# Properties

 Label 630.2.k.d 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: $$\Q(\sqrt{-3})$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + ( 1 - \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} - q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} - q^{20} -3 q^{22} + 9 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} -6 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 6 q^{34} + ( -2 - \zeta_{6} ) q^{35} + 7 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{38} + \zeta_{6} q^{40} -3 q^{41} + 2 q^{43} + 3 \zeta_{6} q^{44} + ( 9 - 9 \zeta_{6} ) q^{46} + 9 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + q^{50} + ( 1 - \zeta_{6} ) q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} + 3 q^{55} + ( -3 + 2 \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} -8 \zeta_{6} q^{61} + 8 q^{62} + q^{64} -\zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + ( -1 + 3 \zeta_{6} ) q^{70} + ( 4 - 4 \zeta_{6} ) q^{73} + ( 7 - 7 \zeta_{6} ) q^{74} - q^{76} + ( -3 + 9 \zeta_{6} ) q^{77} + 10 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + 3 \zeta_{6} q^{82} -6 q^{85} -2 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + ( 3 - 2 \zeta_{6} ) q^{91} -9 q^{92} + ( 9 - 9 \zeta_{6} ) q^{94} + ( -1 + \zeta_{6} ) q^{95} -10 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + q^{5} - 4q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + q^{5} - 4q^{7} + 2q^{8} + q^{10} + 3q^{11} - 2q^{13} + 5q^{14} - q^{16} - 6q^{17} + q^{19} - 2q^{20} - 6q^{22} + 9q^{23} - q^{25} + q^{26} - q^{28} - 12q^{29} - 8q^{31} - q^{32} + 12q^{34} - 5q^{35} + 7q^{37} + q^{38} + q^{40} - 6q^{41} + 4q^{43} + 3q^{44} + 9q^{46} + 9q^{47} + 2q^{49} + 2q^{50} + q^{52} + 9q^{53} + 6q^{55} - 4q^{56} + 6q^{58} - 8q^{61} + 16q^{62} + 2q^{64} - q^{65} - 8q^{67} - 6q^{68} + q^{70} + 4q^{73} + 7q^{74} - 2q^{76} + 3q^{77} + 10q^{79} + q^{80} + 3q^{82} - 12q^{85} - 2q^{86} + 3q^{88} + 6q^{89} + 4q^{91} - 18q^{92} + 9q^{94} - q^{95} - 20q^{97} - 13q^{98} + O(q^{100})$$

## 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.d 2
3.b odd 2 1 70.2.e.d 2
7.c even 3 1 inner 630.2.k.d 2
7.c even 3 1 4410.2.a.x 1
7.d odd 6 1 4410.2.a.bg 1
12.b even 2 1 560.2.q.b 2
15.d odd 2 1 350.2.e.b 2
15.e even 4 2 350.2.j.d 4
21.c even 2 1 490.2.e.g 2
21.g even 6 1 490.2.a.d 1
21.g even 6 1 490.2.e.g 2
21.h odd 6 1 70.2.e.d 2
21.h odd 6 1 490.2.a.a 1
84.j odd 6 1 3920.2.a.e 1
84.n even 6 1 560.2.q.b 2
84.n even 6 1 3920.2.a.bh 1
105.o odd 6 1 350.2.e.b 2
105.o odd 6 1 2450.2.a.bf 1
105.p even 6 1 2450.2.a.v 1
105.w odd 12 2 2450.2.c.e 2
105.x even 12 2 350.2.j.d 4
105.x even 12 2 2450.2.c.q 2

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

## Hecke kernels

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

 $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{13} + 1$$ $$T_{17}^{2} + 6 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$1 - T + T^{2}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 + T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 2 T + 43 T^{2} )^{2}$$
$47$ $$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$
$53$ $$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$