# Properties

 Label 630.2.u.a Level $630$ Weight $2$ Character orbit 630.u Analytic conductor $5.031$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 630.u (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + 3 \zeta_{12}^{2} q^{11} + 5 \zeta_{12}^{3} q^{13} + ( 3 - \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -5 + 5 \zeta_{12}^{2} ) q^{19} + ( -2 - \zeta_{12}^{3} ) q^{20} + 3 \zeta_{12}^{3} q^{22} + 7 \zeta_{12} q^{23} + ( 4 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} + ( -5 + 5 \zeta_{12}^{2} ) q^{26} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{28} -4 q^{29} + 2 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 2 q^{34} + ( -3 + 2 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{35} -\zeta_{12} q^{37} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{38} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{40} -3 q^{41} -2 \zeta_{12}^{3} q^{43} + ( -3 + 3 \zeta_{12}^{2} ) q^{44} + 7 \zeta_{12}^{2} q^{46} -7 \zeta_{12} q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( 4 - 3 \zeta_{12}^{3} ) q^{50} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{52} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{53} + ( -6 - 3 \zeta_{12}^{3} ) q^{55} + ( 1 + 2 \zeta_{12}^{2} ) q^{56} -4 \zeta_{12} q^{58} + 4 \zeta_{12}^{2} q^{59} + ( -6 + 6 \zeta_{12}^{2} ) q^{61} + 2 \zeta_{12}^{3} q^{62} - q^{64} + ( 5 - 10 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{65} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + 2 \zeta_{12} q^{68} + ( -4 - 3 \zeta_{12} + 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{70} + 6 q^{71} + ( 16 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{73} -\zeta_{12}^{2} q^{74} -5 q^{76} + ( 9 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{77} + ( 14 - 14 \zeta_{12}^{2} ) q^{79} + ( \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{80} -3 \zeta_{12} q^{82} -6 \zeta_{12}^{3} q^{83} + ( -2 + 4 \zeta_{12}^{3} ) q^{85} + ( 2 - 2 \zeta_{12}^{2} ) q^{86} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{88} + ( -2 + 2 \zeta_{12}^{2} ) q^{89} + ( 5 + 10 \zeta_{12}^{2} ) q^{91} + 7 \zeta_{12}^{3} q^{92} -7 \zeta_{12}^{2} q^{94} + ( 5 \zeta_{12} - 10 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{95} -12 \zeta_{12}^{3} q^{97} + ( 3 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 4 q^{5} + O(q^{10})$$ $$4 q + 2 q^{4} - 4 q^{5} - 2 q^{10} + 6 q^{11} + 10 q^{14} - 2 q^{16} - 10 q^{19} - 8 q^{20} - 6 q^{25} - 10 q^{26} - 16 q^{29} + 4 q^{31} + 8 q^{34} - 10 q^{35} + 2 q^{40} - 12 q^{41} - 6 q^{44} + 14 q^{46} - 4 q^{49} + 16 q^{50} - 24 q^{55} + 8 q^{56} + 8 q^{59} - 12 q^{61} - 4 q^{64} + 10 q^{65} - 4 q^{70} + 24 q^{71} - 2 q^{74} - 20 q^{76} + 28 q^{79} - 4 q^{80} - 8 q^{85} + 4 q^{86} - 4 q^{89} + 40 q^{91} - 14 q^{94} - 20 q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.133975 2.23205i 0 −1.73205 2.00000i 1.00000i 0 1.23205 + 1.86603i
109.2 0.866025 0.500000i 0 0.500000 0.866025i −1.86603 1.23205i 0 1.73205 + 2.00000i 1.00000i 0 −2.23205 0.133975i
289.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.133975 + 2.23205i 0 −1.73205 + 2.00000i 1.00000i 0 1.23205 1.86603i
289.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.86603 + 1.23205i 0 1.73205 2.00000i 1.00000i 0 −2.23205 + 0.133975i
 $$n$$: 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.u.a 4
3.b odd 2 1 70.2.i.b 4
5.b even 2 1 inner 630.2.u.a 4
7.c even 3 1 inner 630.2.u.a 4
12.b even 2 1 560.2.bw.d 4
15.d odd 2 1 70.2.i.b 4
15.e even 4 1 350.2.e.c 2
15.e even 4 1 350.2.e.j 2
21.c even 2 1 490.2.i.a 4
21.g even 6 1 490.2.c.d 2
21.g even 6 1 490.2.i.a 4
21.h odd 6 1 70.2.i.b 4
21.h odd 6 1 490.2.c.a 2
35.j even 6 1 inner 630.2.u.a 4
60.h even 2 1 560.2.bw.d 4
84.n even 6 1 560.2.bw.d 4
105.g even 2 1 490.2.i.a 4
105.o odd 6 1 70.2.i.b 4
105.o odd 6 1 490.2.c.a 2
105.p even 6 1 490.2.c.d 2
105.p even 6 1 490.2.i.a 4
105.w odd 12 1 2450.2.a.j 1
105.w odd 12 1 2450.2.a.bb 1
105.x even 12 1 350.2.e.c 2
105.x even 12 1 350.2.e.j 2
105.x even 12 1 2450.2.a.k 1
105.x even 12 1 2450.2.a.ba 1
420.ba even 6 1 560.2.bw.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 3.b odd 2 1
70.2.i.b 4 15.d odd 2 1
70.2.i.b 4 21.h odd 6 1
70.2.i.b 4 105.o odd 6 1
350.2.e.c 2 15.e even 4 1
350.2.e.c 2 105.x even 12 1
350.2.e.j 2 15.e even 4 1
350.2.e.j 2 105.x even 12 1
490.2.c.a 2 21.h odd 6 1
490.2.c.a 2 105.o odd 6 1
490.2.c.d 2 21.g even 6 1
490.2.c.d 2 105.p even 6 1
490.2.i.a 4 21.c even 2 1
490.2.i.a 4 21.g even 6 1
490.2.i.a 4 105.g even 2 1
490.2.i.a 4 105.p even 6 1
560.2.bw.d 4 12.b even 2 1
560.2.bw.d 4 60.h even 2 1
560.2.bw.d 4 84.n even 6 1
560.2.bw.d 4 420.ba even 6 1
630.2.u.a 4 1.a even 1 1 trivial
630.2.u.a 4 5.b even 2 1 inner
630.2.u.a 4 7.c even 3 1 inner
630.2.u.a 4 35.j even 6 1 inner
2450.2.a.j 1 105.w odd 12 1
2450.2.a.k 1 105.x even 12 1
2450.2.a.ba 1 105.x even 12 1
2450.2.a.bb 1 105.w odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{37}^{4} - T_{37}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 9 - 3 T + T^{2} )^{2}$$
$13$ $$( 25 + T^{2} )^{2}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 25 + 5 T + T^{2} )^{2}$$
$23$ $$2401 - 49 T^{2} + T^{4}$$
$29$ $$( 4 + T )^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$( 3 + T )^{4}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$2401 - 49 T^{2} + T^{4}$$
$53$ $$6561 - 81 T^{2} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$( 36 + 6 T + T^{2} )^{2}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$65536 - 256 T^{2} + T^{4}$$
$79$ $$( 196 - 14 T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 4 + 2 T + T^{2} )^{2}$$
$97$ $$( 144 + T^{2} )^{2}$$