Embedded newform 630.2.k.g.541.1

• Label: 630.2.k.g.541.1
• Level: $630$
• Weight: $2$
• Character: 630.541
• Analytic conductor: $5.031$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

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:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	630.541
Dual form			630.2.k.g.361.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(-2.50000 - 4.33013i) q^{11} -5.00000 q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-2.00000 - 3.46410i) q^{17} +(3.50000 - 6.06218i) q^{19} -1.00000 q^{20} -5.00000 q^{22} +(0.500000 - 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.50000 + 4.33013i) q^{26} +(2.50000 + 0.866025i) q^{28} +(1.00000 + 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} -4.00000 q^{34} +(0.500000 + 2.59808i) q^{35} +(-0.500000 + 0.866025i) q^{37} +(-3.50000 - 6.06218i) q^{38} +(-0.500000 + 0.866025i) q^{40} -5.00000 q^{41} +12.0000 q^{43} +(-2.50000 + 4.33013i) q^{44} +(-0.500000 - 0.866025i) q^{46} +(-5.50000 + 9.52628i) q^{47} +(1.00000 - 6.92820i) q^{49} -1.00000 q^{50} +(2.50000 + 4.33013i) q^{52} +(-4.50000 - 7.79423i) q^{53} -5.00000 q^{55} +(2.00000 - 1.73205i) q^{56} +(2.00000 + 3.46410i) q^{59} +(-2.00000 + 3.46410i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(-2.50000 + 4.33013i) q^{65} +(6.00000 + 10.3923i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(2.50000 + 0.866025i) q^{70} -2.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(0.500000 + 0.866025i) q^{74} -7.00000 q^{76} +(12.5000 + 4.33013i) q^{77} +(6.00000 - 10.3923i) q^{79} +(0.500000 + 0.866025i) q^{80} +(-2.50000 + 4.33013i) q^{82} +12.0000 q^{83} -4.00000 q^{85} +(6.00000 - 10.3923i) q^{86} +(2.50000 + 4.33013i) q^{88} +(7.00000 - 12.1244i) q^{89} +(10.0000 - 8.66025i) q^{91} -1.00000 q^{92} +(5.50000 + 9.52628i) q^{94} +(-3.50000 - 6.06218i) q^{95} -8.00000 q^{97} +(-5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^4 + q^5 - 4 * q^7 - 2 * q^8 - q^10 - 5 * q^11 - 10 * q^13 + q^14 - q^16 - 4 * q^17 + 7 * q^19 - 2 * q^20 - 10 * q^22 + q^23 - q^25 - 5 * q^26 + 5 * q^28 + 2 * q^31 + q^32 - 8 * q^34 + q^35 - q^37 - 7 * q^38 - q^40 - 10 * q^41 + 24 * q^43 - 5 * q^44 - q^46 - 11 * q^47 + 2 * q^49 - 2 * q^50 + 5 * q^52 - 9 * q^53 - 10 * q^55 + 4 * q^56 + 4 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 - 5 * q^65 + 12 * q^67 - 4 * q^68 + 5 * q^70 - 4 * q^71 - 10 * q^73 + q^74 - 14 * q^76 + 25 * q^77 + 12 * q^79 + q^80 - 5 * q^82 + 24 * q^83 - 8 * q^85 + 12 * q^86 + 5 * q^88 + 14 * q^89 + 20 * q^91 - 2 * q^92 + 11 * q^94 - 7 * q^95 - 16 * q^97 - 11 * q^98

