# 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 $2$

# 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: $$1$$ 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.

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

## 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
 0.5 + 0.866025i 0.5 − 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: 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.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

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$( 9 + T )^{2}$$
$43$ $$( 7 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$256 - 16 T + T^{2}$$
$79$ $$4 + 2 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$225 + 15 T + T^{2}$$
$97$ $$( -14 + T )^{2}$$