# Properties

 Label 630.2.be.b Level 630 Weight 2 Character orbit 630.be Analytic conductor 5.031 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 - \zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0.500000 0.866025i 0 −2.63896 0.189469i 1.00000i 0 −0.866025 + 0.500000i
341.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.500000 0.866025i 0 2.63896 + 0.189469i 1.00000i 0 −0.866025 + 0.500000i
341.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.500000 0.866025i 0 −0.189469 + 2.63896i 1.00000i 0 0.866025 0.500000i
341.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.500000 0.866025i 0 0.189469 2.63896i 1.00000i 0 0.866025 0.500000i
521.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.500000 + 0.866025i 0 −2.63896 + 0.189469i 1.00000i 0 −0.866025 0.500000i
521.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.500000 + 0.866025i 0 2.63896 0.189469i 1.00000i 0 −0.866025 0.500000i
521.3 0.866025 0.500000i 0 0.500000 0.866025i 0.500000 + 0.866025i 0 −0.189469 2.63896i 1.00000i 0 0.866025 + 0.500000i
521.4 0.866025 0.500000i 0 0.500000 0.866025i 0.500000 + 0.866025i 0 0.189469 + 2.63896i 1.00000i 0 0.866025 + 0.500000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 521.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g 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.be.b yes 8
3.b odd 2 1 630.2.be.a 8
5.b even 2 1 3150.2.bf.c 8
5.c odd 4 1 3150.2.bp.c 8
5.c odd 4 1 3150.2.bp.f 8
7.c even 3 1 4410.2.b.b 8
7.d odd 6 1 630.2.be.a 8
7.d odd 6 1 4410.2.b.e 8
15.d odd 2 1 3150.2.bf.b 8
15.e even 4 1 3150.2.bp.a 8
15.e even 4 1 3150.2.bp.d 8
21.g even 6 1 inner 630.2.be.b yes 8
21.g even 6 1 4410.2.b.b 8
21.h odd 6 1 4410.2.b.e 8
35.i odd 6 1 3150.2.bf.b 8
35.k even 12 1 3150.2.bp.a 8
35.k even 12 1 3150.2.bp.d 8
105.p even 6 1 3150.2.bf.c 8
105.w odd 12 1 3150.2.bp.c 8
105.w odd 12 1 3150.2.bp.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.be.a 8 3.b odd 2 1
630.2.be.a 8 7.d odd 6 1
630.2.be.b yes 8 1.a even 1 1 trivial
630.2.be.b yes 8 21.g even 6 1 inner
3150.2.bf.b 8 15.d odd 2 1
3150.2.bf.b 8 35.i odd 6 1
3150.2.bf.c 8 5.b even 2 1
3150.2.bf.c 8 105.p even 6 1
3150.2.bp.a 8 15.e even 4 1
3150.2.bp.a 8 35.k even 12 1
3150.2.bp.c 8 5.c odd 4 1
3150.2.bp.c 8 105.w odd 12 1
3150.2.bp.d 8 15.e even 4 1
3150.2.bp.d 8 35.k even 12 1
3150.2.bp.f 8 5.c odd 4 1
3150.2.bp.f 8 105.w odd 12 1
4410.2.b.b 8 7.c even 3 1
4410.2.b.b 8 21.g even 6 1
4410.2.b.e 8 7.d odd 6 1
4410.2.b.e 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ 1
$5$ $$( 1 - T + T^{2} )^{4}$$
$7$ $$1 - 94 T^{4} + 2401 T^{8}$$
$11$ $$1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 22396176 T^{9} + 60114736 T^{10} - 138732792 T^{11} + 270580321 T^{12} - 432905088 T^{13} + 538554544 T^{14} - 467692104 T^{15} + 214358881 T^{16}$$
$13$ $$1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16}$$
$17$ $$1 - 28 T^{2} + 192 T^{3} + 442 T^{4} - 4320 T^{5} + 15824 T^{6} + 58560 T^{7} - 349805 T^{8} + 995520 T^{9} + 4573136 T^{10} - 21224160 T^{11} + 36916282 T^{12} + 272612544 T^{13} - 675851932 T^{14} + 6975757441 T^{16}$$
$19$ $$1 + 58 T^{2} + 1881 T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 1352496 T^{9} + 16211066 T^{10} - 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16}$$
$23$ $$( 1 - 6 T + 57 T^{2} - 270 T^{3} + 1772 T^{4} - 6210 T^{5} + 30153 T^{6} - 73002 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16}$$
$37$ $$1 - 24 T + 216 T^{2} - 1680 T^{3} + 21025 T^{4} - 178920 T^{5} + 986040 T^{6} - 7528032 T^{7} + 58723920 T^{8} - 278537184 T^{9} + 1349888760 T^{10} - 9062834760 T^{11} + 39404235025 T^{12} - 116497847760 T^{13} + 554196904344 T^{14} - 2278365051192 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 43000 T^{5} + 332820 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 8 T - 78 T^{2} - 240 T^{3} + 6305 T^{4} - 3648 T^{5} - 344398 T^{6} - 95176 T^{7} + 11426244 T^{8} - 4473272 T^{9} - 760775182 T^{10} - 378746304 T^{11} + 30766388705 T^{12} - 55042801680 T^{13} - 840778795662 T^{14} + 4052984963704 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 24 T + 362 T^{2} - 4080 T^{3} + 37881 T^{4} - 336960 T^{5} + 2978650 T^{6} - 25456152 T^{7} + 197950532 T^{8} - 1349176056 T^{9} + 8367027850 T^{10} - 50165593920 T^{11} + 298899310761 T^{12} - 1706237611440 T^{13} + 8023498728698 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 24 T + 224 T^{2} + 1392 T^{3} + 9358 T^{4} + 16056 T^{5} - 613888 T^{6} - 6806952 T^{7} - 48734957 T^{8} - 401610168 T^{9} - 2136944128 T^{10} + 3297565224 T^{11} + 113394264238 T^{12} + 995174624208 T^{13} + 9448439535584 T^{14} + 59727635635656 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 + 24 T + 332 T^{2} + 3696 T^{3} + 30922 T^{4} + 206568 T^{5} + 963056 T^{6} + 1103736 T^{7} - 11820413 T^{8} + 73950312 T^{9} + 4323158384 T^{10} + 62128011384 T^{11} + 623112963562 T^{12} + 4990062395472 T^{13} + 30032182880108 T^{14} + 145457078527752 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 1227034224 T^{10} + 346515336682 T^{12} + 23608139301084 T^{14} + 806460091894081 T^{16}$$
$79$ $$1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 1485339672 T^{9} - 3383620560 T^{10} + 8791871448 T^{11} + 196697909050 T^{12} + 4874057336016 T^{13} + 61258038791292 T^{14} + 460893815667816 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 + 8 T + 148 T^{2} + 1192 T^{3} + 18438 T^{4} + 98936 T^{5} + 1019572 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 + 16 T - 48 T^{2} - 1824 T^{3} + 2078 T^{4} + 135504 T^{5} + 38912 T^{6} - 7970384 T^{7} - 72309597 T^{8} - 709364176 T^{9} + 308221952 T^{10} + 95526119376 T^{11} + 130378376798 T^{12} - 10185324434976 T^{13} - 23855101966128 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16}$$
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