# Properties

 Label 630.2.k.c Level 630 Weight 2 Character orbit 630.k Analytic conductor 5.031 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## 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.

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

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$1 + T + T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4}$$
$23$ $$1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{2}$$
$31$ $$1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 - 11 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4}$$
$53$ $$1 + 11 T + 68 T^{2} + 583 T^{3} + 2809 T^{4}$$
$59$ $$1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4}$$
$67$ $$1 - 67 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 8 T + 83 T^{2} )^{2}$$
$89$ $$1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 16 T + 97 T^{2} )^{2}$$
show more
show less