Properties

Label 630.2.j.h
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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

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

Character Values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\)

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0.500000 + 0.866025i −0.724745 + 1.57313i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.72474 + 0.158919i −0.500000 0.866025i −1.00000 −1.94949 2.28024i −1.00000
211.2 0.500000 + 0.866025i 1.72474 + 0.158919i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.724745 + 1.57313i −0.500000 0.866025i −1.00000 2.94949 + 0.548188i −1.00000
421.1 0.500000 0.866025i −0.724745 1.57313i −0.500000 0.866025i −0.500000 0.866025i −1.72474 0.158919i −0.500000 + 0.866025i −1.00000 −1.94949 + 2.28024i −1.00000
421.2 0.500000 0.866025i 1.72474 0.158919i −0.500000 0.866025i −0.500000 0.866025i 0.724745 1.57313i −0.500000 + 0.866025i −1.00000 2.94949 0.548188i −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 \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{4} - 6 T_{11}^{3} + 33 T_{11}^{2} - 18 T_{11} + 9 \)
\( T_{13}^{4} + 4 T_{13}^{3} + 18 T_{13}^{2} - 8 T_{13} + 4 \)