Properties

Label 630.2.u.b
Level 630
Weight 2
Character orbit 630.u
Analytic conductor 5.031
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{10} + 2 \zeta_{12}^{3} q^{13} + ( -3 + 2 \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -2 + 2 \zeta_{12}^{2} ) q^{19} + ( 1 + 2 \zeta_{12}^{3} ) q^{20} + \zeta_{12} q^{23} + ( 4 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} + ( -2 + 2 \zeta_{12}^{2} ) q^{26} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} - q^{29} -10 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 2 q^{34} + ( -6 + 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + 8 \zeta_{12} q^{37} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{38} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{40} + 3 q^{41} -5 \zeta_{12}^{3} q^{43} + \zeta_{12}^{2} q^{46} + 8 \zeta_{12} q^{47} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} + ( 4 + 3 \zeta_{12}^{3} ) q^{50} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{52} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + ( -2 - \zeta_{12}^{2} ) q^{56} -\zeta_{12} q^{58} -2 \zeta_{12}^{2} q^{59} + ( 9 - 9 \zeta_{12}^{2} ) q^{61} -10 \zeta_{12}^{3} q^{62} - q^{64} + ( -4 + 2 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{65} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{67} + 2 \zeta_{12} q^{68} + ( -1 - 6 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70} -6 q^{71} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{73} + 8 \zeta_{12}^{2} q^{74} -2 q^{76} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{80} + 3 \zeta_{12} q^{82} -9 \zeta_{12}^{3} q^{83} + ( 4 - 2 \zeta_{12}^{3} ) q^{85} + ( 5 - 5 \zeta_{12}^{2} ) q^{86} + ( 7 - 7 \zeta_{12}^{2} ) q^{89} + ( -4 - 2 \zeta_{12}^{2} ) q^{91} + \zeta_{12}^{3} q^{92} + 8 \zeta_{12}^{2} q^{94} + ( -4 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{95} + ( -3 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{5} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{5} + 4q^{10} - 8q^{14} - 2q^{16} - 4q^{19} + 4q^{20} + 6q^{25} - 4q^{26} - 4q^{29} - 20q^{31} + 8q^{34} - 16q^{35} - 4q^{40} + 12q^{41} + 2q^{46} - 22q^{49} + 16q^{50} - 10q^{56} - 4q^{59} + 18q^{61} - 4q^{64} - 8q^{65} + 2q^{70} - 24q^{71} + 16q^{74} - 8q^{76} - 20q^{79} + 2q^{80} + 16q^{85} + 10q^{86} + 14q^{89} - 20q^{91} + 16q^{94} + 4q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −1.23205 + 1.86603i 0 0.866025 + 2.50000i 1.00000i 0 0.133975 2.23205i
109.2 0.866025 0.500000i 0 0.500000 0.866025i 2.23205 0.133975i 0 −0.866025 2.50000i 1.00000i 0 1.86603 1.23205i
289.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.23205 1.86603i 0 0.866025 2.50000i 1.00000i 0 0.133975 + 2.23205i
289.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.23205 + 0.133975i 0 −0.866025 + 2.50000i 1.00000i 0 1.86603 + 1.23205i
\(n\): 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.u.b 4
3.b odd 2 1 70.2.i.a 4
5.b even 2 1 inner 630.2.u.b 4
7.c even 3 1 inner 630.2.u.b 4
12.b even 2 1 560.2.bw.c 4
15.d odd 2 1 70.2.i.a 4
15.e even 4 1 350.2.e.f 2
15.e even 4 1 350.2.e.g 2
21.c even 2 1 490.2.i.b 4
21.g even 6 1 490.2.c.b 2
21.g even 6 1 490.2.i.b 4
21.h odd 6 1 70.2.i.a 4
21.h odd 6 1 490.2.c.c 2
35.j even 6 1 inner 630.2.u.b 4
60.h even 2 1 560.2.bw.c 4
84.n even 6 1 560.2.bw.c 4
105.g even 2 1 490.2.i.b 4
105.o odd 6 1 70.2.i.a 4
105.o odd 6 1 490.2.c.c 2
105.p even 6 1 490.2.c.b 2
105.p even 6 1 490.2.i.b 4
105.w odd 12 1 2450.2.a.c 1
105.w odd 12 1 2450.2.a.bh 1
105.x even 12 1 350.2.e.f 2
105.x even 12 1 350.2.e.g 2
105.x even 12 1 2450.2.a.r 1
105.x even 12 1 2450.2.a.s 1
420.ba even 6 1 560.2.bw.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a 4 3.b odd 2 1
70.2.i.a 4 15.d odd 2 1
70.2.i.a 4 21.h odd 6 1
70.2.i.a 4 105.o odd 6 1
350.2.e.f 2 15.e even 4 1
350.2.e.f 2 105.x even 12 1
350.2.e.g 2 15.e even 4 1
350.2.e.g 2 105.x even 12 1
490.2.c.b 2 21.g even 6 1
490.2.c.b 2 105.p even 6 1
490.2.c.c 2 21.h odd 6 1
490.2.c.c 2 105.o odd 6 1
490.2.i.b 4 21.c even 2 1
490.2.i.b 4 21.g even 6 1
490.2.i.b 4 105.g even 2 1
490.2.i.b 4 105.p even 6 1
560.2.bw.c 4 12.b even 2 1
560.2.bw.c 4 60.h even 2 1
560.2.bw.c 4 84.n even 6 1
560.2.bw.c 4 420.ba even 6 1
630.2.u.b 4 1.a even 1 1 trivial
630.2.u.b 4 5.b even 2 1 inner
630.2.u.b 4 7.c even 3 1 inner
630.2.u.b 4 35.j even 6 1 inner
2450.2.a.c 1 105.w odd 12 1
2450.2.a.r 1 105.x even 12 1
2450.2.a.s 1 105.x even 12 1
2450.2.a.bh 1 105.w odd 12 1

Hecke kernels

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

\( T_{11} \)
\( T_{37}^{4} - 64 T_{37}^{2} + 4096 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ 1
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \)
$79$ \( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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