# Properties

 Label 630.2.u.c 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 210) 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} -5 \zeta_{12}^{2} q^{11} + \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} + ( 7 - 7 \zeta_{12}^{2} ) q^{19} + ( 2 - \zeta_{12}^{3} ) q^{20} -5 \zeta_{12}^{3} q^{22} + 3 \zeta_{12} q^{23} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( -1 + \zeta_{12}^{2} ) q^{26} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{28} + 6 \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} -5 \zeta_{12} q^{37} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{38} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{40} + 9 q^{41} + 10 \zeta_{12}^{3} q^{43} + ( 5 - 5 \zeta_{12}^{2} ) q^{44} + 3 \zeta_{12}^{2} q^{46} + 13 \zeta_{12} q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -4 - 3 \zeta_{12}^{3} ) q^{50} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{52} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{53} + ( -10 + 5 \zeta_{12}^{3} ) q^{55} + ( 1 + 2 \zeta_{12}^{2} ) q^{56} -4 \zeta_{12}^{2} q^{59} + ( 2 - 2 \zeta_{12}^{2} ) q^{61} + 6 \zeta_{12}^{3} q^{62} - q^{64} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{65} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{67} -2 \zeta_{12} q^{68} + ( 4 - 3 \zeta_{12} - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{70} + 2 q^{71} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} -5 \zeta_{12}^{2} q^{74} + 7 q^{76} + ( -15 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{77} + ( -14 + 14 \zeta_{12}^{2} ) q^{79} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{80} + 9 \zeta_{12} q^{82} + 10 \zeta_{12}^{3} q^{83} + ( 2 + 4 \zeta_{12}^{3} ) q^{85} + ( -10 + 10 \zeta_{12}^{2} ) q^{86} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{88} + ( -10 + 10 \zeta_{12}^{2} ) q^{89} + ( 1 + 2 \zeta_{12}^{2} ) q^{91} + 3 \zeta_{12}^{3} q^{92} + 13 \zeta_{12}^{2} q^{94} + ( -7 \zeta_{12} - 14 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{95} -8 \zeta_{12}^{3} q^{97} + ( 3 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 4q^{5} + O(q^{10})$$ $$4q + 2q^{4} + 4q^{5} - 2q^{10} - 10q^{11} + 10q^{14} - 2q^{16} + 14q^{19} + 8q^{20} - 6q^{25} - 2q^{26} + 12q^{31} - 8q^{34} - 10q^{35} + 2q^{40} + 36q^{41} + 10q^{44} + 6q^{46} - 4q^{49} - 16q^{50} - 40q^{55} + 8q^{56} - 8q^{59} + 4q^{61} - 4q^{64} + 2q^{65} + 4q^{70} + 8q^{71} - 10q^{74} + 28q^{76} - 28q^{79} + 4q^{80} + 8q^{85} - 20q^{86} - 20q^{89} + 8q^{91} + 26q^{94} - 28q^{95} + O(q^{100})$$

## 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 1.86603 + 1.23205i 0 −1.73205 2.00000i 1.00000i 0 −2.23205 0.133975i
109.2 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
289.1 −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
289.2 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
 $$n$$: 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.c 4
3.b odd 2 1 210.2.n.a 4
5.b even 2 1 inner 630.2.u.c 4
7.c even 3 1 inner 630.2.u.c 4
12.b even 2 1 1680.2.di.a 4
15.d odd 2 1 210.2.n.a 4
15.e even 4 1 1050.2.i.f 2
15.e even 4 1 1050.2.i.o 2
21.c even 2 1 1470.2.n.i 4
21.g even 6 1 1470.2.g.a 2
21.g even 6 1 1470.2.n.i 4
21.h odd 6 1 210.2.n.a 4
21.h odd 6 1 1470.2.g.f 2
35.j even 6 1 inner 630.2.u.c 4
60.h even 2 1 1680.2.di.a 4
84.n even 6 1 1680.2.di.a 4
105.g even 2 1 1470.2.n.i 4
105.o odd 6 1 210.2.n.a 4
105.o odd 6 1 1470.2.g.f 2
105.p even 6 1 1470.2.g.a 2
105.p even 6 1 1470.2.n.i 4
105.w odd 12 1 7350.2.a.b 1
105.w odd 12 1 7350.2.a.ch 1
105.x even 12 1 1050.2.i.f 2
105.x even 12 1 1050.2.i.o 2
105.x even 12 1 7350.2.a.t 1
105.x even 12 1 7350.2.a.bn 1
420.ba even 6 1 1680.2.di.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 3.b odd 2 1
210.2.n.a 4 15.d odd 2 1
210.2.n.a 4 21.h odd 6 1
210.2.n.a 4 105.o odd 6 1
630.2.u.c 4 1.a even 1 1 trivial
630.2.u.c 4 5.b even 2 1 inner
630.2.u.c 4 7.c even 3 1 inner
630.2.u.c 4 35.j even 6 1 inner
1050.2.i.f 2 15.e even 4 1
1050.2.i.f 2 105.x even 12 1
1050.2.i.o 2 15.e even 4 1
1050.2.i.o 2 105.x even 12 1
1470.2.g.a 2 21.g even 6 1
1470.2.g.a 2 105.p even 6 1
1470.2.g.f 2 21.h odd 6 1
1470.2.g.f 2 105.o odd 6 1
1470.2.n.i 4 21.c even 2 1
1470.2.n.i 4 21.g even 6 1
1470.2.n.i 4 105.g even 2 1
1470.2.n.i 4 105.p even 6 1
1680.2.di.a 4 12.b even 2 1
1680.2.di.a 4 60.h even 2 1
1680.2.di.a 4 84.n even 6 1
1680.2.di.a 4 420.ba even 6 1
7350.2.a.b 1 105.w odd 12 1
7350.2.a.t 1 105.x even 12 1
7350.2.a.bn 1 105.x even 12 1
7350.2.a.ch 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{37}^{4} - 25 T_{37}^{2} + 625$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ 1
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 25 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 9 T + 41 T^{2} )^{4}$$
$43$ $$( 1 + 14 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 105 T^{2} + 8216 T^{4} + 294945 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 98 T^{2} + 5115 T^{4} + 439922 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{4}$$
$73$ $$1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 66 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )^{2}( 1 + 18 T + 97 T^{2} )^{2}$$