Self dual Yes
Analytic conductor 44.796
Analytic rank 0
Dimension 4
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: 4.4.17428.1
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 4 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 1\)

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
2.36865
−0.787711
−2.10710
1.52616
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −1.81471 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 −0.662077 −1.00000 1.00000 1.00000
1.3 −1.00000 1.00000 1.00000 −1.00000 −1.00000 2.92682 −1.00000 1.00000 1.00000
1.4 −1.00000 1.00000 1.00000 −1.00000 −1.00000 4.54997 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{4} - 5 T_{7}^{3} - 4 T_{7}^{2} + 24 T_{7} + 16 \)
\( T_{13}^{4} - 6 T_{13}^{3} - 12 T_{13}^{2} + 56 T_{13} + 64 \)
\( T_{19}^{4} - 8 T_{19}^{3} - 16 T_{19}^{2} + 208 T_{19} - 256 \)
\( T_{23}^{4} - T_{23}^{3} - 36 T_{23}^{2} - 80 T_{23} - 48 \)