Embedded newform 560.2.bj.c.97.1

• Label: 560.2.bj.c.97.1
• Level: $560$
• Weight: $2$
• Character: 560.97
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.bj (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{16})\)
Defining polynomial:	\( x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			97.1
Root			\(-0.923880 - 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	560.97
Dual form			560.2.bj.c.433.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30656 - 1.30656i) q^{3} +(0.158513 + 2.23044i) q^{5} +(0.941740 + 2.47247i) q^{7} +0.414214i q^{9} -2.82843 q^{11} +(-4.23671 - 4.23671i) q^{13} +(2.70711 - 3.12132i) q^{15} +(-3.69552 + 3.69552i) q^{17} -1.39942 q^{19} +(2.00000 - 4.46088i) q^{21} +(-0.414214 + 0.414214i) q^{23} +(-4.94975 + 0.707107i) q^{25} +(-3.37849 + 3.37849i) q^{27} -0.828427i q^{29} -1.53073i q^{31} +(3.69552 + 3.69552i) q^{33} +(-5.36543 + 2.49242i) q^{35} +(2.58579 + 2.58579i) q^{37} +11.0711i q^{39} +3.69552i q^{41} +(-4.00000 + 4.00000i) q^{43} +(-0.923880 + 0.0656581i) q^{45} +(-1.08239 + 1.08239i) q^{47} +(-5.22625 + 4.65685i) q^{49} +9.65685 q^{51} +(-8.24264 + 8.24264i) q^{53} +(-0.448342 - 6.30864i) q^{55} +(1.82843 + 1.82843i) q^{57} +9.23880 q^{59} -6.43996i q^{61} +(-1.02413 + 0.390081i) q^{63} +(8.77817 - 10.1213i) q^{65} +(-10.4853 - 10.4853i) q^{67} +1.08239 q^{69} +0.585786 q^{71} +(4.14386 + 4.14386i) q^{73} +(7.39104 + 5.54328i) q^{75} +(-2.66364 - 6.99321i) q^{77} -5.07107i q^{79} +10.0711 q^{81} +(-5.31911 - 5.31911i) q^{83} +(-8.82843 - 7.65685i) q^{85} +(-1.08239 + 1.08239i) q^{87} +11.3492 q^{89} +(6.48528 - 14.4650i) q^{91} +(-2.00000 + 2.00000i) q^{93} +(-0.221825 - 3.12132i) q^{95} +(-4.59220 + 4.59220i) q^{97} -1.17157i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 8 * q^7 + 16 * q^15 + 16 * q^21 + 8 * q^23 + 8 * q^35 + 32 * q^37 - 32 * q^43 + 32 * q^51 - 32 * q^53 - 8 * q^57 + 8 * q^65 - 16 * q^67 + 16 * q^71 - 16 * q^77 + 24 * q^81 - 48 * q^85 - 16 * q^91 - 16 * q^93 - 64 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(1\)