Character values

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

\(n\)	\(127\)	\(281\)	\(451\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	−	0.866025i	0.353553	−	0.612372i
							\(3\)		0			0
\(5\)	0.500000	−	0.866025i	0.223607	−	0.387298i
							\(7\)	−2.00000	+	1.73205i	−0.755929	+	0.654654i
\(11\)	−2.50000	−	4.33013i	−0.753778	−	1.30558i								−0.945979	−	0.324227i	\(-0.894896\pi\)
							0.192201	−	0.981356i	\(-0.438437\pi\)
\(13\)	−5.00000			−1.38675			−0.693375	−	0.720577i	\(-0.743877\pi\)
							−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	−2.00000	−	3.46410i	−0.485071	−	0.840168i	0.514782	−	0.857321i	\(-0.327873\pi\)
							−0.999853	+	0.0171533i	\(0.994540\pi\)
\(19\)	3.50000	−	6.06218i	0.802955	−	1.39076i	−0.114708	−	0.993399i	\(-0.536593\pi\)
							0.917663	−	0.397360i	\(-0.130073\pi\)
\(23\)	0.500000	−	0.866025i	0.104257	−	0.180579i	−0.809177	−	0.587565i	\(-0.800087\pi\)
							0.913434	+	0.406986i	\(0.133420\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	\(-0.109185\pi\)
							−0.762140	+	0.647412i	\(0.775851\pi\)
\(37\)	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	−0.904194	−	0.427121i	\(-0.859528\pi\)
							0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)	−5.00000			−0.780869			−0.390434	−	0.920631i	\(-0.627675\pi\)
							−0.390434	+	0.920631i	\(0.627675\pi\)
\(43\)	12.0000			1.82998			0.914991	−	0.403473i	\(-0.132197\pi\)
							0.914991	+	0.403473i	\(0.132197\pi\)
\(47\)	−5.50000	+	9.52628i	−0.802257	+	1.38955i	0.115870	+	0.993264i	\(0.463035\pi\)
							−0.918127	+	0.396286i	\(0.870299\pi\)
\(53\)	−4.50000	−	7.79423i	−0.618123	−	1.07062i	−0.989828	−	0.142269i	\(-0.954560\pi\)
							0.371706	−	0.928351i	\(-0.378773\pi\)
\(59\)	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	\(-0.0828195\pi\)
							−0.705965	+	0.708247i	\(0.749486\pi\)
\(61\)	−2.00000	+	3.46410i	−0.256074	+	0.443533i	−0.965187	−	0.261562i	\(-0.915762\pi\)
							0.709113	+	0.705095i	\(0.249096\pi\)
\(67\)	6.00000	+	10.3923i	0.733017	+	1.26962i	0.955588	+	0.294706i	\(0.0952216\pi\)
							−0.222571	+	0.974916i	\(0.571445\pi\)
\(71\)	−2.00000			−0.237356			−0.118678	−	0.992933i	\(-0.537866\pi\)
							−0.118678	+	0.992933i	\(0.537866\pi\)
\(73\)	−5.00000	−	8.66025i	−0.585206	−	1.01361i	−0.994850	−	0.101361i	\(-0.967680\pi\)
							0.409644	−	0.912245i	\(-0.365653\pi\)
\(79\)	6.00000	−	10.3923i	0.675053	−	1.16923i	−0.301401	−	0.953498i	\(-0.597454\pi\)
							0.976453	−	0.215728i	\(-0.0692125\pi\)
\(83\)	12.0000			1.31717			0.658586	−	0.752506i	\(-0.271155\pi\)
							0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	7.00000	−	12.1244i	0.741999	−	1.28518i	−0.209585	−	0.977790i	\(-0.567211\pi\)
							0.951584	−	0.307389i	\(-0.0994552\pi\)
\(97\)	−8.00000			−0.812277			−0.406138	−	0.913812i	\(-0.633125\pi\)
							−0.406138	+	0.913812i	\(0.633125\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.2.k.g.541.1		2
3.2	odd	2		210.2.i.b.121.1	✓	2
7.2	even	3		4410.2.a.j.1.1		1
7.4	even	3	inner	630.2.k.g.361.1		2
7.5	odd	6		4410.2.a.u.1.1		1
12.11	even	2		1680.2.bg.d.961.1		2
15.2	even	4		1050.2.o.g.499.1		4
15.8	even	4		1050.2.o.g.499.2		4
15.14	odd	2		1050.2.i.p.751.1		2
21.2	odd	6		1470.2.a.l.1.1		1
21.5	even	6		1470.2.a.o.1.1		1
21.11	odd	6		210.2.i.b.151.1	yes	2
21.17	even	6		1470.2.i.e.361.1		2
21.20	even	2		1470.2.i.e.961.1		2
84.11	even	6		1680.2.bg.d.1201.1		2
105.32	even	12		1050.2.o.g.949.2		4
105.44	odd	6		7350.2.a.u.1.1		1
105.53	even	12		1050.2.o.g.949.1		4
105.74	odd	6		1050.2.i.p.151.1		2
105.89	even	6		7350.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.b.121.1	✓	2	3.2	odd	2
210.2.i.b.151.1	yes	2	21.11	odd	6
630.2.k.g.361.1		2	7.4	even	3	inner
630.2.k.g.541.1		2	1.1	even	1	trivial
1050.2.i.p.151.1		2	105.74	odd	6
1050.2.i.p.751.1		2	15.14	odd	2
1050.2.o.g.499.1		4	15.2	even	4
1050.2.o.g.499.2		4	15.8	even	4
1050.2.o.g.949.1		4	105.53	even	12
1050.2.o.g.949.2		4	105.32	even	12
1470.2.a.l.1.1		1	21.2	odd	6
1470.2.a.o.1.1		1	21.5	even	6
1470.2.i.e.361.1		2	21.17	even	6
1470.2.i.e.961.1		2	21.20	even	2
1680.2.bg.d.961.1		2	12.11	even	2
1680.2.bg.d.1201.1		2	84.11	even	6
4410.2.a.j.1.1		1	7.2	even	3
4410.2.a.u.1.1		1	7.5	odd	6
7350.2.a.a.1.1		1	105.89	even	6
7350.2.a.u.1.1		1	105.44	odd	6