Properties

Label 65.2.e.a
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

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

Character Values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
−0.809017 1.40126i 1.11803 + 1.93649i −0.309017 + 0.535233i 1.00000 1.80902 3.13331i 0.118034 0.204441i −2.23607 −1.00000 + 1.73205i −0.809017 1.40126i
16.2 0.309017 + 0.535233i −1.11803 1.93649i 0.809017 1.40126i 1.00000 0.690983 1.19682i −2.11803 + 3.66854i 2.23607 −1.00000 + 1.73205i 0.309017 + 0.535233i
61.1 −0.809017 + 1.40126i 1.11803 1.93649i −0.309017 0.535233i 1.00000 1.80902 + 3.13331i 0.118034 + 0.204441i −2.23607 −1.00000 1.73205i −0.809017 + 1.40126i
61.2 0.309017 0.535233i −1.11803 + 1.93649i 0.809017 + 1.40126i 1.00000 0.690983 + 1.19682i −2.11803 3.66854i 2.23607 −1.00000 1.73205i 0.309017 0.535233i
\(n\): 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 \( T_{2}^{4} + T_{2}^{3} + 2 T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\).