Properties

Label 630.2.k.b
Level 630
Weight 2
Character orbit 630.k
Analytic conductor 5.031
Analytic rank 1
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: \(1\)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{10} + ( -6 + 6 \zeta_{6} ) q^{11} -4 q^{13} + ( -3 + 2 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} -2 \zeta_{6} q^{19} + q^{20} + 6 q^{22} -3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + ( 2 + \zeta_{6} ) q^{28} + 3 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + ( -3 + 2 \zeta_{6} ) q^{35} + 4 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} -9 q^{41} -7 q^{43} -6 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} + ( -8 + 3 \zeta_{6} ) q^{49} + q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + 6 q^{55} + ( 1 - 3 \zeta_{6} ) q^{56} -3 \zeta_{6} q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 4 \zeta_{6} q^{65} + ( -5 + 5 \zeta_{6} ) q^{67} + ( 2 + \zeta_{6} ) q^{70} + 6 q^{71} + ( 16 - 16 \zeta_{6} ) q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + 2 q^{76} + ( 12 + 6 \zeta_{6} ) q^{77} -2 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + 9 \zeta_{6} q^{82} -3 q^{83} + 7 \zeta_{6} q^{86} + ( -6 + 6 \zeta_{6} ) q^{88} -15 \zeta_{6} q^{89} + ( -4 + 12 \zeta_{6} ) q^{91} + 3 q^{92} + ( -2 + 2 \zeta_{6} ) q^{95} + 14 q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - q^{5} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - q^{5} - q^{7} + 2q^{8} - q^{10} - 6q^{11} - 8q^{13} - 4q^{14} - q^{16} - 2q^{19} + 2q^{20} + 12q^{22} - 3q^{23} - q^{25} + 4q^{26} + 5q^{28} + 6q^{29} - 8q^{31} - q^{32} - 4q^{35} + 4q^{37} - 2q^{38} - q^{40} - 18q^{41} - 14q^{43} - 6q^{44} - 3q^{46} - 13q^{49} + 2q^{50} + 4q^{52} - 6q^{53} + 12q^{55} - q^{56} - 3q^{58} - 6q^{59} - 5q^{61} + 16q^{62} + 2q^{64} + 4q^{65} - 5q^{67} + 5q^{70} + 12q^{71} + 16q^{73} + 4q^{74} + 4q^{76} + 30q^{77} - 2q^{79} - q^{80} + 9q^{82} - 6q^{83} + 7q^{86} - 6q^{88} - 15q^{89} + 4q^{91} + 6q^{92} - 2q^{95} + 28q^{97} + 11q^{98} + O(q^{100}) \)

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

\( T_{11}^{2} + 6 T_{11} + 36 \)
\( T_{13} + 4 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ 1
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 9 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 7 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 3 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2} \)
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