Properties

Label 65.2.m.a
Level 65
Weight 2
Character orbit 65.m
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.m (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + \nu^{5} + 4 \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 8 \nu - 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - \nu^{5} - 4 \nu^{4} + 3 \nu^{3} + 10 \nu^{2} - 16 \nu + 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} - 3 \nu^{2} + 18 \nu - 16 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{7} - 5 \nu^{6} + 2 \nu^{5} + 7 \nu^{4} - 8 \nu^{3} - 9 \nu^{2} + 28 \nu - 20 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 7 \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 40 \nu + 32 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{7} - 20 \nu^{6} + 11 \nu^{5} + 30 \nu^{4} - 45 \nu^{3} - 28 \nu^{2} + 116 \nu - 88 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(\beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 1\)
\(\nu^{6}\)\(=\)\(-\beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 4 \beta_{1} - 1\)
\(\nu^{7}\)\(=\)\(-6 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + \beta_{1} + 6\)

Character Values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(1 - \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
1.20036 + 0.747754i
−1.27597 + 0.609843i
1.40994 0.109843i
0.665665 1.24775i
1.20036 0.747754i
−1.27597 0.609843i
1.40994 + 0.109843i
0.665665 + 1.24775i
−2.16117 + 1.24775i 1.41342 + 2.44811i 2.11378 3.66117i 1.00000i −6.10929 3.52720i −1.64996 0.952606i 5.55889i −2.49551 + 4.32235i −1.24775 2.16117i
36.2 −0.190254 + 0.109843i 0.800098 + 1.38581i −0.975869 + 1.69025i 1.00000i −0.304444 0.175771i −0.287734 0.166123i 0.868145i 0.219687 0.380509i 0.109843 + 0.190254i
36.3 1.05628 0.609843i −1.16612 2.01978i −0.256182 + 0.443720i 1.00000i −2.46350 1.42231i 3.11786 + 1.80010i 3.06430i −1.21969 + 2.11256i −0.609843 1.05628i
36.4 1.29515 0.747754i −0.0473938 0.0820885i 0.118272 0.204852i 1.00000i −0.122764 0.0708778i −4.18016 2.41342i 2.63726i 1.49551 2.59030i 0.747754 + 1.29515i
56.1 −2.16117 1.24775i 1.41342 2.44811i 2.11378 + 3.66117i 1.00000i −6.10929 + 3.52720i −1.64996 + 0.952606i 5.55889i −2.49551 4.32235i −1.24775 + 2.16117i
56.2 −0.190254 0.109843i 0.800098 1.38581i −0.975869 1.69025i 1.00000i −0.304444 + 0.175771i −0.287734 + 0.166123i 0.868145i 0.219687 + 0.380509i 0.109843 0.190254i
56.3 1.05628 + 0.609843i −1.16612 + 2.01978i −0.256182 0.443720i 1.00000i −2.46350 + 1.42231i 3.11786 1.80010i 3.06430i −1.21969 2.11256i −0.609843 + 1.05628i
56.4 1.29515 + 0.747754i −0.0473938 + 0.0820885i 0.118272 + 0.204852i 1.00000i −0.122764 + 0.0708778i −4.18016 + 2.41342i 2.63726i 1.49551 + 2.59030i 0.747754 1.29515i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(65, [\chi])\).