# Properties

 Label 630.2.bk.a Level 630 Weight 2 Character orbit 630.bk Analytic conductor 5.031 Analytic rank 1 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.bk (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} - q^{4} -\zeta_{12}^{2} q^{5} + ( -2 + \zeta_{12}^{2} ) q^{6} + ( -2 - \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} - q^{4} -\zeta_{12}^{2} q^{5} + ( -2 + \zeta_{12}^{2} ) q^{6} + ( -2 - \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{10} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{11} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{12} + ( -2 - 2 \zeta_{12}^{2} ) q^{13} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{14} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} + q^{16} -3 \zeta_{12} q^{18} + ( 1 - 3 \zeta_{12} + \zeta_{12}^{2} ) q^{19} + \zeta_{12}^{2} q^{20} + ( -\zeta_{12} - 4 \zeta_{12}^{3} ) q^{21} + ( 3 + \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{22} + ( -4 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{23} + ( 2 - \zeta_{12}^{2} ) q^{24} + ( -1 + \zeta_{12}^{2} ) q^{25} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 2 + \zeta_{12}^{2} ) q^{28} + ( -4 - 3 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{29} + ( 1 + \zeta_{12}^{2} ) q^{30} + ( -3 + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{31} + \zeta_{12}^{3} q^{32} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{33} + ( -1 + 3 \zeta_{12}^{2} ) q^{35} + ( 3 - 3 \zeta_{12}^{2} ) q^{36} + ( -1 - 3 \zeta_{12} + \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{37} + ( 3 - \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{38} -6 \zeta_{12}^{3} q^{39} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{40} + ( 9 - 9 \zeta_{12}^{2} ) q^{41} + ( 5 - \zeta_{12}^{2} ) q^{42} + ( -3 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{43} + ( 1 + 3 \zeta_{12} + \zeta_{12}^{2} ) q^{44} + 3 q^{45} + ( -2 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{46} -9 q^{47} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{48} + ( 3 + 5 \zeta_{12}^{2} ) q^{49} -\zeta_{12} q^{50} + ( 2 + 2 \zeta_{12}^{2} ) q^{52} + ( -2 + 9 \zeta_{12} + \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{54} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{55} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{56} + ( 3 - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{57} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{58} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{60} + ( -4 + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{61} + ( -3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{62} + ( 9 - 6 \zeta_{12}^{2} ) q^{63} - q^{64} + ( -2 + 4 \zeta_{12}^{2} ) q^{65} + ( 3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{66} -4 q^{67} + ( 6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{69} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{70} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{71} + 3 \zeta_{12} q^{72} + ( 8 - 4 \zeta_{12}^{2} ) q^{73} + ( -3 - \zeta_{12} - 3 \zeta_{12}^{2} ) q^{74} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{75} + ( -1 + 3 \zeta_{12} - \zeta_{12}^{2} ) q^{76} + ( 1 + 6 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{77} + 6 q^{78} + ( -1 + 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{80} -9 \zeta_{12}^{2} q^{81} + 9 \zeta_{12} q^{82} + ( -3 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{83} + ( \zeta_{12} + 4 \zeta_{12}^{3} ) q^{84} + ( 6 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{86} + ( -3 - 6 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{87} + ( -3 - \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( -6 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{89} + 3 \zeta_{12}^{3} q^{90} + ( 2 + 8 \zeta_{12}^{2} ) q^{91} + ( 4 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{92} + ( -6 - 9 \zeta_{12} + 3 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{93} -9 \zeta_{12}^{3} q^{94} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{95} + ( -2 + \zeta_{12}^{2} ) q^{96} + ( 4 - 12 \zeta_{12} - 2 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{97} + ( -5 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{98} + ( 6 + 9 \zeta_{12} - 3 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 2q^{5} - 6q^{6} - 10q^{7} - 6q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 2q^{5} - 6q^{6} - 10q^{7} - 6q^{9} - 6q^{11} - 12q^{13} + 4q^{16} + 6q^{19} + 2q^{20} + 6q^{22} - 12q^{23} + 6q^{24} - 2q^{25} + 10q^{28} - 12q^{29} + 6q^{30} + 2q^{35} + 6q^{36} - 2q^{37} + 6q^{38} + 18q^{41} + 18q^{42} + 4q^{43} + 6q^{44} + 12q^{45} + 12q^{46} - 36q^{47} + 22q^{49} + 12q^{52} - 6q^{53} - 6q^{58} - 12q^{59} - 12q^{62} + 24q^{63} - 4q^{64} + 12q^{66} - 16q^{67} + 36q^{69} + 24q^{73} - 18q^{74} - 6q^{76} + 12q^{77} + 24q^{78} - 4q^{79} - 2q^{80} - 18q^{81} - 12q^{83} + 18q^{86} - 18q^{87} - 6q^{88} - 12q^{89} + 24q^{91} + 12q^{92} - 18q^{93} - 6q^{96} + 12q^{97} + 18q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
1.00000i 0.866025 1.50000i −1.00000 −0.500000 + 0.866025i −1.50000 0.866025i −2.50000 + 0.866025i 1.00000i −1.50000 2.59808i 0.866025 + 0.500000i
101.2 1.00000i −0.866025 + 1.50000i −1.00000 −0.500000 + 0.866025i −1.50000 0.866025i −2.50000 + 0.866025i 1.00000i −1.50000 2.59808i −0.866025 0.500000i
131.1 1.00000i −0.866025 1.50000i −1.00000 −0.500000 0.866025i −1.50000 + 0.866025i −2.50000 0.866025i 1.00000i −1.50000 + 2.59808i −0.866025 + 0.500000i
131.2 1.00000i 0.866025 + 1.50000i −1.00000 −0.500000 0.866025i −1.50000 + 0.866025i −2.50000 0.866025i 1.00000i −1.50000 + 2.59808i 0.866025 0.500000i
 $$n$$: 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
63.i 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.bk.a yes 4
3.b odd 2 1 1890.2.bk.a 4
7.d odd 6 1 630.2.t.a 4
9.c even 3 1 1890.2.t.a 4
9.d odd 6 1 630.2.t.a 4
21.g even 6 1 1890.2.t.a 4
63.i even 6 1 inner 630.2.bk.a yes 4
63.t odd 6 1 1890.2.bk.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.t.a 4 7.d odd 6 1
630.2.t.a 4 9.d odd 6 1
630.2.bk.a yes 4 1.a even 1 1 trivial
630.2.bk.a yes 4 63.i even 6 1 inner
1890.2.t.a 4 9.c even 3 1
1890.2.t.a 4 21.g even 6 1
1890.2.bk.a 4 3.b odd 2 1
1890.2.bk.a 4 63.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 6 T_{11}^{3} + 6 T_{11}^{2} - 36 T_{11} + 36$$ acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 1056 T^{5} + 3388 T^{6} + 7986 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 6 T + 25 T^{2} + 78 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$
$19$ $$1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 3648 T^{5} + 15884 T^{6} - 41154 T^{7} + 130321 T^{8}$$
$23$ $$1 + 12 T + 70 T^{2} + 264 T^{3} + 1059 T^{4} + 6072 T^{5} + 37030 T^{6} + 146004 T^{7} + 279841 T^{8}$$
$29$ $$1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 21228 T^{5} + 91669 T^{6} + 292668 T^{7} + 707281 T^{8}$$
$31$ $$1 - 52 T^{2} + 1626 T^{4} - 49972 T^{6} + 923521 T^{8}$$
$37$ $$1 + 2 T - 44 T^{2} - 52 T^{3} + 787 T^{4} - 1924 T^{5} - 60236 T^{6} + 101306 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 4 T - 47 T^{2} + 92 T^{3} + 1432 T^{4} + 3956 T^{5} - 86903 T^{6} - 318028 T^{7} + 3418801 T^{8}$$
$47$ $$( 1 + 9 T + 47 T^{2} )^{4}$$
$53$ $$1 + 6 T + 40 T^{2} + 168 T^{3} - 1389 T^{4} + 8904 T^{5} + 112360 T^{6} + 893262 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 + 6 T + 100 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$1 - 76 T^{2} + 1974 T^{4} - 282796 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{4}$$
$71$ $$1 - 260 T^{2} + 26874 T^{4} - 1310660 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 2 T - 84 T^{2} + 158 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 12 T - 31 T^{2} + 108 T^{3} + 11784 T^{4} + 8964 T^{5} - 213559 T^{6} + 6861444 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 12 T + 38 T^{2} - 864 T^{3} - 9501 T^{4} - 76896 T^{5} + 300998 T^{6} + 8459628 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 12 T + 110 T^{2} - 744 T^{3} - 909 T^{4} - 72168 T^{5} + 1034990 T^{6} - 10952076 T^{7} + 88529281 T^{8}$$