Properties

Label 630.2.i.a
Level 630
Weight 2
Character orbit 630.i
Analytic conductor 5.031
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.i (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 −1.50000 + 0.866025i 1.00000 −0.500000 0.866025i 1.50000 0.866025i −0.500000 + 2.59808i −1.00000 1.50000 2.59808i 0.500000 + 0.866025i
151.1 −1.00000 −1.50000 0.866025i 1.00000 −0.500000 + 0.866025i 1.50000 + 0.866025i −0.500000 2.59808i −1.00000 1.50000 + 2.59808i 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
63.h Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11} \)
\( T_{13}^{2} + 2 T_{13} + 4 \)