Properties

Label 630.2.g.c
Level 630
Weight 2
Character orbit 630.g
Analytic conductor 5.031
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -1 - 2 i ) q^{5} + i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -1 - 2 i ) q^{5} + i q^{7} -i q^{8} + ( 2 - i ) q^{10} -2 q^{11} + 6 i q^{13} - q^{14} + q^{16} + 4 i q^{17} + 6 q^{19} + ( 1 + 2 i ) q^{20} -2 i q^{22} + 8 i q^{23} + ( -3 + 4 i ) q^{25} -6 q^{26} -i q^{28} + 6 q^{29} -2 q^{31} + i q^{32} -4 q^{34} + ( 2 - i ) q^{35} -4 i q^{37} + 6 i q^{38} + ( -2 + i ) q^{40} -2 q^{41} + 4 i q^{43} + 2 q^{44} -8 q^{46} + 8 i q^{47} - q^{49} + ( -4 - 3 i ) q^{50} -6 i q^{52} -6 i q^{53} + ( 2 + 4 i ) q^{55} + q^{56} + 6 i q^{58} -8 q^{59} -10 q^{61} -2 i q^{62} - q^{64} + ( 12 - 6 i ) q^{65} -8 i q^{67} -4 i q^{68} + ( 1 + 2 i ) q^{70} + 6 q^{71} -14 i q^{73} + 4 q^{74} -6 q^{76} -2 i q^{77} + 12 q^{79} + ( -1 - 2 i ) q^{80} -2 i q^{82} + 8 i q^{83} + ( 8 - 4 i ) q^{85} -4 q^{86} + 2 i q^{88} -10 q^{89} -6 q^{91} -8 i q^{92} -8 q^{94} + ( -6 - 12 i ) q^{95} + 10 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} + 4q^{10} - 4q^{11} - 2q^{14} + 2q^{16} + 12q^{19} + 2q^{20} - 6q^{25} - 12q^{26} + 12q^{29} - 4q^{31} - 8q^{34} + 4q^{35} - 4q^{40} - 4q^{41} + 4q^{44} - 16q^{46} - 2q^{49} - 8q^{50} + 4q^{55} + 2q^{56} - 16q^{59} - 20q^{61} - 2q^{64} + 24q^{65} + 2q^{70} + 12q^{71} + 8q^{74} - 12q^{76} + 24q^{79} - 2q^{80} + 16q^{85} - 8q^{86} - 20q^{89} - 12q^{91} - 16q^{94} - 12q^{95} + O(q^{100}) \)

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
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 + 2.00000i 0 1.00000i 1.00000i 0 2.00000 + 1.00000i
379.2 1.00000i 0 −1.00000 −1.00000 2.00000i 0 1.00000i 1.00000i 0 2.00000 1.00000i
\(n\): 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.c 2
3.b odd 2 1 210.2.g.b 2
4.b odd 2 1 5040.2.t.h 2
5.b even 2 1 inner 630.2.g.c 2
5.c odd 4 1 3150.2.a.d 1
5.c odd 4 1 3150.2.a.bk 1
12.b even 2 1 1680.2.t.e 2
15.d odd 2 1 210.2.g.b 2
15.e even 4 1 1050.2.a.d 1
15.e even 4 1 1050.2.a.p 1
20.d odd 2 1 5040.2.t.h 2
21.c even 2 1 1470.2.g.b 2
21.g even 6 2 1470.2.n.f 4
21.h odd 6 2 1470.2.n.b 4
60.h even 2 1 1680.2.t.e 2
60.l odd 4 1 8400.2.a.w 1
60.l odd 4 1 8400.2.a.bp 1
105.g even 2 1 1470.2.g.b 2
105.k odd 4 1 7350.2.a.bk 1
105.k odd 4 1 7350.2.a.bz 1
105.o odd 6 2 1470.2.n.b 4
105.p even 6 2 1470.2.n.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 3.b odd 2 1
210.2.g.b 2 15.d odd 2 1
630.2.g.c 2 1.a even 1 1 trivial
630.2.g.c 2 5.b even 2 1 inner
1050.2.a.d 1 15.e even 4 1
1050.2.a.p 1 15.e even 4 1
1470.2.g.b 2 21.c even 2 1
1470.2.g.b 2 105.g even 2 1
1470.2.n.b 4 21.h odd 6 2
1470.2.n.b 4 105.o odd 6 2
1470.2.n.f 4 21.g even 6 2
1470.2.n.f 4 105.p even 6 2
1680.2.t.e 2 12.b even 2 1
1680.2.t.e 2 60.h even 2 1
3150.2.a.d 1 5.c odd 4 1
3150.2.a.bk 1 5.c odd 4 1
5040.2.t.h 2 4.b odd 2 1
5040.2.t.h 2 20.d odd 2 1
7350.2.a.bk 1 105.k odd 4 1
7350.2.a.bz 1 105.k odd 4 1
8400.2.a.w 1 60.l odd 4 1
8400.2.a.bp 1 60.l odd 4 1

Hecke kernels

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

\( T_{11} + 2 \)
\( T_{29} - 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ 1
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( 1 - 18 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 6 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 58 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 8 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 70 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 50 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 12 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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