Properties

Label 630.2.k.a
Level 630
Weight 2
Character orbit 630.k
Analytic conductor 5.031
Analytic rank 0
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.k (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + 5 q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} -5 \zeta_{6} q^{19} + q^{20} -3 q^{22} -9 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -5 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} + ( 10 - 10 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + ( 2 + \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( -5 + 5 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} -9 q^{41} + 8 q^{43} + 3 \zeta_{6} q^{44} + ( -9 + 9 \zeta_{6} ) q^{46} + 3 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} -3 q^{55} + ( -3 + 2 \zeta_{6} ) q^{56} + ( 12 - 12 \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} -10 q^{62} + q^{64} -5 \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} + ( 1 - 3 \zeta_{6} ) q^{70} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + ( 1 - \zeta_{6} ) q^{74} + 5 q^{76} + ( -3 + 9 \zeta_{6} ) q^{77} -8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + 9 \zeta_{6} q^{82} -8 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + ( -15 + 10 \zeta_{6} ) q^{91} + 9 q^{92} + ( 3 - 3 \zeta_{6} ) q^{94} + ( -5 + 5 \zeta_{6} ) q^{95} + 8 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - q^{5} - 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - q^{5} - 4q^{7} + 2q^{8} - q^{10} + 3q^{11} + 10q^{13} + 5q^{14} - q^{16} - 5q^{19} + 2q^{20} - 6q^{22} - 9q^{23} - q^{25} - 5q^{26} - q^{28} + 10q^{31} - q^{32} + 5q^{35} + q^{37} - 5q^{38} - q^{40} - 18q^{41} + 16q^{43} + 3q^{44} - 9q^{46} + 3q^{47} + 2q^{49} + 2q^{50} - 5q^{52} - 3q^{53} - 6q^{55} - 4q^{56} + 12q^{59} - 8q^{61} - 20q^{62} + 2q^{64} - 5q^{65} - 8q^{67} - q^{70} + 12q^{71} - 2q^{73} + q^{74} + 10q^{76} + 3q^{77} - 8q^{79} - q^{80} + 9q^{82} - 8q^{86} + 3q^{88} + 6q^{89} - 20q^{91} + 18q^{92} + 3q^{94} - 5q^{95} + 16q^{97} - 13q^{98} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −2.00000 + 1.73205i 1.00000 0 −0.500000 + 0.866025i
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −2.00000 1.73205i 1.00000 0 −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
7.c Even 1 yes

Hecke kernels

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

\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{13} - 5 \)
\( T_{17} \)