# Properties

 Label 630.2.g.e 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: yes 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.

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

## 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.e yes 2
3.b odd 2 1 630.2.g.b 2
4.b odd 2 1 5040.2.t.j 2
5.b even 2 1 inner 630.2.g.e yes 2
5.c odd 4 1 3150.2.a.k 1
5.c odd 4 1 3150.2.a.br 1
12.b even 2 1 5040.2.t.i 2
15.d odd 2 1 630.2.g.b 2
15.e even 4 1 3150.2.a.l 1
15.e even 4 1 3150.2.a.v 1
20.d odd 2 1 5040.2.t.j 2
60.h even 2 1 5040.2.t.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.g.b 2 3.b odd 2 1
630.2.g.b 2 15.d odd 2 1
630.2.g.e yes 2 1.a even 1 1 trivial
630.2.g.e yes 2 5.b even 2 1 inner
3150.2.a.k 1 5.c odd 4 1
3150.2.a.l 1 15.e even 4 1
3150.2.a.v 1 15.e even 4 1
3150.2.a.br 1 5.c odd 4 1
5040.2.t.i 2 12.b even 2 1
5040.2.t.i 2 60.h even 2 1
5040.2.t.j 2 4.b odd 2 1
5040.2.t.j 2 20.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$4 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 12 + T )^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$324 + T^{2}$$