# 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

# Related objects

## 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.

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

## 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
 0.5 + 0.866025i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ 1
$5$ $$1 + T + T^{2}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 10 T + 43 T^{2} )^{2}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 14 T + 97 T^{2} )^{2}$$