Properties

Label 62.2.d.b
Level 62
Weight 2
Character orbit 62.d
Analytic conductor 0.495
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 62 = 2 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 62.d (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.495072492532\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1903140625.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 6 x^{6} + x^{5} + 29 x^{4} + 43 x^{3} + 194 x^{2} + 209 x + 361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 16228 \nu^{7} + 164686 \nu^{6} - 1074875 \nu^{5} + 2192108 \nu^{4} + 7497629 \nu^{3} - 47195553 \nu^{2} + 40440235 \nu + 55384240 \)\()/ 205159891 \)
\(\beta_{3}\)\(=\)\((\)\( -294510 \nu^{7} + 348813 \nu^{6} + 3198495 \nu^{5} - 17987557 \nu^{4} + 10405212 \nu^{3} - 6474129 \nu^{2} - 49033872 \nu - 65903001 \)\()/ 205159891 \)
\(\beta_{4}\)\(=\)\((\)\( -28143 \nu^{7} + 261345 \nu^{6} - 931213 \nu^{5} + 997158 \nu^{4} + 325779 \nu^{3} + 426372 \nu^{2} - 228969 \nu + 5595690 \)\()/10797889\)
\(\beta_{5}\)\(=\)\((\)\( -45075 \nu^{7} - 64855 \nu^{6} + 453785 \nu^{5} - 989293 \nu^{4} - 2983130 \nu^{3} - 5142509 \nu^{2} - 5877897 \nu - 23973516 \)\()/10797889\)
\(\beta_{6}\)\(=\)\((\)\( 967254 \nu^{7} - 4292904 \nu^{6} + 9536834 \nu^{5} - 8103878 \nu^{4} + 28199801 \nu^{3} - 8897747 \nu^{2} + 98040673 \nu - 119034259 \)\()/ 205159891 \)
\(\beta_{7}\)\(=\)\((\)\( 56305 \nu^{7} + 3158 \nu^{6} - 339265 \nu^{5} + 1359136 \nu^{4} + 462427 \nu^{3} + 7105246 \nu^{2} + 8614349 \nu + 23665184 \)\()/10797889\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} - \beta_{3} - 5 \beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(-7 \beta_{7} - \beta_{6} - 7 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-12 \beta_{7} - 15 \beta_{6} - 15 \beta_{5} - 15 \beta_{4} - 22 \beta_{3} + 3 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(-63 \beta_{6} - 30 \beta_{5} - 64 \beta_{4} + 63 \beta_{2} - 87\)
\(\nu^{6}\)\(=\)\(127 \beta_{7} - 166 \beta_{6} - 127 \beta_{4} + 166 \beta_{3} + 350 \beta_{2} - 54 \beta_{1} - 350\)
\(\nu^{7}\)\(=\)\(658 \beta_{7} + 365 \beta_{5} + 689 \beta_{3} + 1032 \beta_{2} - 365 \beta_{1} - 689\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.671745 + 2.06742i
−0.480762 1.47963i
2.68070 1.94764i
−1.37168 + 0.996583i
2.68070 + 1.94764i
−1.37168 0.996583i
0.671745 2.06742i
−0.480762 + 1.47963i
0.809017 + 0.587785i −1.75865 + 1.27773i 0.309017 + 0.951057i 1.89927 −2.17381 −0.500000 1.53884i −0.309017 + 0.951057i 0.533195 1.64101i 1.53654 + 1.11636i
33.2 0.809017 + 0.587785i 1.25865 0.914463i 0.309017 + 0.951057i −4.13533 1.55578 −0.500000 1.53884i −0.309017 + 0.951057i −0.179093 + 0.551192i −3.34556 2.43069i
35.1 −0.309017 0.951057i −1.02393 + 3.15135i −0.809017 + 0.587785i 2.66590 3.31352 −0.500000 + 0.363271i 0.809017 + 0.587785i −6.45549 4.69019i −0.823809 2.53542i
35.2 −0.309017 0.951057i 0.523934 1.61250i −0.809017 + 0.587785i −0.429835 −1.69549 −0.500000 + 0.363271i 0.809017 + 0.587785i 0.101388 + 0.0736626i 0.132826 + 0.408797i
39.1 −0.309017 + 0.951057i −1.02393 3.15135i −0.809017 0.587785i 2.66590 3.31352 −0.500000 0.363271i 0.809017 0.587785i −6.45549 + 4.69019i −0.823809 + 2.53542i
39.2 −0.309017 + 0.951057i 0.523934 + 1.61250i −0.809017 0.587785i −0.429835 −1.69549 −0.500000 0.363271i 0.809017 0.587785i 0.101388 0.0736626i 0.132826 0.408797i
47.1 0.809017 0.587785i −1.75865 1.27773i 0.309017 0.951057i 1.89927 −2.17381 −0.500000 + 1.53884i −0.309017 0.951057i 0.533195 + 1.64101i 1.53654 1.11636i
47.2 0.809017 0.587785i 1.25865 + 0.914463i 0.309017 0.951057i −4.13533 1.55578 −0.500000 + 1.53884i −0.309017 0.951057i −0.179093 0.551192i −3.34556 + 2.43069i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(62, [\chi])\).