Properties

Label 630.2.g.e
Level 630
Weight 2
Character orbit 630.g
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.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( 1 + 2 i ) q^{5} + i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( 1 + 2 i ) q^{5} + i q^{7} -i q^{8} + ( -2 + i ) q^{10} + 6 q^{11} + 2 i q^{13} - q^{14} + q^{16} -2 i q^{17} -4 q^{19} + ( -1 - 2 i ) q^{20} + 6 i q^{22} + 4 i q^{23} + ( -3 + 4 i ) q^{25} -2 q^{26} -i q^{28} -2 q^{29} -2 q^{31} + i q^{32} + 2 q^{34} + ( -2 + i ) q^{35} + 10 i q^{37} -4 i q^{38} + ( 2 - i ) q^{40} + 6 q^{41} -2 i q^{43} -6 q^{44} -4 q^{46} + 2 i q^{47} - q^{49} + ( -4 - 3 i ) q^{50} -2 i q^{52} -6 i q^{53} + ( 6 + 12 i ) q^{55} + q^{56} -2 i q^{58} + 4 q^{59} -12 q^{61} -2 i q^{62} - q^{64} + ( -4 + 2 i ) q^{65} -10 i q^{67} + 2 i q^{68} + ( -1 - 2 i ) q^{70} + 12 q^{71} -2 i q^{73} -10 q^{74} + 4 q^{76} + 6 i q^{77} + 16 q^{79} + ( 1 + 2 i ) q^{80} + 6 i q^{82} + 12 i q^{83} + ( 4 - 2 i ) q^{85} + 2 q^{86} -6 i q^{88} -14 q^{89} -2 q^{91} -4 i q^{92} -2 q^{94} + ( -4 - 8 i ) q^{95} -18 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{5} - 4q^{10} + 12q^{11} - 2q^{14} + 2q^{16} - 8q^{19} - 2q^{20} - 6q^{25} - 4q^{26} - 4q^{29} - 4q^{31} + 4q^{34} - 4q^{35} + 4q^{40} + 12q^{41} - 12q^{44} - 8q^{46} - 2q^{49} - 8q^{50} + 12q^{55} + 2q^{56} + 8q^{59} - 24q^{61} - 2q^{64} - 8q^{65} - 2q^{70} + 24q^{71} - 20q^{74} + 8q^{76} + 32q^{79} + 2q^{80} + 8q^{85} + 4q^{86} - 28q^{89} - 4q^{91} - 4q^{94} - 8q^{95} + O(q^{100}) \)

Character Values

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

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

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
1.00000i
1.00000i
1.00000i 0 −1.00000 1.00000 2.00000i 0 1.00000i 1.00000i 0 −2.00000 1.00000i
379.2 1.00000i 0 −1.00000 1.00000 + 2.00000i 0 1.00000i 1.00000i 0 −2.00000 + 1.00000i
\(n\): 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 \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11} - 6 \)
\( T_{29} + 2 \)