Embedded newform 630.2.be.a.521.4

• Label: 630.2.be.a.521.4
• Level: $630$
• Weight: $2$
• Character: 630.521
• Analytic conductor: $5.031$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

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:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.4
Root			\(-0.965926 - 0.258819i\) of defining polynomial
Character	\(\chi\)	\(=\)	630.521
Dual form			630.2.be.a.341.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(2.63896 - 0.189469i) q^{7} -1.00000i q^{8} +(-0.866025 - 0.500000i) q^{10} +(-1.32697 - 0.766125i) q^{11} -1.48236i q^{13} +(2.19067 - 1.48356i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.21441 - 2.10342i) q^{17} +(4.21209 - 2.43185i) q^{19} -1.00000 q^{20} -1.53225 q^{22} +(0.232051 - 0.133975i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.741181 - 1.28376i) q^{26} +(1.15539 - 2.38014i) q^{28} +0.898979i q^{29} +(-0.717439 - 0.414214i) q^{31} +(-0.866025 - 0.500000i) q^{32} -2.42883i q^{34} +(-1.48356 - 2.19067i) q^{35} +(2.74118 + 4.74786i) q^{37} +(2.43185 - 4.21209i) q^{38} +(-0.866025 + 0.500000i) q^{40} -8.76028 q^{41} +1.86370 q^{43} +(-1.32697 + 0.766125i) q^{44} +(0.133975 - 0.232051i) q^{46} +(-3.72973 - 6.46008i) q^{47} +(6.92820 - 1.00000i) q^{49} +1.00000i q^{50} +(-1.28376 - 0.741181i) q^{52} +(3.00524 + 1.73508i) q^{53} +1.53225i q^{55} +(-0.189469 - 2.63896i) q^{56} +(0.449490 + 0.778539i) q^{58} +(3.12837 - 5.41849i) q^{59} +(5.73445 - 3.31079i) q^{61} -0.828427 q^{62} -1.00000 q^{64} +(-1.28376 + 0.741181i) q^{65} +(-8.01702 + 13.8859i) q^{67} +(-1.21441 - 2.10342i) q^{68} +(-2.38014 - 1.15539i) q^{70} +12.7627i q^{71} +(0.297173 + 0.171573i) q^{73} +(4.74786 + 2.74118i) q^{74} -4.86370i q^{76} +(-3.64697 - 1.77035i) q^{77} +(5.22438 + 9.04889i) q^{79} +(-0.500000 + 0.866025i) q^{80} +(-7.58662 + 4.38014i) q^{82} -5.45001 q^{83} -2.42883 q^{85} +(1.61401 - 0.931852i) q^{86} +(-0.766125 + 1.32697i) q^{88} +(7.98502 + 13.8305i) q^{89} +(-0.280861 - 3.91189i) q^{91} -0.267949i q^{92} +(-6.46008 - 3.72973i) q^{94} +(-4.21209 - 2.43185i) q^{95} +14.9481i q^{97} +(5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 4 * q^4 - 4 * q^5 - 24 * q^11 - 4 * q^16 - 8 * q^20 - 12 * q^23 - 4 * q^25 - 8 * q^26 + 24 * q^37 + 4 * q^38 - 32 * q^41 - 16 * q^43 - 24 * q^44 + 8 * q^46 + 8 * q^47 - 24 * q^53 - 16 * q^58 + 24 * q^59 + 16 * q^62 - 8 * q^64 - 24 * q^67 + 16 * q^77 - 24 * q^79 - 4 * q^80 + 16 * q^83 + 16 * q^89 - 20 * q^91 - 12 * q^94 + 44 * q^98

