# Properties

 Label 630.2.g.a 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: $$\Q(\sqrt{-1})$$ 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.

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

## 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 −2.00000 + 1.00000i 0 1.00000i 1.00000i 0 1.00000 + 2.00000i
379.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000 2.00000i
 $$n$$: 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.a 2
3.b odd 2 1 630.2.g.f yes 2
4.b odd 2 1 5040.2.t.b 2
5.b even 2 1 inner 630.2.g.a 2
5.c odd 4 1 3150.2.a.e 1
5.c odd 4 1 3150.2.a.bn 1
12.b even 2 1 5040.2.t.q 2
15.d odd 2 1 630.2.g.f yes 2
15.e even 4 1 3150.2.a.p 1
15.e even 4 1 3150.2.a.z 1
20.d odd 2 1 5040.2.t.b 2
60.h even 2 1 5040.2.t.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.g.a 2 1.a even 1 1 trivial
630.2.g.a 2 5.b even 2 1 inner
630.2.g.f yes 2 3.b odd 2 1
630.2.g.f yes 2 15.d odd 2 1
3150.2.a.e 1 5.c odd 4 1
3150.2.a.p 1 15.e even 4 1
3150.2.a.z 1 15.e even 4 1
3150.2.a.bn 1 5.c odd 4 1
5040.2.t.b 2 4.b odd 2 1
5040.2.t.b 2 20.d odd 2 1
5040.2.t.q 2 12.b even 2 1
5040.2.t.q 2 60.h even 2 1

## Hecke kernels

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

 $$T_{11}$$ $$T_{29} + 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ $$1 + 4 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 8 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 12 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 - 78 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 8 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{2}$$
$67$ $$1 - 70 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 130 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{2}$$
$89$ $$( 1 - 4 T + 89 T^{2} )^{2}$$
$97$ $$1 - 50 T^{2} + 9409 T^{4}$$