# Properties

 Label 630.2.g.g Level 630 Weight 2 Character orbit 630.g Analytic conductor 5.031 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## 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
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
1.00000i 0 −1.00000 −2.22474 0.224745i 0 1.00000i 1.00000i 0 −0.224745 + 2.22474i
379.2 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 1.00000i 1.00000i 0 2.22474 0.224745i
379.3 1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 1.00000i 1.00000i 0 −0.224745 2.22474i
379.4 1.00000i 0 −1.00000 0.224745 2.22474i 0 1.00000i 1.00000i 0 2.22474 + 0.224745i
 $$n$$: 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.g 4
3.b odd 2 1 70.2.c.a 4
4.b odd 2 1 5040.2.t.t 4
5.b even 2 1 inner 630.2.g.g 4
5.c odd 4 1 3150.2.a.bs 2
5.c odd 4 1 3150.2.a.bt 2
12.b even 2 1 560.2.g.e 4
15.d odd 2 1 70.2.c.a 4
15.e even 4 1 350.2.a.g 2
15.e even 4 1 350.2.a.h 2
20.d odd 2 1 5040.2.t.t 4
21.c even 2 1 490.2.c.e 4
21.g even 6 2 490.2.i.f 8
21.h odd 6 2 490.2.i.c 8
24.f even 2 1 2240.2.g.i 4
24.h odd 2 1 2240.2.g.j 4
60.h even 2 1 560.2.g.e 4
60.l odd 4 1 2800.2.a.bl 2
60.l odd 4 1 2800.2.a.bm 2
105.g even 2 1 490.2.c.e 4
105.k odd 4 1 2450.2.a.bl 2
105.k odd 4 1 2450.2.a.bq 2
105.o odd 6 2 490.2.i.c 8
105.p even 6 2 490.2.i.f 8
120.i odd 2 1 2240.2.g.j 4
120.m even 2 1 2240.2.g.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 3.b odd 2 1
70.2.c.a 4 15.d odd 2 1
350.2.a.g 2 15.e even 4 1
350.2.a.h 2 15.e even 4 1
490.2.c.e 4 21.c even 2 1
490.2.c.e 4 105.g even 2 1
490.2.i.c 8 21.h odd 6 2
490.2.i.c 8 105.o odd 6 2
490.2.i.f 8 21.g even 6 2
490.2.i.f 8 105.p even 6 2
560.2.g.e 4 12.b even 2 1
560.2.g.e 4 60.h even 2 1
630.2.g.g 4 1.a even 1 1 trivial
630.2.g.g 4 5.b even 2 1 inner
2240.2.g.i 4 24.f even 2 1
2240.2.g.i 4 120.m even 2 1
2240.2.g.j 4 24.h odd 2 1
2240.2.g.j 4 120.i odd 2 1
2450.2.a.bl 2 105.k odd 4 1
2450.2.a.bq 2 105.k odd 4 1
2800.2.a.bl 2 60.l odd 4 1
2800.2.a.bm 2 60.l odd 4 1
3150.2.a.bs 2 5.c odd 4 1
3150.2.a.bt 2 5.c odd 4 1
5040.2.t.t 4 4.b odd 2 1
5040.2.t.t 4 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} - 24$$ $$T_{29}^{2} - 4 T_{29} - 20$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ 1
$5$ $$1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$
$13$ $$1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 36 T^{2} + 998 T^{4} - 19044 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 12 T + 94 T^{2} - 492 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 108 T^{2} + 5798 T^{4} - 238572 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 92 T^{2} + 4278 T^{4} - 258428 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 70 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 12 T + 154 T^{2} - 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 236 T^{2} + 24198 T^{4} - 1257644 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 4 T + 138 T^{2} + 316 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 160 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{4}$$
$97$ $$1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8}$$