Properties

Label 630.2.k.f
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 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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} - q^{8} + ( 1 - \zeta_{6} ) q^{10} + ( -2 + 2 \zeta_{6} ) q^{11} + ( -3 + 2 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} + q^{20} -2 q^{22} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -2 - \zeta_{6} ) q^{28} -9 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -4 q^{34} + ( 3 - 2 \zeta_{6} ) q^{35} + 4 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + \zeta_{6} q^{40} + 7 q^{41} -5 q^{43} -2 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} + 8 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} - q^{50} + ( -2 + 2 \zeta_{6} ) q^{53} + 2 q^{55} + ( 1 - 3 \zeta_{6} ) q^{56} -9 \zeta_{6} q^{58} + ( 10 - 10 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} + 4 q^{62} + q^{64} + ( 9 - 9 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} + ( 2 + \zeta_{6} ) q^{70} -2 q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} -6 q^{76} + ( -4 - 2 \zeta_{6} ) q^{77} -10 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + 7 \zeta_{6} q^{82} + 7 q^{83} + 4 q^{85} -5 \zeta_{6} q^{86} + ( 2 - 2 \zeta_{6} ) q^{88} + \zeta_{6} q^{89} -3 q^{92} + ( -8 + 8 \zeta_{6} ) q^{94} + ( 6 - 6 \zeta_{6} ) q^{95} + 14 q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - q^{5} + q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - q^{5} + q^{7} - 2q^{8} + q^{10} - 2q^{11} - 4q^{14} - q^{16} - 4q^{17} + 6q^{19} + 2q^{20} - 4q^{22} + 3q^{23} - q^{25} - 5q^{28} - 18q^{29} + 4q^{31} + q^{32} - 8q^{34} + 4q^{35} + 4q^{37} - 6q^{38} + q^{40} + 14q^{41} - 10q^{43} - 2q^{44} - 3q^{46} + 8q^{47} - 13q^{49} - 2q^{50} - 2q^{53} + 4q^{55} - q^{56} - 9q^{58} + 10q^{59} - q^{61} + 8q^{62} + 2q^{64} + 9q^{67} - 4q^{68} + 5q^{70} - 4q^{71} + 4q^{73} - 4q^{74} - 12q^{76} - 10q^{77} - 10q^{79} - q^{80} + 7q^{82} + 14q^{83} + 8q^{85} - 5q^{86} + 2q^{88} + q^{89} - 6q^{92} - 8q^{94} + 6q^{95} + 28q^{97} - 11q^{98} + O(q^{100}) \)

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 0.500000 + 2.59808i −1.00000 0 0.500000 0.866025i
541.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 2.59808i −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.f 2
3.b odd 2 1 70.2.e.a 2
7.c even 3 1 inner 630.2.k.f 2
7.c even 3 1 4410.2.a.r 1
7.d odd 6 1 4410.2.a.h 1
12.b even 2 1 560.2.q.i 2
15.d odd 2 1 350.2.e.l 2
15.e even 4 2 350.2.j.f 4
21.c even 2 1 490.2.e.f 2
21.g even 6 1 490.2.a.e 1
21.g even 6 1 490.2.e.f 2
21.h odd 6 1 70.2.e.a 2
21.h odd 6 1 490.2.a.k 1
84.j odd 6 1 3920.2.a.bk 1
84.n even 6 1 560.2.q.i 2
84.n even 6 1 3920.2.a.b 1
105.o odd 6 1 350.2.e.l 2
105.o odd 6 1 2450.2.a.b 1
105.p even 6 1 2450.2.a.q 1
105.w odd 12 2 2450.2.c.a 2
105.x even 12 2 350.2.j.f 4
105.x even 12 2 2450.2.c.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 3.b odd 2 1
70.2.e.a 2 21.h odd 6 1
350.2.e.l 2 15.d odd 2 1
350.2.e.l 2 105.o odd 6 1
350.2.j.f 4 15.e even 4 2
350.2.j.f 4 105.x even 12 2
490.2.a.e 1 21.g even 6 1
490.2.a.k 1 21.h odd 6 1
490.2.e.f 2 21.c even 2 1
490.2.e.f 2 21.g even 6 1
560.2.q.i 2 12.b even 2 1
560.2.q.i 2 84.n even 6 1
630.2.k.f 2 1.a even 1 1 trivial
630.2.k.f 2 7.c even 3 1 inner
2450.2.a.b 1 105.o odd 6 1
2450.2.a.q 1 105.p even 6 1
2450.2.c.a 2 105.w odd 12 2
2450.2.c.s 2 105.x even 12 2
3920.2.a.b 1 84.n even 6 1
3920.2.a.bk 1 84.j odd 6 1
4410.2.a.h 1 7.d odd 6 1
4410.2.a.r 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 \(S_{2}^{\mathrm{new}}(630, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ 1
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 9 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 7 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 7 T + 83 T^{2} )^{2} \)
$89$ \( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2} \)
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