# Properties

 Label 630.2.k.b Level $630$ Weight $2$ Character orbit 630.k Analytic conductor $5.031$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10})$$ q - z * q^2 + (z - 1) * q^4 - z * q^5 + (-3*z + 1) * q^7 + q^8 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} + (\zeta_{6} - 1) q^{10} + (6 \zeta_{6} - 6) q^{11} - 4 q^{13} + (2 \zeta_{6} - 3) q^{14} - \zeta_{6} q^{16} - 2 \zeta_{6} q^{19} + q^{20} + 6 q^{22} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 4 \zeta_{6} q^{26} + (\zeta_{6} + 2) q^{28} + 3 q^{29} + (8 \zeta_{6} - 8) q^{31} + (\zeta_{6} - 1) q^{32} + (2 \zeta_{6} - 3) q^{35} + 4 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{38} - \zeta_{6} q^{40} - 9 q^{41} - 7 q^{43} - 6 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + (3 \zeta_{6} - 8) q^{49} + q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + (6 \zeta_{6} - 6) q^{53} + 6 q^{55} + ( - 3 \zeta_{6} + 1) q^{56} - 3 \zeta_{6} q^{58} + (6 \zeta_{6} - 6) q^{59} - 5 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 4 \zeta_{6} q^{65} + (5 \zeta_{6} - 5) q^{67} + (\zeta_{6} + 2) q^{70} + 6 q^{71} + ( - 16 \zeta_{6} + 16) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + 2 q^{76} + (6 \zeta_{6} + 12) q^{77} - 2 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + 9 \zeta_{6} q^{82} - 3 q^{83} + 7 \zeta_{6} q^{86} + (6 \zeta_{6} - 6) q^{88} - 15 \zeta_{6} q^{89} + (12 \zeta_{6} - 4) q^{91} + 3 q^{92} + (2 \zeta_{6} - 2) q^{95} + 14 q^{97} + (5 \zeta_{6} + 3) q^{98} +O(q^{100})$$ q - z * q^2 + (z - 1) * q^4 - z * q^5 + (-3*z + 1) * q^7 + q^8 + (z - 1) * q^10 + (6*z - 6) * q^11 - 4 * q^13 + (2*z - 3) * q^14 - z * q^16 - 2*z * q^19 + q^20 + 6 * q^22 - 3*z * q^23 + (z - 1) * q^25 + 4*z * q^26 + (z + 2) * q^28 + 3 * q^29 + (8*z - 8) * q^31 + (z - 1) * q^32 + (2*z - 3) * q^35 + 4*z * q^37 + (2*z - 2) * q^38 - z * q^40 - 9 * q^41 - 7 * q^43 - 6*z * q^44 + (3*z - 3) * q^46 + (3*z - 8) * q^49 + q^50 + (-4*z + 4) * q^52 + (6*z - 6) * q^53 + 6 * q^55 + (-3*z + 1) * q^56 - 3*z * q^58 + (6*z - 6) * q^59 - 5*z * q^61 + 8 * q^62 + q^64 + 4*z * q^65 + (5*z - 5) * q^67 + (z + 2) * q^70 + 6 * q^71 + (-16*z + 16) * q^73 + (-4*z + 4) * q^74 + 2 * q^76 + (6*z + 12) * q^77 - 2*z * q^79 + (z - 1) * q^80 + 9*z * q^82 - 3 * q^83 + 7*z * q^86 + (6*z - 6) * q^88 - 15*z * q^89 + (12*z - 4) * q^91 + 3 * q^92 + (2*z - 2) * q^95 + 14 * q^97 + (5*z + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - q^{5} - q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 - q^5 - q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} - q^{5} - q^{7} + 2 q^{8} - q^{10} - 6 q^{11} - 8 q^{13} - 4 q^{14} - q^{16} - 2 q^{19} + 2 q^{20} + 12 q^{22} - 3 q^{23} - q^{25} + 4 q^{26} + 5 q^{28} + 6 q^{29} - 8 q^{31} - q^{32} - 4 q^{35} + 4 q^{37} - 2 q^{38} - q^{40} - 18 q^{41} - 14 q^{43} - 6 q^{44} - 3 q^{46} - 13 q^{49} + 2 q^{50} + 4 q^{52} - 6 q^{53} + 12 q^{55} - q^{56} - 3 q^{58} - 6 q^{59} - 5 q^{61} + 16 q^{62} + 2 q^{64} + 4 q^{65} - 5 q^{67} + 5 q^{70} + 12 q^{71} + 16 q^{73} + 4 q^{74} + 4 q^{76} + 30 q^{77} - 2 q^{79} - q^{80} + 9 q^{82} - 6 q^{83} + 7 q^{86} - 6 q^{88} - 15 q^{89} + 4 q^{91} + 6 q^{92} - 2 q^{95} + 28 q^{97} + 11 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - q^5 - q^7 + 2 * q^8 - q^10 - 6 * q^11 - 8 * q^13 - 4 * q^14 - q^16 - 2 * q^19 + 2 * q^20 + 12 * q^22 - 3 * q^23 - q^25 + 4 * q^26 + 5 * q^28 + 6 * q^29 - 8 * q^31 - q^32 - 4 * q^35 + 4 * q^37 - 2 * q^38 - q^40 - 18 * q^41 - 14 * q^43 - 6 * q^44 - 3 * q^46 - 13 * q^49 + 2 * q^50 + 4 * q^52 - 6 * q^53 + 12 * q^55 - q^56 - 3 * q^58 - 6 * q^59 - 5 * q^61 + 16 * q^62 + 2 * q^64 + 4 * q^65 - 5 * q^67 + 5 * q^70 + 12 * q^71 + 16 * q^73 + 4 * q^74 + 4 * q^76 + 30 * q^77 - 2 * q^79 - q^80 + 9 * q^82 - 6 * q^83 + 7 * q^86 - 6 * q^88 - 15 * q^89 + 4 * q^91 + 6 * q^92 - 2 * q^95 + 28 * q^97 + 11 * q^98

