Properties

Label 630.2.b.a
Level 630
Weight 2
Character orbit 630.b
Analytic conductor 5.031
Analytic rank 0
Dimension 8
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 26 x^{6} + 205 x^{4} + 540 x^{2} + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 20 \nu^{5} + 12 \nu^{4} + 73 \nu^{3} + 156 \nu^{2} - 198 \nu + 216 \)\()/144\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 20 \nu^{5} + 12 \nu^{4} - 73 \nu^{3} + 156 \nu^{2} + 198 \nu + 216 \)\()/144\)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{7} + 112 \nu^{5} + 665 \nu^{3} + 954 \nu \)\()/432\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} - 20 \nu^{4} - 73 \nu^{2} + 54 \)\()/24\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 6 \nu^{6} + 20 \nu^{5} + 132 \nu^{4} + 73 \nu^{3} + 738 \nu^{2} - 54 \nu + 900 \)\()/144\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{6} + 20 \nu^{5} - 132 \nu^{4} + 73 \nu^{3} - 738 \nu^{2} - 54 \nu - 900 \)\()/144\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 26 \nu^{5} - 187 \nu^{3} - 306 \nu \)\()/36\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_{1} - 14\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} + 12 \beta_{3} - 7 \beta_{2} + 7 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(13 \beta_{6} - 13 \beta_{5} - 26 \beta_{4} + 25 \beta_{2} + 25 \beta_{1} + 146\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-50 \beta_{7} + 163 \beta_{6} + 163 \beta_{5} - 228 \beta_{3} + 73 \beta_{2} - 73 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-187 \beta_{6} + 187 \beta_{5} + 326 \beta_{4} - 427 \beta_{2} - 427 \beta_{1} - 1790\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(854 \beta_{7} - 2113 \beta_{6} - 2113 \beta_{5} + 3684 \beta_{3} - 895 \beta_{2} + 895 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.916813i
2.73923i
3.73923i
1.91681i
0.916813i
2.73923i
3.73923i
1.91681i
1.00000i 0 −1.00000 −1.00000 0 −2.56510 + 0.648285i 1.00000i 0 1.00000i
251.2 1.00000i 0 −1.00000 −1.00000 0 −1.80230 1.93693i 1.00000i 0 1.00000i
251.3 1.00000i 0 −1.00000 −1.00000 0 0.0951965 + 2.64404i 1.00000i 0 1.00000i
251.4 1.00000i 0 −1.00000 −1.00000 0 2.27220 1.35539i 1.00000i 0 1.00000i
251.5 1.00000i 0 −1.00000 −1.00000 0 −2.56510 0.648285i 1.00000i 0 1.00000i
251.6 1.00000i 0 −1.00000 −1.00000 0 −1.80230 + 1.93693i 1.00000i 0 1.00000i
251.7 1.00000i 0 −1.00000 −1.00000 0 0.0951965 2.64404i 1.00000i 0 1.00000i
251.8 1.00000i 0 −1.00000 −1.00000 0 2.27220 + 1.35539i 1.00000i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
21.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{17}^{4} - 50 T_{17}^{2} + 120 T_{17} - 72 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\).