Properties

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

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Newspace parameters

Level: \( N \) = \( 62 = 2 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 62.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.495072492532\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 −1.50000 + 2.59808i 1.00000 −0.500000 0.866025i −1.50000 + 2.59808i 1.50000 2.59808i 1.00000 −3.00000 5.19615i −0.500000 0.866025i
25.1 1.00000 −1.50000 2.59808i 1.00000 −0.500000 + 0.866025i −1.50000 2.59808i 1.50000 + 2.59808i 1.00000 −3.00000 + 5.19615i −0.500000 + 0.866025i
\(n\): 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
31.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3 T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(62, [\chi])\).