Properties

Label 65.2.e.b
Level 65
Weight 2
Character orbit 65.e
Analytic conductor 0.519
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 65 = 5 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 65.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 4 x^{2} + 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 4 \nu^{2} - 4 \nu - 3 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 7 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(4 \beta_{3} - 7\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−0.651388 1.12824i
1.15139 + 1.99426i
−0.651388 + 1.12824i
1.15139 1.99426i
−0.651388 1.12824i −0.500000 0.866025i 0.151388 0.262211i −1.00000 −0.651388 + 1.12824i 0.500000 0.866025i −3.00000 1.00000 1.73205i 0.651388 + 1.12824i
16.2 1.15139 + 1.99426i −0.500000 0.866025i −1.65139 + 2.86029i −1.00000 1.15139 1.99426i 0.500000 0.866025i −3.00000 1.00000 1.73205i −1.15139 1.99426i
61.1 −0.651388 + 1.12824i −0.500000 + 0.866025i 0.151388 + 0.262211i −1.00000 −0.651388 1.12824i 0.500000 + 0.866025i −3.00000 1.00000 + 1.73205i 0.651388 1.12824i
61.2 1.15139 1.99426i −0.500000 + 0.866025i −1.65139 2.86029i −1.00000 1.15139 + 1.99426i 0.500000 + 0.866025i −3.00000 1.00000 + 1.73205i −1.15139 + 1.99426i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\).