# Properties

 Label 630.2.g.c Level $630$ Weight $2$ Character orbit 630.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 −1.00000 + 2.00000i 0 1.00000i 1.00000i 0 2.00000 + 1.00000i
379.2 1.00000i 0 −1.00000 −1.00000 2.00000i 0 1.00000i 1.00000i 0 2.00000 1.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.g.c 2
3.b odd 2 1 210.2.g.b 2
4.b odd 2 1 5040.2.t.h 2
5.b even 2 1 inner 630.2.g.c 2
5.c odd 4 1 3150.2.a.d 1
5.c odd 4 1 3150.2.a.bk 1
12.b even 2 1 1680.2.t.e 2
15.d odd 2 1 210.2.g.b 2
15.e even 4 1 1050.2.a.d 1
15.e even 4 1 1050.2.a.p 1
20.d odd 2 1 5040.2.t.h 2
21.c even 2 1 1470.2.g.b 2
21.g even 6 2 1470.2.n.f 4
21.h odd 6 2 1470.2.n.b 4
60.h even 2 1 1680.2.t.e 2
60.l odd 4 1 8400.2.a.w 1
60.l odd 4 1 8400.2.a.bp 1
105.g even 2 1 1470.2.g.b 2
105.k odd 4 1 7350.2.a.bk 1
105.k odd 4 1 7350.2.a.bz 1
105.o odd 6 2 1470.2.n.b 4
105.p even 6 2 1470.2.n.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 3.b odd 2 1
210.2.g.b 2 15.d odd 2 1
630.2.g.c 2 1.a even 1 1 trivial
630.2.g.c 2 5.b even 2 1 inner
1050.2.a.d 1 15.e even 4 1
1050.2.a.p 1 15.e even 4 1
1470.2.g.b 2 21.c even 2 1
1470.2.g.b 2 105.g even 2 1
1470.2.n.b 4 21.h odd 6 2
1470.2.n.b 4 105.o odd 6 2
1470.2.n.f 4 21.g even 6 2
1470.2.n.f 4 105.p even 6 2
1680.2.t.e 2 12.b even 2 1
1680.2.t.e 2 60.h even 2 1
3150.2.a.d 1 5.c odd 4 1
3150.2.a.bk 1 5.c odd 4 1
5040.2.t.h 2 4.b odd 2 1
5040.2.t.h 2 20.d odd 2 1
7350.2.a.bk 1 105.k odd 4 1
7350.2.a.bz 1 105.k odd 4 1
8400.2.a.w 1 60.l odd 4 1
8400.2.a.bp 1 60.l odd 4 1

## Hecke kernels

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

 $$T_{11} + 2$$ $$T_{29} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( -6 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( -12 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$100 + T^{2}$$