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 2.59808i 1.00000 0 −0.500000 + 0.866025i
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 2.59808i 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.b 2
3.b odd 2 1 70.2.e.c 2
7.c even 3 1 inner 630.2.k.b 2
7.c even 3 1 4410.2.a.bm 1
7.d odd 6 1 4410.2.a.bd 1
12.b even 2 1 560.2.q.g 2
15.d odd 2 1 350.2.e.e 2
15.e even 4 2 350.2.j.b 4
21.c even 2 1 490.2.e.h 2
21.g even 6 1 490.2.a.b 1
21.g even 6 1 490.2.e.h 2
21.h odd 6 1 70.2.e.c 2
21.h odd 6 1 490.2.a.c 1
84.j odd 6 1 3920.2.a.bc 1
84.n even 6 1 560.2.q.g 2
84.n even 6 1 3920.2.a.p 1
105.o odd 6 1 350.2.e.e 2
105.o odd 6 1 2450.2.a.w 1
105.p even 6 1 2450.2.a.bc 1
105.w odd 12 2 2450.2.c.l 2
105.x even 12 2 350.2.j.b 4
105.x even 12 2 2450.2.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 3.b odd 2 1
70.2.e.c 2 21.h odd 6 1
350.2.e.e 2 15.d odd 2 1
350.2.e.e 2 105.o odd 6 1
350.2.j.b 4 15.e even 4 2
350.2.j.b 4 105.x even 12 2
490.2.a.b 1 21.g even 6 1
490.2.a.c 1 21.h odd 6 1
490.2.e.h 2 21.c even 2 1
490.2.e.h 2 21.g even 6 1
560.2.q.g 2 12.b even 2 1
560.2.q.g 2 84.n even 6 1
630.2.k.b 2 1.a even 1 1 trivial
630.2.k.b 2 7.c even 3 1 inner
2450.2.a.w 1 105.o odd 6 1
2450.2.a.bc 1 105.p even 6 1
2450.2.c.g 2 105.x even 12 2
2450.2.c.l 2 105.w odd 12 2
3920.2.a.p 1 84.n even 6 1
3920.2.a.bc 1 84.j odd 6 1
4410.2.a.bd 1 7.d odd 6 1
4410.2.a.bm 1 7.c even 3 1

## Hecke kernels

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

 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36 $$T_{13} + 4$$ T13 + 4 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$T^{2} - 4T + 16$$
$41$ $$(T + 9)^{2}$$
$43$ $$(T + 7)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 16T + 256$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$(T - 14)^{2}$$