Properties

Label 630.2.k.g
Level 630
Weight 2
Character orbit 630.k
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.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{10} + ( -5 + 5 \zeta_{6} ) q^{11} -5 q^{13} + ( 2 - 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} - q^{20} -5 q^{22} + \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -5 \zeta_{6} q^{26} + ( 3 - \zeta_{6} ) q^{28} + ( 2 - 2 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -4 q^{34} + ( 2 - 3 \zeta_{6} ) q^{35} -\zeta_{6} q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} -5 q^{41} + 12 q^{43} -5 \zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{46} -11 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} - q^{50} + ( 5 - 5 \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} -5 q^{55} + ( 1 + 2 \zeta_{6} ) q^{56} + ( 4 - 4 \zeta_{6} ) q^{59} -4 \zeta_{6} q^{61} + 2 q^{62} + q^{64} -5 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} + ( 3 - \zeta_{6} ) q^{70} -2 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} + ( 1 - \zeta_{6} ) q^{74} -7 q^{76} + ( 15 - 5 \zeta_{6} ) q^{77} + 12 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} -5 \zeta_{6} q^{82} + 12 q^{83} -4 q^{85} + 12 \zeta_{6} q^{86} + ( 5 - 5 \zeta_{6} ) q^{88} + 14 \zeta_{6} q^{89} + ( 5 + 10 \zeta_{6} ) q^{91} - q^{92} + ( 11 - 11 \zeta_{6} ) q^{94} + ( -7 + 7 \zeta_{6} ) q^{95} -8 q^{97} + ( -8 + 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + q^{5} - 4q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + q^{5} - 4q^{7} - 2q^{8} - q^{10} - 5q^{11} - 10q^{13} + q^{14} - q^{16} - 4q^{17} + 7q^{19} - 2q^{20} - 10q^{22} + q^{23} - q^{25} - 5q^{26} + 5q^{28} + 2q^{31} + q^{32} - 8q^{34} + q^{35} - q^{37} - 7q^{38} - q^{40} - 10q^{41} + 24q^{43} - 5q^{44} - q^{46} - 11q^{47} + 2q^{49} - 2q^{50} + 5q^{52} - 9q^{53} - 10q^{55} + 4q^{56} + 4q^{59} - 4q^{61} + 4q^{62} + 2q^{64} - 5q^{65} + 12q^{67} - 4q^{68} + 5q^{70} - 4q^{71} - 10q^{73} + q^{74} - 14q^{76} + 25q^{77} + 12q^{79} + q^{80} - 5q^{82} + 24q^{83} - 8q^{85} + 12q^{86} + 5q^{88} + 14q^{89} + 20q^{91} - 2q^{92} + 11q^{94} - 7q^{95} - 16q^{97} - 11q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −2.00000 1.73205i −1.00000 0 −0.500000 + 0.866025i
541.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −2.00000 + 1.73205i −1.00000 0 −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.k.g 2
3.b odd 2 1 210.2.i.b 2
7.c even 3 1 inner 630.2.k.g 2
7.c even 3 1 4410.2.a.j 1
7.d odd 6 1 4410.2.a.u 1
12.b even 2 1 1680.2.bg.d 2
15.d odd 2 1 1050.2.i.p 2
15.e even 4 2 1050.2.o.g 4
21.c even 2 1 1470.2.i.e 2
21.g even 6 1 1470.2.a.o 1
21.g even 6 1 1470.2.i.e 2
21.h odd 6 1 210.2.i.b 2
21.h odd 6 1 1470.2.a.l 1
84.n even 6 1 1680.2.bg.d 2
105.o odd 6 1 1050.2.i.p 2
105.o odd 6 1 7350.2.a.u 1
105.p even 6 1 7350.2.a.a 1
105.x even 12 2 1050.2.o.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.b 2 3.b odd 2 1
210.2.i.b 2 21.h odd 6 1
630.2.k.g 2 1.a even 1 1 trivial
630.2.k.g 2 7.c even 3 1 inner
1050.2.i.p 2 15.d odd 2 1
1050.2.i.p 2 105.o odd 6 1
1050.2.o.g 4 15.e even 4 2
1050.2.o.g 4 105.x even 12 2
1470.2.a.l 1 21.h odd 6 1
1470.2.a.o 1 21.g even 6 1
1470.2.i.e 2 21.c even 2 1
1470.2.i.e 2 21.g even 6 1
1680.2.bg.d 2 12.b even 2 1
1680.2.bg.d 2 84.n even 6 1
4410.2.a.j 1 7.c even 3 1
4410.2.a.u 1 7.d odd 6 1
7350.2.a.a 1 105.p even 6 1
7350.2.a.u 1 105.o odd 6 1

Hecke kernels

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

\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{13} + 5 \)
\( T_{17}^{2} + 4 T_{17} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ 1
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \)
$41$ \( ( 1 + 5 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 11 T + 74 T^{2} + 517 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{2} \)
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