Embedded newform 630.2.p.a.307.4

• Label: 630.2.p.a.307.4
• Level: $630$
• Weight: $2$
• Character: 630.307
• Analytic conductor: $5.031$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.03057532734\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{16})\)
Defining polynomial:	\( x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			307.4
Root			\(-0.923880 - 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	630.307
Dual form			630.2.p.a.433.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{2} +1.00000i q^{4} +(0.158513 + 2.23044i) q^{5} +(-2.47247 - 0.941740i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-1.46508 + 1.68925i) q^{10} -2.82843 q^{11} +(4.23671 + 4.23671i) q^{13} +(-1.08239 - 2.41421i) q^{14} -1.00000 q^{16} +(-3.69552 + 3.69552i) q^{17} -1.39942 q^{19} +(-2.23044 + 0.158513i) q^{20} +(-2.00000 - 2.00000i) q^{22} +(-0.414214 + 0.414214i) q^{23} +(-4.94975 + 0.707107i) q^{25} +5.99162i q^{26} +(0.941740 - 2.47247i) q^{28} +0.828427i q^{29} -1.53073i q^{31} +(-0.707107 - 0.707107i) q^{32} -5.22625 q^{34} +(1.70858 - 5.66399i) q^{35} +(2.58579 + 2.58579i) q^{37} +(-0.989538 - 0.989538i) q^{38} +(-1.68925 - 1.46508i) q^{40} +3.69552i q^{41} +(4.00000 - 4.00000i) q^{43} -2.82843i q^{44} -0.585786 q^{46} +(1.08239 - 1.08239i) q^{47} +(5.22625 + 4.65685i) q^{49} +(-4.00000 - 3.00000i) q^{50} +(-4.23671 + 4.23671i) q^{52} +(8.24264 - 8.24264i) q^{53} +(-0.448342 - 6.30864i) q^{55} +(2.41421 - 1.08239i) q^{56} +(-0.585786 + 0.585786i) q^{58} -9.23880 q^{59} +6.43996i q^{61} +(1.08239 - 1.08239i) q^{62} -1.00000i q^{64} +(-8.77817 + 10.1213i) q^{65} +(10.4853 + 10.4853i) q^{67} +(-3.69552 - 3.69552i) q^{68} +(5.21319 - 2.79690i) q^{70} +0.585786 q^{71} +(-4.14386 - 4.14386i) q^{73} +3.65685i q^{74} -1.39942i q^{76} +(6.99321 + 2.66364i) q^{77} +5.07107i q^{79} +(-0.158513 - 2.23044i) q^{80} +(-2.61313 + 2.61313i) q^{82} +(5.31911 + 5.31911i) q^{83} +(-8.82843 - 7.65685i) q^{85} +5.65685 q^{86} +(2.00000 - 2.00000i) q^{88} +11.3492 q^{89} +(-6.48528 - 14.4650i) q^{91} +(-0.414214 - 0.414214i) q^{92} +1.53073 q^{94} +(-0.221825 - 3.12132i) q^{95} +(4.59220 - 4.59220i) q^{97} +(0.402625 + 6.98841i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q - 8 * q^7 - 8 * q^16 - 16 * q^22 + 8 * q^23 + 8 * q^28 + 8 * q^35 + 32 * q^37 + 32 * q^43 - 16 * q^46 - 32 * q^50 + 32 * q^53 + 8 * q^56 - 16 * q^58 - 8 * q^65 + 16 * q^67 - 24 * q^70 + 16 * q^71 + 16 * q^77 - 48 * q^85 + 16 * q^88 + 16 * q^91 + 8 * q^92 - 64 * q^95 - 32 * q^98

