Properties

Label 630.2.g.g
Level 630
Weight 2
Character orbit 630.g
Analytic conductor 5.031
Analytic rank 0
Dimension 4
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.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} - q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} - q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{8} + ( 1 + \beta_{2} + \beta_{3} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{13} + q^{14} + q^{16} + 2 \beta_{2} q^{17} + ( -4 + \beta_{1} - \beta_{3} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} ) q^{20} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( 2 + \beta_{1} - \beta_{3} ) q^{26} -\beta_{2} q^{28} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{31} -\beta_{2} q^{32} + 2 q^{34} + ( -1 - \beta_{2} - \beta_{3} ) q^{35} -2 \beta_{2} q^{37} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{38} + ( -1 - \beta_{2} - \beta_{3} ) q^{40} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{44} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{46} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{47} - q^{49} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{50} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{52} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -6 - 6 \beta_{2} + 4 \beta_{3} ) q^{55} - q^{56} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{58} + ( -4 + \beta_{1} - \beta_{3} ) q^{59} + ( 6 - \beta_{1} + \beta_{3} ) q^{61} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{62} - q^{64} + ( 1 - 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{65} + 8 \beta_{2} q^{67} -2 \beta_{2} q^{68} + ( -1 - \beta_{1} + \beta_{2} ) q^{70} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{73} -2 q^{74} + ( 4 - \beta_{1} + \beta_{3} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{77} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} ) q^{80} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{82} + ( \beta_{1} + \beta_{3} ) q^{83} + ( -2 - 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{86} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{88} -10 q^{89} + ( -2 - \beta_{1} + \beta_{3} ) q^{91} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{94} + ( 1 + 4 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{97} + \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{5} + 4q^{10} + 4q^{14} + 4q^{16} - 16q^{19} + 4q^{20} + 8q^{26} + 8q^{29} + 16q^{31} + 8q^{34} - 4q^{35} - 4q^{40} + 24q^{41} + 8q^{46} - 4q^{49} + 4q^{50} - 24q^{55} - 4q^{56} - 16q^{59} + 24q^{61} - 4q^{64} + 4q^{65} - 4q^{70} + 24q^{71} - 8q^{74} + 16q^{76} - 8q^{79} - 4q^{80} - 8q^{85} + 16q^{86} - 40q^{89} - 8q^{91} + 16q^{94} + 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character Values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
1.00000i 0 −1.00000 −2.22474 0.224745i 0 1.00000i 1.00000i 0 −0.224745 + 2.22474i
379.2 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 1.00000i 1.00000i 0 2.22474 0.224745i
379.3 1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 1.00000i 1.00000i 0 −0.224745 2.22474i
379.4 1.00000i 0 −1.00000 0.224745 2.22474i 0 1.00000i 1.00000i 0 2.22474 + 0.224745i
\(n\): 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
5.b Even 1 yes

Hecke kernels

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

\( T_{11}^{2} - 24 \)
\( T_{29}^{2} - 4 T_{29} - 20 \)