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			0
							\(3\)	−1.30656	−	1.30656i	−0.754344	−	0.754344i	0.220942	−	0.975287i	\(-0.429087\pi\)
−0.975287	+	0.220942i	\(0.929087\pi\)
\(5\)	0.158513	+	2.23044i	0.0708890	+	0.997484i
							\(7\)	0.941740	+	2.47247i	0.355944	+	0.934507i
\(11\)	−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	\(-0.937833\pi\)
							−0.194064	−	0.980989i	\(-0.562167\pi\)
\(17\)	−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	\(-0.975492\pi\)
							0.997037	−	0.0769175i	\(-0.0245078\pi\)
\(31\)		−	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	−0.942944	−	0.332950i	\(-0.891956\pi\)
							0.332950	+	0.942944i	\(0.391956\pi\)
\(47\)	−1.08239	+	1.08239i	−0.157883	+	0.157883i	−0.781628	−	0.623745i	\(-0.785610\pi\)
							0.623745	+	0.781628i	\(0.285610\pi\)
\(53\)	−8.24264	+	8.24264i	−1.13221	+	1.13221i	−0.142405	+	0.989808i	\(0.545484\pi\)
							−0.989808	+	0.142405i	\(0.954516\pi\)
\(59\)	9.23880			1.20279			0.601394	−	0.798952i	\(-0.294612\pi\)
							0.601394	+	0.798952i	\(0.294612\pi\)
\(61\)		−	6.43996i		−	0.824552i	−0.911059	−	0.412276i	\(-0.864734\pi\)
							0.911059	−	0.412276i	\(-0.135266\pi\)
\(67\)	−10.4853	−	10.4853i	−1.28098	−	1.28098i	−0.940110	−	0.340871i	\(-0.889278\pi\)
							−0.340871	−	0.940110i	\(-0.610722\pi\)
\(71\)	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	0.906725	−	0.421723i	\(-0.138574\pi\)
							−0.421723	+	0.906725i	\(0.638574\pi\)
\(79\)		−	5.07107i		−	0.570540i	−0.958447	−	0.285270i	\(-0.907917\pi\)
							0.958447	−	0.285270i	\(-0.0920832\pi\)
\(83\)	−5.31911	−	5.31911i	−0.583848	−	0.583848i	0.352111	−	0.935958i	\(-0.385464\pi\)
							−0.935958	+	0.352111i	\(0.885464\pi\)
\(89\)	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	−0.900703	−	0.434436i	\(-0.856948\pi\)
							0.434436	+	0.900703i	\(0.356948\pi\)

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	560.2.bj.c.97.1		8
4.3	odd	2		70.2.g.a.27.2	yes	8
5.3	odd	4	inner	560.2.bj.c.433.4		8
7.6	odd	2	inner	560.2.bj.c.97.4		8
12.11	even	2		630.2.p.a.307.3		8
20.3	even	4		70.2.g.a.13.1	✓	8
20.7	even	4		350.2.g.a.293.4		8
20.19	odd	2		350.2.g.a.307.3		8
28.3	even	6		490.2.l.a.117.3		16
28.11	odd	6		490.2.l.a.117.4		16
28.19	even	6		490.2.l.a.227.2		16
28.23	odd	6		490.2.l.a.227.1		16
28.27	even	2		70.2.g.a.27.1	yes	8
35.13	even	4	inner	560.2.bj.c.433.1		8
60.23	odd	4		630.2.p.a.433.4		8
84.83	odd	2		630.2.p.a.307.4		8
140.3	odd	12		490.2.l.a.313.1		16
140.23	even	12		490.2.l.a.423.3		16
140.27	odd	4		350.2.g.a.293.3		8
140.83	odd	4		70.2.g.a.13.2	yes	8
140.103	odd	12		490.2.l.a.423.4		16
140.123	even	12		490.2.l.a.313.2		16
140.139	even	2		350.2.g.a.307.4		8
420.83	even	4		630.2.p.a.433.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.g.a.13.1	✓	8	20.3	even	4
70.2.g.a.13.2	yes	8	140.83	odd	4
70.2.g.a.27.1	yes	8	28.27	even	2
70.2.g.a.27.2	yes	8	4.3	odd	2
350.2.g.a.293.3		8	140.27	odd	4
350.2.g.a.293.4		8	20.7	even	4
350.2.g.a.307.3		8	20.19	odd	2
350.2.g.a.307.4		8	140.139	even	2
490.2.l.a.117.3		16	28.3	even	6
490.2.l.a.117.4		16	28.11	odd	6
490.2.l.a.227.1		16	28.23	odd	6
490.2.l.a.227.2		16	28.19	even	6
490.2.l.a.313.1		16	140.3	odd	12
490.2.l.a.313.2		16	140.123	even	12
490.2.l.a.423.3		16	140.23	even	12
490.2.l.a.423.4		16	140.103	odd	12
560.2.bj.c.97.1		8	1.1	even	1	trivial
560.2.bj.c.97.4		8	7.6	odd	2	inner
560.2.bj.c.433.1		8	35.13	even	4	inner
560.2.bj.c.433.4		8	5.3	odd	4	inner
630.2.p.a.307.3		8	12.11	even	2
630.2.p.a.307.4		8	84.83	odd	2
630.2.p.a.433.3		8	420.83	even	4
630.2.p.a.433.4		8	60.23	odd	4