Properties

Label 63.2.i.a
Level 63
Weight 2
Character orbit 63.i
Analytic conductor 0.503
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 63.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.503057532734\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(\zeta_{6}\) \(\zeta_{6}\)

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
1.73205i 1.73205i −1.00000 −1.50000 2.59808i −3.00000 2.00000 1.73205i 1.73205i −3.00000 4.50000 2.59808i
38.1 1.73205i 1.73205i −1.00000 −1.50000 + 2.59808i −3.00000 2.00000 + 1.73205i 1.73205i −3.00000 4.50000 + 2.59808i
\(n\): 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
63.i Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\).