Properties

Label 630.2.g.g
Level $630$
Weight $2$
Character orbit 630.g
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{8} + (\beta_{3} + \beta_{2} + 1) q^{10} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + q^{14} + q^{16} + 2 \beta_{2} q^{17} + ( - \beta_{3} + \beta_1 - 4) q^{19} + ( - \beta_{2} + \beta_1 + 1) q^{20} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{3} + \beta_1 + 2) q^{26} - \beta_{2} q^{28} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{31} - \beta_{2} q^{32} + 2 q^{34} + ( - \beta_{3} - \beta_{2} - 1) q^{35} - 2 \beta_{2} q^{37} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{38} + ( - \beta_{3} - \beta_{2} - 1) q^{40} + (2 \beta_{3} - 2 \beta_1 + 6) q^{41} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{3} - 2 \beta_1) q^{44} + (2 \beta_{3} - 2 \beta_1 + 2) q^{46} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{47} - q^{49} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{50} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{52} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{53} + (4 \beta_{3} - 6 \beta_{2} - 6) q^{55} - q^{56} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{58} + ( - \beta_{3} + \beta_1 - 4) q^{59} + (\beta_{3} - \beta_1 + 6) q^{61} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{62} - q^{64} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 1) q^{65} + 8 \beta_{2} q^{67} - 2 \beta_{2} q^{68} + (\beta_{2} - \beta_1 - 1) q^{70} + (2 \beta_{3} - 2 \beta_1 + 6) q^{71} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{73} - 2 q^{74} + (\beta_{3} - \beta_1 + 4) q^{76} + (2 \beta_{3} + 2 \beta_1) q^{77} + (2 \beta_{3} - 2 \beta_1 - 2) q^{79} + (\beta_{2} - \beta_1 - 1) q^{80} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{82} + (\beta_{3} + \beta_1) q^{83} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{85} + (2 \beta_{3} - 2 \beta_1 + 4) q^{86} + (2 \beta_{3} + 2 \beta_1) q^{88} - 10 q^{89} + (\beta_{3} - \beta_1 - 2) q^{91} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{92} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{94} + (2 \beta_{3} - 7 \beta_{2} + 4 \beta_1 + 1) q^{95} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{97} + \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{5} + 4 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{19} + 4 q^{20} + 8 q^{26} + 8 q^{29} + 16 q^{31} + 8 q^{34} - 4 q^{35} - 4 q^{40} + 24 q^{41} + 8 q^{46} - 4 q^{49} + 4 q^{50} - 24 q^{55} - 4 q^{56} - 16 q^{59} + 24 q^{61} - 4 q^{64} + 4 q^{65} - 4 q^{70} + 24 q^{71} - 8 q^{74} + 16 q^{76} - 8 q^{79} - 4 q^{80} - 8 q^{85} + 16 q^{86} - 40 q^{89} - 8 q^{91} + 16 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
1.00000i 0 −1.00000 −2.22474 0.224745i 0 1.00000i 1.00000i 0 −0.224745 + 2.22474i
379.2 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 1.00000i 1.00000i 0 2.22474 0.224745i
379.3 1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 1.00000i 1.00000i 0 −0.224745 2.22474i
379.4 1.00000i 0 −1.00000 0.224745 2.22474i 0 1.00000i 1.00000i 0 2.22474 + 0.224745i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.g.g 4
3.b odd 2 1 70.2.c.a 4
4.b odd 2 1 5040.2.t.t 4
5.b even 2 1 inner 630.2.g.g 4
5.c odd 4 1 3150.2.a.bs 2
5.c odd 4 1 3150.2.a.bt 2
12.b even 2 1 560.2.g.e 4
15.d odd 2 1 70.2.c.a 4
15.e even 4 1 350.2.a.g 2
15.e even 4 1 350.2.a.h 2
20.d odd 2 1 5040.2.t.t 4
21.c even 2 1 490.2.c.e 4
21.g even 6 2 490.2.i.f 8
21.h odd 6 2 490.2.i.c 8
24.f even 2 1 2240.2.g.i 4
24.h odd 2 1 2240.2.g.j 4
60.h even 2 1 560.2.g.e 4
60.l odd 4 1 2800.2.a.bl 2
60.l odd 4 1 2800.2.a.bm 2
105.g even 2 1 490.2.c.e 4
105.k odd 4 1 2450.2.a.bl 2
105.k odd 4 1 2450.2.a.bq 2
105.o odd 6 2 490.2.i.c 8
105.p even 6 2 490.2.i.f 8
120.i odd 2 1 2240.2.g.j 4
120.m even 2 1 2240.2.g.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 3.b odd 2 1
70.2.c.a 4 15.d odd 2 1
350.2.a.g 2 15.e even 4 1
350.2.a.h 2 15.e even 4 1
490.2.c.e 4 21.c even 2 1
490.2.c.e 4 105.g even 2 1
490.2.i.c 8 21.h odd 6 2
490.2.i.c 8 105.o odd 6 2
490.2.i.f 8 21.g even 6 2
490.2.i.f 8 105.p even 6 2
560.2.g.e 4 12.b even 2 1
560.2.g.e 4 60.h even 2 1
630.2.g.g 4 1.a even 1 1 trivial
630.2.g.g 4 5.b even 2 1 inner
2240.2.g.i 4 24.f even 2 1
2240.2.g.i 4 120.m even 2 1
2240.2.g.j 4 24.h odd 2 1
2240.2.g.j 4 120.i odd 2 1
2450.2.a.bl 2 105.k odd 4 1
2450.2.a.bq 2 105.k odd 4 1
2800.2.a.bl 2 60.l odd 4 1
2800.2.a.bm 2 60.l odd 4 1
3150.2.a.bs 2 5.c odd 4 1
3150.2.a.bt 2 5.c odd 4 1
5040.2.t.t 4 4.b odd 2 1
5040.2.t.t 4 20.d odd 2 1

Hecke kernels

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

\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{29}^{2} - 4T_{29} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 8 T^{2} + 20 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 264T^{2} + 3600 \) Copy content Toggle raw display
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