Properties

Label 630.2.d.d
Level 630
Weight 2
Character orbit 630.d
Analytic conductor 5.031
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{8} -\beta_{3} q^{10} -\beta_{2} q^{11} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + ( -\beta_{1} - \beta_{2} ) q^{14} + q^{16} -2 \beta_{3} q^{17} -\beta_{3} q^{20} -\beta_{2} q^{22} + 6 q^{23} -5 q^{25} + ( -2 \beta_{1} + \beta_{2} ) q^{26} + ( -\beta_{1} - \beta_{2} ) q^{28} + 2 \beta_{2} q^{29} + q^{32} -2 \beta_{3} q^{34} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{35} + 3 \beta_{2} q^{37} -\beta_{3} q^{40} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{41} -6 \beta_{2} q^{43} -\beta_{2} q^{44} + 6 q^{46} -2 \beta_{3} q^{47} + ( -2 + 3 \beta_{3} ) q^{49} -5 q^{50} + ( -2 \beta_{1} + \beta_{2} ) q^{52} + 6 q^{53} + ( -2 \beta_{1} + \beta_{2} ) q^{55} + ( -\beta_{1} - \beta_{2} ) q^{56} + 2 \beta_{2} q^{58} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{59} + 6 \beta_{3} q^{61} + q^{64} + 5 \beta_{2} q^{65} -2 \beta_{3} q^{68} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{70} -4 \beta_{2} q^{71} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{73} + 3 \beta_{2} q^{74} + ( -3 + \beta_{3} ) q^{77} -4 q^{79} -\beta_{3} q^{80} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{82} + 4 \beta_{3} q^{83} -10 q^{85} -6 \beta_{2} q^{86} -\beta_{2} q^{88} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 5 + 3 \beta_{3} ) q^{91} + 6 q^{92} -2 \beta_{3} q^{94} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -2 + 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + 4q^{16} + 24q^{23} - 20q^{25} + 4q^{32} + 24q^{46} - 8q^{49} - 20q^{50} + 24q^{53} + 4q^{64} - 12q^{77} - 16q^{79} - 40q^{85} + 20q^{91} + 24q^{92} - 8q^{98} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
629.1
1.58114 + 0.707107i
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
1.00000 0 1.00000 2.23607i 0 −1.58114 2.12132i 1.00000 0 2.23607i
629.2 1.00000 0 1.00000 2.23607i 0 1.58114 + 2.12132i 1.00000 0 2.23607i
629.3 1.00000 0 1.00000 2.23607i 0 −1.58114 + 2.12132i 1.00000 0 2.23607i
629.4 1.00000 0 1.00000 2.23607i 0 1.58114 2.12132i 1.00000 0 2.23607i
\(n\): 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
7.b Odd 1 yes
15.d Odd 1 yes
105.g Even 1 yes

Hecke kernels

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

\( T_{11}^{2} + 2 \)
\( T_{23} - 6 \)