Properties

Label 630.2.j.k
Level 630
Weight 2
Character orbit 630.j
Analytic conductor 5.031
Analytic rank 0
Dimension 6
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 5 \nu^{4} + \nu^{3} + 9 \nu^{2} - 6 \nu - 45 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 9 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 2 \nu^{3} - 6 \nu - 18 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} - 24 \nu - 72 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} + 4\)

Character Values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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
−1.62241 0.606458i
0.403374 + 1.68443i
1.71903 0.211943i
−1.62241 + 0.606458i
0.403374 1.68443i
1.71903 + 0.211943i
−0.500000 0.866025i −1.62241 0.606458i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.285997 + 1.70828i −0.500000 0.866025i 1.00000 2.26442 + 1.96784i 1.00000
211.2 −0.500000 0.866025i 0.403374 + 1.68443i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.25707 1.19154i −0.500000 0.866025i 1.00000 −2.67458 + 1.35891i 1.00000
211.3 −0.500000 0.866025i 1.71903 0.211943i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.04307 1.38276i −0.500000 0.866025i 1.00000 2.91016 0.728674i 1.00000
421.1 −0.500000 + 0.866025i −1.62241 + 0.606458i −0.500000 0.866025i −0.500000 0.866025i 0.285997 1.70828i −0.500000 + 0.866025i 1.00000 2.26442 1.96784i 1.00000
421.2 −0.500000 + 0.866025i 0.403374 1.68443i −0.500000 0.866025i −0.500000 0.866025i 1.25707 + 1.19154i −0.500000 + 0.866025i 1.00000 −2.67458 1.35891i 1.00000
421.3 −0.500000 + 0.866025i 1.71903 + 0.211943i −0.500000 0.866025i −0.500000 0.866025i −1.04307 + 1.38276i −0.500000 + 0.866025i 1.00000 2.91016 + 0.728674i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.3
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 \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{6} - T_{11}^{5} + 6 T_{11}^{4} - T_{11}^{3} + 28 T_{11}^{2} - 15 T_{11} + 9 \)
\( T_{13}^{6} - 2 T_{13}^{5} + 8 T_{13}^{4} + 4 T_{13}^{3} + 20 T_{13}^{2} - 8 T_{13} + 4 \)