# Properties

 Label 630.2.k.i Level 630 Weight 2 Character orbit 630.k 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.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 − 2.29129i −1.32288 + 2.29129i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −2.64575 1.00000 0 0.500000 0.866025i
361.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 2.64575 1.00000 0 0.500000 0.866025i
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −2.64575 1.00000 0 0.500000 + 0.866025i
541.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 2.64575 1.00000 0 0.500000 + 0.866025i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.k.i 4
3.b odd 2 1 630.2.k.j yes 4
7.c even 3 1 inner 630.2.k.i 4
7.c even 3 1 4410.2.a.bv 2
7.d odd 6 1 4410.2.a.ca 2
21.g even 6 1 4410.2.a.bo 2
21.h odd 6 1 630.2.k.j yes 4
21.h odd 6 1 4410.2.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.k.i 4 1.a even 1 1 trivial
630.2.k.i 4 7.c even 3 1 inner
630.2.k.j yes 4 3.b odd 2 1
630.2.k.j yes 4 21.h odd 6 1
4410.2.a.bo 2 21.g even 6 1
4410.2.a.bq 2 21.h odd 6 1
4410.2.a.bv 2 7.c even 3 1
4410.2.a.ca 2 7.d odd 6 1

## Hecke kernels

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

 $$T_{11}^{4} + 4 T_{11}^{3} + 19 T_{11}^{2} - 12 T_{11} + 9$$ $$T_{13}^{2} + 4 T_{13} - 3$$ $$T_{17}^{4} - 4 T_{17}^{3} + 40 T_{17}^{2} + 96 T_{17} + 576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ 1
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 + 4 T - 3 T^{2} - 12 T^{3} + 64 T^{4} - 132 T^{5} - 363 T^{6} + 5324 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 - 4 T + 6 T^{2} + 96 T^{3} - 461 T^{4} + 1632 T^{5} + 1734 T^{6} - 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 + 2 T - 7 T^{2} - 54 T^{3} - 316 T^{4} - 1026 T^{5} - 2527 T^{6} + 13718 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 4 T + T^{2} - 236 T^{3} - 1736 T^{4} - 8732 T^{5} + 1369 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 - 8 T + 91 T^{2} - 328 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 12 T + 94 T^{2} - 516 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 + 2 T + 21 T^{2} - 222 T^{3} - 2108 T^{4} - 10434 T^{5} + 46389 T^{6} + 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 10 T - 3 T^{2} - 30 T^{3} + 2500 T^{4} - 1590 T^{5} - 8427 T^{6} + 1488770 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 8 T - 42 T^{2} + 96 T^{3} + 3979 T^{4} + 5664 T^{5} - 146202 T^{6} - 1643032 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 12 T + 14 T^{2} + 96 T^{3} + 4395 T^{4} + 5856 T^{5} + 52094 T^{6} + 2723772 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 20 T + 194 T^{2} + 1440 T^{3} + 11147 T^{4} + 96480 T^{5} + 870866 T^{6} + 6015260 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 16 T + 178 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 + 12 T - 22 T^{2} + 96 T^{3} + 9939 T^{4} + 7584 T^{5} - 137302 T^{6} + 5916468 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 20 T + 238 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 68352 T^{5} - 142578 T^{6} - 5639752 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 4 T + 97 T^{2} )^{4}$$