Character values

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

\(n\)	\(127\)	\(281\)	\(451\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.866025	−	0.500000i	0.612372	−	0.353553i
							\(3\)		0			0
\(5\)	−0.500000	−	0.866025i	−0.223607	−	0.387298i
							\(7\)	2.63896	−	0.189469i	0.997433	−	0.0716124i
\(11\)	−1.32697	−	0.766125i	−0.400096	−	0.230995i								0.286430	−	0.958101i	\(-0.407532\pi\)
							−0.686525	+	0.727106i	\(0.740865\pi\)
\(13\)		−	1.48236i		−	0.411133i	−0.978643	−	0.205567i	\(-0.934096\pi\)
							0.978643	−	0.205567i	\(-0.0659037\pi\)
\(17\)	1.21441	−	2.10342i	0.294538	−	0.510155i	−0.680339	−	0.732898i	\(-0.738167\pi\)
							0.974877	+	0.222742i	\(0.0715008\pi\)
\(19\)	4.21209	−	2.43185i	0.966320	−	0.557905i	0.0682075	−	0.997671i	\(-0.478272\pi\)
							0.898112	+	0.439766i	\(0.144939\pi\)
\(23\)	0.232051	−	0.133975i	0.0483859	−	0.0279356i	−0.475612	−	0.879655i	\(-0.657773\pi\)
							0.523998	+	0.851720i	\(0.324440\pi\)
\(29\)			0.898979i			0.166936i	0.996510	+	0.0834681i	\(0.0265997\pi\)
							−0.996510	+	0.0834681i	\(0.973400\pi\)
\(31\)	−0.717439	−	0.414214i	−0.128856	−	0.0743950i	0.434187	−	0.900823i	\(-0.357036\pi\)
							−0.563042	+	0.826428i	\(0.690369\pi\)
\(37\)	2.74118	+	4.74786i	0.450647	+	0.780544i	0.998426	−	0.0560790i	\(-0.0178599\pi\)
							−0.547779	+	0.836623i	\(0.684527\pi\)
\(41\)	−8.76028			−1.36813			−0.684063	−	0.729423i	\(-0.739789\pi\)
							−0.684063	+	0.729423i	\(0.739789\pi\)
\(43\)	1.86370			0.284212			0.142106	−	0.989851i	\(-0.454613\pi\)
							0.142106	+	0.989851i	\(0.454613\pi\)
\(47\)	−3.72973	−	6.46008i	−0.544037	−	0.942299i	−0.998667	−	0.0516191i	\(-0.983562\pi\)
							0.454630	−	0.890680i	\(-0.349771\pi\)
\(53\)	3.00524	+	1.73508i	0.412802	+	0.238331i	0.691993	−	0.721904i	\(-0.256733\pi\)
							−0.279191	+	0.960236i	\(0.590066\pi\)
\(59\)	3.12837	−	5.41849i	0.407279	−	0.705428i	−0.587305	−	0.809366i	\(-0.699811\pi\)
							0.994584	+	0.103938i	\(0.0331444\pi\)
\(61\)	5.73445	−	3.31079i	0.734222	−	0.423903i	−0.0857429	−	0.996317i	\(-0.527326\pi\)
							0.819965	+	0.572414i	\(0.193993\pi\)
\(67\)	−8.01702	+	13.8859i	−0.979434	+	1.69643i	−0.314985	+	0.949097i	\(0.602000\pi\)
							−0.664449	+	0.747334i	\(0.731334\pi\)
\(71\)			12.7627i			1.51465i	0.653037	+	0.757326i	\(0.273495\pi\)
							−0.653037	+	0.757326i	\(0.726505\pi\)
\(73\)	0.297173	+	0.171573i	0.0347815	+	0.0200811i	0.517290	−	0.855810i	\(-0.326941\pi\)
							−0.482508	+	0.875891i	\(0.660274\pi\)
\(79\)	5.22438	+	9.04889i	0.587789	+	1.01808i	0.994521	+	0.104533i	\(0.0333347\pi\)
							−0.406733	+	0.913547i	\(0.633332\pi\)
\(83\)	−5.45001			−0.598216			−0.299108	−	0.954219i	\(-0.596689\pi\)
							−0.299108	+	0.954219i	\(0.596689\pi\)
\(89\)	7.98502	+	13.8305i	0.846411	+	1.46603i	0.884390	+	0.466748i	\(0.154574\pi\)
							−0.0379795	+	0.999279i	\(0.512092\pi\)
\(97\)			14.9481i			1.51775i	0.651234	+	0.758877i	\(0.274252\pi\)
							−0.651234	+	0.758877i	\(0.725748\pi\)

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.be.a.521.4	yes	8
3.2	odd	2		630.2.be.b.521.2	yes	8
5.2	odd	4		3150.2.bp.d.899.3		8
5.3	odd	4		3150.2.bp.a.899.2		8
5.4	even	2		3150.2.bf.b.1151.1		8
7.3	odd	6		4410.2.b.b.881.7		8
7.4	even	3		4410.2.b.e.881.7		8
7.5	odd	6		630.2.be.b.341.2	yes	8
15.2	even	4		3150.2.bp.c.899.3		8
15.8	even	4		3150.2.bp.f.899.2		8
15.14	odd	2		3150.2.bf.c.1151.3		8
21.5	even	6	inner	630.2.be.a.341.4	✓	8
21.11	odd	6		4410.2.b.b.881.2		8
21.17	even	6		4410.2.b.e.881.2		8
35.12	even	12		3150.2.bp.f.1349.2		8
35.19	odd	6		3150.2.bf.c.1601.3		8
35.33	even	12		3150.2.bp.c.1349.3		8
105.47	odd	12		3150.2.bp.a.1349.2		8
105.68	odd	12		3150.2.bp.d.1349.3		8
105.89	even	6		3150.2.bf.b.1601.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.4	✓	8	21.5	even	6	inner
630.2.be.a.521.4	yes	8	1.1	even	1	trivial
630.2.be.b.341.2	yes	8	7.5	odd	6
630.2.be.b.521.2	yes	8	3.2	odd	2
3150.2.bf.b.1151.1		8	5.4	even	2
3150.2.bf.b.1601.1		8	105.89	even	6
3150.2.bf.c.1151.3		8	15.14	odd	2
3150.2.bf.c.1601.3		8	35.19	odd	6
3150.2.bp.a.899.2		8	5.3	odd	4
3150.2.bp.a.1349.2		8	105.47	odd	12
3150.2.bp.c.899.3		8	15.2	even	4
3150.2.bp.c.1349.3		8	35.33	even	12
3150.2.bp.d.899.3		8	5.2	odd	4
3150.2.bp.d.1349.3		8	105.68	odd	12
3150.2.bp.f.899.2		8	15.8	even	4
3150.2.bp.f.1349.2		8	35.12	even	12
4410.2.b.b.881.2		8	21.11	odd	6
4410.2.b.b.881.7		8	7.3	odd	6
4410.2.b.e.881.2		8	21.17	even	6
4410.2.b.e.881.7		8	7.4	even	3