# Properties

 Label 630.2.k.h 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 210) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{10} + ( -1 + \zeta_{6} ) q^{11} + 7 q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} - q^{20} - q^{22} + \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 7 \zeta_{6} q^{26} + ( -1 + 3 \zeta_{6} ) q^{28} + 8 q^{29} + ( -6 + 6 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -4 q^{34} + ( 2 + \zeta_{6} ) q^{35} + 3 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{38} -\zeta_{6} q^{40} -9 q^{41} -4 q^{43} -\zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{46} -3 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} - q^{50} + ( -7 + 7 \zeta_{6} ) q^{52} + ( -1 + \zeta_{6} ) q^{53} - q^{55} + ( -3 + 2 \zeta_{6} ) q^{56} + 8 \zeta_{6} q^{58} + ( 12 - 12 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} -6 q^{62} + q^{64} + 7 \zeta_{6} q^{65} + ( -12 + 12 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} + ( -1 + 3 \zeta_{6} ) q^{70} + 14 q^{71} + ( 14 - 14 \zeta_{6} ) q^{73} + ( -3 + 3 \zeta_{6} ) q^{74} + q^{76} + ( -1 + 3 \zeta_{6} ) q^{77} -4 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} -9 \zeta_{6} q^{82} -12 q^{83} -4 q^{85} -4 \zeta_{6} q^{86} + ( 1 - \zeta_{6} ) q^{88} -2 \zeta_{6} q^{89} + ( 21 - 14 \zeta_{6} ) q^{91} - q^{92} + ( 3 - 3 \zeta_{6} ) q^{94} + ( 1 - \zeta_{6} ) q^{95} -16 q^{97} + ( 8 - 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + q^{5} + 4q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + q^{5} + 4q^{7} - 2q^{8} - q^{10} - q^{11} + 14q^{13} + 5q^{14} - q^{16} - 4q^{17} - q^{19} - 2q^{20} - 2q^{22} + q^{23} - q^{25} + 7q^{26} + q^{28} + 16q^{29} - 6q^{31} + q^{32} - 8q^{34} + 5q^{35} + 3q^{37} + q^{38} - q^{40} - 18q^{41} - 8q^{43} - q^{44} - q^{46} - 3q^{47} + 2q^{49} - 2q^{50} - 7q^{52} - q^{53} - 2q^{55} - 4q^{56} + 8q^{58} + 12q^{59} + 4q^{61} - 12q^{62} + 2q^{64} + 7q^{65} - 12q^{67} - 4q^{68} + q^{70} + 28q^{71} + 14q^{73} - 3q^{74} + 2q^{76} + q^{77} - 4q^{79} + q^{80} - 9q^{82} - 24q^{83} - 8q^{85} - 4q^{86} + q^{88} - 2q^{89} + 28q^{91} - 2q^{92} + 3q^{94} + q^{95} - 32q^{97} + 13q^{98} + O(q^{100})$$

## 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.h 2
3.b odd 2 1 210.2.i.a 2
7.c even 3 1 inner 630.2.k.h 2
7.c even 3 1 4410.2.a.g 1
7.d odd 6 1 4410.2.a.q 1
12.b even 2 1 1680.2.bg.k 2
15.d odd 2 1 1050.2.i.s 2
15.e even 4 2 1050.2.o.j 4
21.c even 2 1 1470.2.i.i 2
21.g even 6 1 1470.2.a.k 1
21.g even 6 1 1470.2.i.i 2
21.h odd 6 1 210.2.i.a 2
21.h odd 6 1 1470.2.a.r 1
84.n even 6 1 1680.2.bg.k 2
105.o odd 6 1 1050.2.i.s 2
105.o odd 6 1 7350.2.a.j 1
105.p even 6 1 7350.2.a.ba 1
105.x even 12 2 1050.2.o.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 3.b odd 2 1
210.2.i.a 2 21.h odd 6 1
630.2.k.h 2 1.a even 1 1 trivial
630.2.k.h 2 7.c even 3 1 inner
1050.2.i.s 2 15.d odd 2 1
1050.2.i.s 2 105.o odd 6 1
1050.2.o.j 4 15.e even 4 2
1050.2.o.j 4 105.x even 12 2
1470.2.a.k 1 21.g even 6 1
1470.2.a.r 1 21.h odd 6 1
1470.2.i.i 2 21.c even 2 1
1470.2.i.i 2 21.g even 6 1
1680.2.bg.k 2 12.b even 2 1
1680.2.bg.k 2 84.n even 6 1
4410.2.a.g 1 7.c even 3 1
4410.2.a.q 1 7.d odd 6 1
7350.2.a.j 1 105.o odd 6 1
7350.2.a.ba 1 105.p even 6 1

## Hecke kernels

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

 $$T_{11}^{2} + T_{11} + 1$$ $$T_{13} - 7$$ $$T_{17}^{2} + 4 T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ 1
$5$ $$1 - T + T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{2}$$
$31$ $$1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4}$$
$59$ $$1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 14 T + 71 T^{2} )^{2}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} )$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 16 T + 97 T^{2} )^{2}$$