Character values

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

\(n\)	\(127\)	\(281\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	+	0.707107i	0.500000	+	0.500000i
							\(3\)		0			0
\(5\)	0.158513	+	2.23044i	0.0708890	+	0.997484i
							\(7\)	−2.47247	−	0.941740i	−0.934507	−	0.355944i
\(11\)	−2.82843			−0.852803										−0.426401	−	0.904534i	\(-0.640219\pi\)
							−0.426401	+	0.904534i	\(0.640219\pi\)
\(13\)	4.23671	+	4.23671i	1.17505	+	1.17505i	0.980989	+	0.194064i	\(0.0621670\pi\)
							0.194064	+	0.980989i	\(0.437833\pi\)
\(17\)	−3.69552	+	3.69552i	−0.896295	+	0.896295i	−0.995106	−	0.0988114i	\(-0.968496\pi\)
							0.0988114	+	0.995106i	\(0.468496\pi\)
\(19\)	−1.39942			−0.321048			−0.160524	−	0.987032i	\(-0.551318\pi\)
							−0.160524	+	0.987032i	\(0.551318\pi\)
\(23\)	−0.414214	+	0.414214i	−0.0863695	+	0.0863695i	−0.748972	−	0.662602i	\(-0.769452\pi\)
							0.662602	+	0.748972i	\(0.269452\pi\)
\(29\)			0.828427i			0.153835i	0.997037	+	0.0769175i	\(0.0245078\pi\)
							−0.997037	+	0.0769175i	\(0.975492\pi\)
\(31\)		−	1.53073i		−	0.274928i	−0.990507	−	0.137464i	\(-0.956105\pi\)
							0.990507	−	0.137464i	\(-0.0438951\pi\)
\(37\)	2.58579	+	2.58579i	0.425101	+	0.425101i	0.886956	−	0.461855i	\(-0.152816\pi\)
							−0.461855	+	0.886956i	\(0.652816\pi\)
\(41\)			3.69552i			0.577143i	0.957458	+	0.288571i	\(0.0931803\pi\)
							−0.957458	+	0.288571i	\(0.906820\pi\)
\(43\)	4.00000	−	4.00000i	0.609994	−	0.609994i	−0.332950	−	0.942944i	\(-0.608044\pi\)
							0.942944	+	0.332950i	\(0.108044\pi\)
\(47\)	1.08239	−	1.08239i	0.157883	−	0.157883i	−0.623745	−	0.781628i	\(-0.714390\pi\)
							0.781628	+	0.623745i	\(0.214390\pi\)
\(53\)	8.24264	−	8.24264i	1.13221	−	1.13221i	0.142405	−	0.989808i	\(-0.454516\pi\)
							0.989808	−	0.142405i	\(-0.0454837\pi\)
\(59\)	−9.23880			−1.20279			−0.601394	−	0.798952i	\(-0.705388\pi\)
							−0.601394	+	0.798952i	\(0.705388\pi\)
\(61\)			6.43996i			0.824552i	0.911059	+	0.412276i	\(0.135266\pi\)
							−0.911059	+	0.412276i	\(0.864734\pi\)
\(67\)	10.4853	+	10.4853i	1.28098	+	1.28098i	0.940110	+	0.340871i	\(0.110722\pi\)
							0.340871	+	0.940110i	\(0.389278\pi\)
\(71\)	0.585786			0.0695201			0.0347600	−	0.999396i	\(-0.488933\pi\)
							0.0347600	+	0.999396i	\(0.488933\pi\)
\(73\)	−4.14386	−	4.14386i	−0.485002	−	0.485002i	0.421723	−	0.906725i	\(-0.361426\pi\)
							−0.906725	+	0.421723i	\(0.861426\pi\)
\(79\)			5.07107i			0.570540i	0.958447	+	0.285270i	\(0.0920832\pi\)
							−0.958447	+	0.285270i	\(0.907917\pi\)
\(83\)	5.31911	+	5.31911i	0.583848	+	0.583848i	0.935958	−	0.352111i	\(-0.114536\pi\)
							−0.352111	+	0.935958i	\(0.614536\pi\)
\(89\)	11.3492			1.20301			0.601506	−	0.798869i	\(-0.294568\pi\)
							0.601506	+	0.798869i	\(0.294568\pi\)
\(97\)	4.59220	−	4.59220i	0.466267	−	0.466267i	−0.434436	−	0.900703i	\(-0.643052\pi\)
							0.900703	+	0.434436i	\(0.143052\pi\)

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.p.a.307.4		8
3.2	odd	2		70.2.g.a.27.1	yes	8
5.3	odd	4	inner	630.2.p.a.433.3		8
7.6	odd	2	inner	630.2.p.a.307.3		8
12.11	even	2		560.2.bj.c.97.4		8
15.2	even	4		350.2.g.a.293.3		8
15.8	even	4		70.2.g.a.13.2	yes	8
15.14	odd	2		350.2.g.a.307.4		8
21.2	odd	6		490.2.l.a.227.2		16
21.5	even	6		490.2.l.a.227.1		16
21.11	odd	6		490.2.l.a.117.3		16
21.17	even	6		490.2.l.a.117.4		16
21.20	even	2		70.2.g.a.27.2	yes	8
35.13	even	4	inner	630.2.p.a.433.4		8
60.23	odd	4		560.2.bj.c.433.1		8
84.83	odd	2		560.2.bj.c.97.1		8
105.23	even	12		490.2.l.a.423.4		16
105.38	odd	12		490.2.l.a.313.2		16
105.53	even	12		490.2.l.a.313.1		16
105.62	odd	4		350.2.g.a.293.4		8
105.68	odd	12		490.2.l.a.423.3		16
105.83	odd	4		70.2.g.a.13.1	✓	8
105.104	even	2		350.2.g.a.307.3		8
420.83	even	4		560.2.bj.c.433.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.g.a.13.1	✓	8	105.83	odd	4
70.2.g.a.13.2	yes	8	15.8	even	4
70.2.g.a.27.1	yes	8	3.2	odd	2
70.2.g.a.27.2	yes	8	21.20	even	2
350.2.g.a.293.3		8	15.2	even	4
350.2.g.a.293.4		8	105.62	odd	4
350.2.g.a.307.3		8	105.104	even	2
350.2.g.a.307.4		8	15.14	odd	2
490.2.l.a.117.3		16	21.11	odd	6
490.2.l.a.117.4		16	21.17	even	6
490.2.l.a.227.1		16	21.5	even	6
490.2.l.a.227.2		16	21.2	odd	6
490.2.l.a.313.1		16	105.53	even	12
490.2.l.a.313.2		16	105.38	odd	12
490.2.l.a.423.3		16	105.68	odd	12
490.2.l.a.423.4		16	105.23	even	12
560.2.bj.c.97.1		8	84.83	odd	2
560.2.bj.c.97.4		8	12.11	even	2
560.2.bj.c.433.1		8	60.23	odd	4
560.2.bj.c.433.4		8	420.83	even	4
630.2.p.a.307.3		8	7.6	odd	2	inner
630.2.p.a.307.4		8	1.1	even	1	trivial
630.2.p.a.433.3		8	5.3	odd	4	inner
630.2.p.a.433.4		8	35.13	even	4	inner