Properties

Label 630.2.d.b
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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} - q^{8} -\beta_{3} q^{10} + 4 \beta_{1} q^{11} + 2 \beta_{3} q^{13} + ( \beta_{1} + \beta_{3} ) q^{14} + q^{16} + \beta_{2} q^{17} -\beta_{2} q^{19} + \beta_{3} q^{20} -4 \beta_{1} q^{22} + 4 q^{23} + 5 q^{25} -2 \beta_{3} q^{26} + ( -\beta_{1} - \beta_{3} ) q^{28} + 2 \beta_{1} q^{29} -2 \beta_{2} q^{31} - q^{32} -\beta_{2} q^{34} + ( -5 - \beta_{2} ) q^{35} + 7 \beta_{1} q^{37} + \beta_{2} q^{38} -\beta_{3} q^{40} + 2 \beta_{3} q^{41} + \beta_{1} q^{43} + 4 \beta_{1} q^{44} -4 q^{46} -3 \beta_{2} q^{47} + ( 3 + 2 \beta_{2} ) q^{49} -5 q^{50} + 2 \beta_{3} q^{52} + 4 q^{53} + 4 \beta_{2} q^{55} + ( \beta_{1} + \beta_{3} ) q^{56} -2 \beta_{1} q^{58} + 2 \beta_{3} q^{59} -3 \beta_{2} q^{61} + 2 \beta_{2} q^{62} + q^{64} + 10 q^{65} + 5 \beta_{1} q^{67} + \beta_{2} q^{68} + ( 5 + \beta_{2} ) q^{70} + \beta_{1} q^{71} + 6 \beta_{3} q^{73} -7 \beta_{1} q^{74} -\beta_{2} q^{76} + ( 8 - 4 \beta_{2} ) q^{77} + 6 q^{79} + \beta_{3} q^{80} -2 \beta_{3} q^{82} + 4 \beta_{2} q^{83} + 5 \beta_{1} q^{85} -\beta_{1} q^{86} -4 \beta_{1} q^{88} -2 \beta_{3} q^{89} + ( -10 - 2 \beta_{2} ) q^{91} + 4 q^{92} + 3 \beta_{2} q^{94} -5 \beta_{1} q^{95} + ( -3 - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} + 4q^{16} + 16q^{23} + 20q^{25} - 4q^{32} - 20q^{35} - 16q^{46} + 12q^{49} - 20q^{50} + 16q^{53} + 4q^{64} + 40q^{65} + 20q^{70} + 32q^{77} + 24q^{79} - 40q^{91} + 16q^{92} - 12q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 8 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 4 \beta_{1}\)

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
2.28825i
2.28825i
0.874032i
0.874032i
−1.00000 0 1.00000 −2.23607 0 2.23607 1.41421i −1.00000 0 2.23607
629.2 −1.00000 0 1.00000 −2.23607 0 2.23607 + 1.41421i −1.00000 0 2.23607
629.3 −1.00000 0 1.00000 2.23607 0 −2.23607 1.41421i −1.00000 0 −2.23607
629.4 −1.00000 0 1.00000 2.23607 0 −2.23607 + 1.41421i −1.00000 0 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.d.b 4
3.b odd 2 1 630.2.d.c yes 4
4.b odd 2 1 5040.2.k.b 4
5.b even 2 1 630.2.d.c yes 4
5.c odd 4 2 3150.2.b.b 8
7.b odd 2 1 inner 630.2.d.b 4
12.b even 2 1 5040.2.k.c 4
15.d odd 2 1 inner 630.2.d.b 4
15.e even 4 2 3150.2.b.b 8
20.d odd 2 1 5040.2.k.c 4
21.c even 2 1 630.2.d.c yes 4
28.d even 2 1 5040.2.k.b 4
35.c odd 2 1 630.2.d.c yes 4
35.f even 4 2 3150.2.b.b 8
60.h even 2 1 5040.2.k.b 4
84.h odd 2 1 5040.2.k.c 4
105.g even 2 1 inner 630.2.d.b 4
105.k odd 4 2 3150.2.b.b 8
140.c even 2 1 5040.2.k.c 4
420.o odd 2 1 5040.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.b 4 1.a even 1 1 trivial
630.2.d.b 4 7.b odd 2 1 inner
630.2.d.b 4 15.d odd 2 1 inner
630.2.d.b 4 105.g even 2 1 inner
630.2.d.c yes 4 3.b odd 2 1
630.2.d.c yes 4 5.b even 2 1
630.2.d.c yes 4 21.c even 2 1
630.2.d.c yes 4 35.c odd 2 1
3150.2.b.b 8 5.c odd 4 2
3150.2.b.b 8 15.e even 4 2
3150.2.b.b 8 35.f even 4 2
3150.2.b.b 8 105.k odd 4 2
5040.2.k.b 4 4.b odd 2 1
5040.2.k.b 4 28.d even 2 1
5040.2.k.b 4 60.h even 2 1
5040.2.k.b 4 420.o odd 2 1
5040.2.k.c 4 12.b even 2 1
5040.2.k.c 4 20.d odd 2 1
5040.2.k.c 4 84.h odd 2 1
5040.2.k.c 4 140.c even 2 1

Hecke kernels

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

\( T_{11}^{2} + 32 \)
\( T_{23} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( 1 - 6 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 6 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 24 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 28 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 22 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 24 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 62 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 84 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 4 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 32 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 84 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 34 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 6 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 6 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 158 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 97 T^{2} )^{4} \)
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