Properties

Label 630.2.k.e
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} - q^{8} + ( 1 - \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + 5 q^{13} + ( 2 - 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + q^{20} + 3 q^{22} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 \zeta_{6} q^{26} + ( 3 - \zeta_{6} ) q^{28} + 6 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 6 q^{34} + ( -2 + 3 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{38} + \zeta_{6} q^{40} -3 q^{41} -10 q^{43} + 3 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} + 3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} - q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} -3 q^{55} + ( 1 + 2 \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} + 4 \zeta_{6} q^{61} + 4 q^{62} + q^{64} -5 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} + 6 \zeta_{6} q^{68} + ( -3 + \zeta_{6} ) q^{70} -12 q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} + ( 11 - 11 \zeta_{6} ) q^{74} - q^{76} + ( -9 + 3 \zeta_{6} ) q^{77} + 10 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} -3 \zeta_{6} q^{82} + 12 q^{83} -6 q^{85} -10 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + ( -5 - 10 \zeta_{6} ) q^{91} -3 q^{92} + ( -3 + 3 \zeta_{6} ) q^{94} + ( 1 - \zeta_{6} ) q^{95} + 14 q^{97} + ( -8 + 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - q^{5} - 4q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - q^{5} - 4q^{7} - 2q^{8} + q^{10} + 3q^{11} + 10q^{13} + q^{14} - q^{16} + 6q^{17} + q^{19} + 2q^{20} + 6q^{22} + 3q^{23} - q^{25} + 5q^{26} + 5q^{28} + 12q^{29} + 4q^{31} + q^{32} + 12q^{34} - q^{35} - 11q^{37} - q^{38} + q^{40} - 6q^{41} - 20q^{43} + 3q^{44} - 3q^{46} + 3q^{47} + 2q^{49} - 2q^{50} - 5q^{52} + 3q^{53} - 6q^{55} + 4q^{56} + 6q^{58} + 4q^{61} + 8q^{62} + 2q^{64} - 5q^{65} + 4q^{67} + 6q^{68} - 5q^{70} - 24q^{71} + 4q^{73} + 11q^{74} - 2q^{76} - 15q^{77} + 10q^{79} - q^{80} - 3q^{82} + 24q^{83} - 12q^{85} - 10q^{86} - 3q^{88} + 6q^{89} - 20q^{91} - 6q^{92} - 3q^{94} + q^{95} + 28q^{97} - 11q^{98} + O(q^{100}) \)

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.e 2
3.b odd 2 1 70.2.e.b 2
7.c even 3 1 inner 630.2.k.e 2
7.c even 3 1 4410.2.a.m 1
7.d odd 6 1 4410.2.a.c 1
12.b even 2 1 560.2.q.d 2
15.d odd 2 1 350.2.e.h 2
15.e even 4 2 350.2.j.a 4
21.c even 2 1 490.2.e.a 2
21.g even 6 1 490.2.a.j 1
21.g even 6 1 490.2.e.a 2
21.h odd 6 1 70.2.e.b 2
21.h odd 6 1 490.2.a.g 1
84.j odd 6 1 3920.2.a.g 1
84.n even 6 1 560.2.q.d 2
84.n even 6 1 3920.2.a.be 1
105.o odd 6 1 350.2.e.h 2
105.o odd 6 1 2450.2.a.p 1
105.p even 6 1 2450.2.a.f 1
105.w odd 12 2 2450.2.c.p 2
105.x even 12 2 350.2.j.a 4
105.x even 12 2 2450.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 3.b odd 2 1
70.2.e.b 2 21.h odd 6 1
350.2.e.h 2 15.d odd 2 1
350.2.e.h 2 105.o odd 6 1
350.2.j.a 4 15.e even 4 2
350.2.j.a 4 105.x even 12 2
490.2.a.g 1 21.h odd 6 1
490.2.a.j 1 21.g even 6 1
490.2.e.a 2 21.c even 2 1
490.2.e.a 2 21.g even 6 1
560.2.q.d 2 12.b even 2 1
560.2.q.d 2 84.n even 6 1
630.2.k.e 2 1.a even 1 1 trivial
630.2.k.e 2 7.c even 3 1 inner
2450.2.a.f 1 105.p even 6 1
2450.2.a.p 1 105.o odd 6 1
2450.2.c.f 2 105.x even 12 2
2450.2.c.p 2 105.w odd 12 2
3920.2.a.g 1 84.j odd 6 1
3920.2.a.be 1 84.n even 6 1
4410.2.a.c 1 7.d odd 6 1
4410.2.a.m 1 7.c even 3 1

Hecke kernels

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

\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{13} - 5 \)
\( T_{17}^{2} - 6 T_{17} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( ( -5 + T )^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 121 + 11 T + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( 9 - 3 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 16 - 4 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 16 - 4 T + T^{2} \)
$79$ \( 100 - 10 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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