Properties

Label 630.2.i.e
Level 630
Weight 2
Character orbit 630.i
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.i (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: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{3} + 2 \beta_{2}\)\()/3\)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
1.00000 −1.50000 + 0.866025i 1.00000 0.500000 + 0.866025i −1.50000 + 0.866025i −2.62132 + 0.358719i 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
121.2 1.00000 −1.50000 + 0.866025i 1.00000 0.500000 + 0.866025i −1.50000 + 0.866025i 1.62132 2.09077i 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
151.1 1.00000 −1.50000 0.866025i 1.00000 0.500000 0.866025i −1.50000 0.866025i −2.62132 0.358719i 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
151.2 1.00000 −1.50000 0.866025i 1.00000 0.500000 0.866025i −1.50000 0.866025i 1.62132 + 2.09077i 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
\(n\): 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
63.h Even 1 yes

Hecke kernels

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

\( T_{11}^{4} + 18 T_{11}^{2} + 324 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)