Properties

Label 630.2.j.f
Level 630
Weight 2
Character orbit 630.j
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.j (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{6} + \zeta_{12}^{2} q^{7} - q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12}^{2} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{6} + \zeta_{12}^{2} q^{7} - q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} - q^{10} + ( \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{11} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{12} + ( -3 + \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} + ( -1 + \zeta_{12}^{2} ) q^{14} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + q^{17} -3 q^{18} + ( -3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{19} -\zeta_{12}^{2} q^{20} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{21} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{22} + ( -3 - \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{23} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{24} -\zeta_{12}^{2} q^{25} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} - q^{28} + ( -3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{29} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{30} + ( -2 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{31} + ( 1 - \zeta_{12}^{2} ) q^{32} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{33} + \zeta_{12}^{2} q^{34} - q^{35} -3 \zeta_{12}^{2} q^{36} + ( -1 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{37} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{38} + ( 3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{39} + ( 1 - \zeta_{12}^{2} ) q^{40} + ( -4 + \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{41} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{42} + ( 6 \zeta_{12} - \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{43} + ( 2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{44} -3 \zeta_{12}^{2} q^{45} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{46} -4 \zeta_{12}^{2} q^{47} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{48} + ( -1 + \zeta_{12}^{2} ) q^{49} + ( 1 - \zeta_{12}^{2} ) q^{50} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( \zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{52} + ( 5 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{53} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{54} + ( 2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} -\zeta_{12}^{2} q^{56} + ( -3 \zeta_{12} + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{57} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{58} + ( -5 + 2 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{59} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{60} + ( -\zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{61} + ( -2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{62} -3 q^{63} + q^{64} + ( \zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{65} + ( -3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{66} + ( 9 - 9 \zeta_{12}^{2} ) q^{67} + ( -1 + \zeta_{12}^{2} ) q^{68} + ( -3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{69} -\zeta_{12}^{2} q^{70} + ( 7 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{71} + ( 3 - 3 \zeta_{12}^{2} ) q^{72} + ( 5 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + ( 5 \zeta_{12} - \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{74} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{75} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{77} + ( -3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{78} + 14 \zeta_{12}^{2} q^{79} + q^{80} -9 \zeta_{12}^{2} q^{81} + ( -4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{82} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{83} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{84} + ( -1 + \zeta_{12}^{2} ) q^{85} + ( 1 - 6 \zeta_{12} - \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{86} + ( 9 - 3 \zeta_{12} - 9 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{87} + ( -\zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{88} + ( -4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{89} + ( 3 - 3 \zeta_{12}^{2} ) q^{90} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{91} + ( -\zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{92} + ( 12 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{93} + ( 4 - 4 \zeta_{12}^{2} ) q^{94} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{95} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{96} + \zeta_{12}^{2} q^{97} - q^{98} + ( 6 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 2q^{5} + 2q^{7} - 4q^{8} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 2q^{5} + 2q^{7} - 4q^{8} - 6q^{9} - 4q^{10} - 4q^{11} - 6q^{13} - 2q^{14} - 2q^{16} + 4q^{17} - 12q^{18} - 12q^{19} - 2q^{20} + 4q^{22} - 6q^{23} - 2q^{25} - 12q^{26} - 4q^{28} + 6q^{29} - 4q^{31} + 2q^{32} - 6q^{33} + 2q^{34} - 4q^{35} - 6q^{36} - 4q^{37} - 6q^{38} + 12q^{39} + 2q^{40} - 8q^{41} - 2q^{43} + 8q^{44} - 6q^{45} - 12q^{46} - 8q^{47} - 2q^{49} + 2q^{50} - 6q^{52} + 20q^{53} + 8q^{55} - 2q^{56} + 12q^{57} - 6q^{58} - 10q^{59} - 2q^{61} - 8q^{62} - 12q^{63} + 4q^{64} - 6q^{65} - 12q^{66} + 18q^{67} - 2q^{68} - 12q^{69} - 2q^{70} + 28q^{71} + 6q^{72} + 20q^{73} - 2q^{74} + 6q^{76} + 4q^{77} + 6q^{78} + 28q^{79} + 4q^{80} - 18q^{81} - 16q^{82} + 4q^{83} - 2q^{85} + 2q^{86} + 18q^{87} + 4q^{88} - 16q^{89} + 6q^{90} - 12q^{91} - 6q^{92} + 48q^{93} + 8q^{94} + 6q^{95} + 2q^{97} - 4q^{98} + 24q^{99} + O(q^{100}) \)

Character Values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.500000 + 0.866025i −0.866025 1.50000i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.866025 1.50000i 0.500000 + 0.866025i −1.00000 −1.50000 + 2.59808i −1.00000
211.2 0.500000 + 0.866025i 0.866025 + 1.50000i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.866025 + 1.50000i 0.500000 + 0.866025i −1.00000 −1.50000 + 2.59808i −1.00000
421.1 0.500000 0.866025i −0.866025 + 1.50000i −0.500000 0.866025i −0.500000 0.866025i 0.866025 + 1.50000i 0.500000 0.866025i −1.00000 −1.50000 2.59808i −1.00000
421.2 0.500000 0.866025i 0.866025 1.50000i −0.500000 0.866025i −0.500000 0.866025i −0.866025 1.50000i 0.500000 0.866025i −1.00000 −1.50000 2.59808i −1.00000
\(n\): 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
9.c Even 1 yes

Hecke kernels

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

\( T_{11}^{4} + 4 T_{11}^{3} + 15 T_{11}^{2} + 4 T_{11} + 1 \)
\( T_{13}^{4} + 6 T_{13}^{3} + 30 T_{13}^{2} + 36 T_{13} + 36 \)