Properties

Label 65.2.k.b
Level 65
Weight 2
Character orbit 65.k
Analytic conductor 0.519
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 65 = 5 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 65.k (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.619810816.2
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{5} + 14 x^{4} - 8 x^{3} + 2 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 64 \nu^{7} + 16 \nu^{6} + 4 \nu^{5} - 127 \nu^{4} + 944 \nu^{3} - 276 \nu^{2} + 378 \nu + 63 \)\()/319\)
\(\beta_{2}\)\(=\)\((\)\( -63 \nu^{7} + 64 \nu^{6} + 16 \nu^{5} + 130 \nu^{4} - 1009 \nu^{3} + 1448 \nu^{2} - 402 \nu - 67 \)\()/319\)
\(\beta_{3}\)\(=\)\((\)\( -67 \nu^{7} + 63 \nu^{6} - 64 \nu^{5} + 118 \nu^{4} - 1068 \nu^{3} + 1545 \nu^{2} - 1263 \nu + 268 \)\()/319\)
\(\beta_{4}\)\(=\)\((\)\( 83 \nu^{7} - 59 \nu^{6} + 65 \nu^{5} - 70 \nu^{4} + 1304 \nu^{3} - 1614 \nu^{2} + 1198 \nu + 306 \)\()/319\)
\(\beta_{5}\)\(=\)\((\)\( -172 \nu^{7} - 43 \nu^{6} + 69 \nu^{5} + 441 \nu^{4} - 2218 \nu^{3} + 662 \nu^{2} + 619 \nu - 269 \)\()/319\)
\(\beta_{6}\)\(=\)\((\)\( -196 \nu^{7} - 49 \nu^{6} - 92 \nu^{5} + 369 \nu^{4} - 2572 \nu^{3} + 1244 \nu^{2} - 1038 \nu - 173 \)\()/319\)
\(\beta_{7}\)\(=\)\( \nu^{7} - 2 \nu^{4} + 14 \nu^{3} - 8 \nu^{2} + \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{5} + \beta_{4} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2}\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + 5 \beta_{5} - 5 \beta_{4} - 2 \beta_{2} + 3 \beta_{1} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\(-\beta_{7} - \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 7\)
\(\nu^{5}\)\(=\)\((\)\(11 \beta_{7} + 2 \beta_{6} + 9 \beta_{5} - 11 \beta_{4} - 12 \beta_{3} + 12 \beta_{2} - 11 \beta_{1} + 12\)\()/2\)
\(\nu^{6}\)\(=\)\(-15 \beta_{6} + 16 \beta_{5} - 16 \beta_{4} + 16 \beta_{3} - 28 \beta_{2} + 7 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(-43 \beta_{7} + 16 \beta_{6} - 89 \beta_{5} + 105 \beta_{4} + 60 \beta_{2} - 43 \beta_{1} - 60\)\()/2\)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.18254 1.18254i
−0.252709 + 0.252709i
−1.49094 + 1.49094i
0.561103 0.561103i
1.18254 + 1.18254i
−0.252709 0.252709i
−1.49094 1.49094i
0.561103 + 0.561103i
−2.31627 −0.240275 0.240275i 3.36509 −1.60536 1.55654i 0.556540 + 0.556540i 3.95872i −3.16190 2.88454i 3.71844 + 3.60536i
8.2 −1.57942 0.725850 + 0.725850i 0.494582 2.23127 + 0.146426i −1.14643 1.14643i 4.24997i 2.37769 1.94628i −3.52412 0.231269i
8.3 −0.134632 −2.15558 2.15558i −1.98187 1.82630 1.29021i 0.290209 + 0.290209i 1.90970i 0.536087 6.29303i −0.245878 + 0.173703i
8.4 2.03032 −1.33000 1.33000i 2.12221 −1.45220 + 1.70032i −2.70032 2.70032i 1.61845i 0.248119 0.537789i −2.94844 + 3.45220i
57.1 −2.31627 −0.240275 + 0.240275i 3.36509 −1.60536 + 1.55654i 0.556540 0.556540i 3.95872i −3.16190 2.88454i 3.71844 3.60536i
57.2 −1.57942 0.725850 0.725850i 0.494582 2.23127 0.146426i −1.14643 + 1.14643i 4.24997i 2.37769 1.94628i −3.52412 + 0.231269i
57.3 −0.134632 −2.15558 + 2.15558i −1.98187 1.82630 + 1.29021i 0.290209 0.290209i 1.90970i 0.536087 6.29303i −0.245878 0.173703i
57.4 2.03032 −1.33000 + 1.33000i 2.12221 −1.45220 1.70032i −2.70032 + 2.70032i 1.61845i 0.248119 0.537789i −2.94844 3.45220i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
65.k Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\).