Embedded newform 91.2.u.b.88.4

• Label: 91.2.u.b.88.4
• Level: $91$
• Weight: $2$
• Character: 91.88
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.2346760387617129.1
Defining polynomial:	x^12 - 3*x^11 + x^10 + 10*x^9 - 15*x^8 - 10*x^7 + 45*x^6 - 20*x^5 - 60*x^4 + 80*x^3 + 16*x^2 - 96*x + 64
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			88.4
Root			\(-1.18541 - 0.771231i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.88
Dual form			91.2.u.b.30.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.433001 + 0.249993i) q^{2} +0.849601 q^{3} +(-0.875007 - 1.51556i) q^{4} +(0.902810 - 0.521238i) q^{5} +(0.367878 + 0.212395i) q^{6} +(1.52469 + 2.16225i) q^{7} -1.87496i q^{8} -2.27818 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 4 q^{4} + 3 q^{5} - 9 q^{6} + 3 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.433001	+	0.249993i	0.306178	+	0.176772i	0.645215	−	0.764001i	\(-0.276768\pi\)
							−0.339037	+	0.940773i	\(0.610101\pi\)
\(3\)	0.849601			0.490518			0.245259	−	0.969458i	\(-0.421127\pi\)
							0.245259	+	0.969458i	\(0.421127\pi\)
\(5\)	0.902810	−	0.521238i	0.403749	−	0.233105i	−0.284351	−	0.958720i	\(-0.591778\pi\)
							0.688100	+	0.725616i	\(0.258445\pi\)
\(7\)	1.52469	+	2.16225i	0.576280	+	0.817252i
							\(11\)			3.96730i			1.19619i	0.801426	+	0.598094i	\(0.204075\pi\)
−0.801426	+	0.598094i	\(0.795925\pi\)
\(13\)	−3.57504	−	0.468096i	−0.991537	−	0.129827i
							\(17\)	−0.0710177	−	0.123006i	−0.0172243	−	0.0298334i	0.857285	−	0.514843i	\(-0.172150\pi\)
−0.874509	+	0.485009i	\(0.838816\pi\)
\(19\)		−	5.50977i		−	1.26403i	−0.774957	−	0.632014i	\(-0.782229\pi\)
							0.774957	−	0.632014i	\(-0.217771\pi\)
\(23\)	2.19549	−	3.80270i	0.457791	−	0.792917i	−0.541053	−	0.840989i	\(-0.681974\pi\)
							0.998844	+	0.0480711i	\(0.0153074\pi\)
\(29\)	4.19880	+	7.27253i	0.779697	+	1.35047i	0.932116	+	0.362159i	\(0.117960\pi\)
							−0.152419	+	0.988316i	\(0.548706\pi\)
\(31\)	−2.46516	−	1.42326i	−0.442756	−	0.255625i	0.262010	−	0.965065i	\(-0.415615\pi\)
							−0.704766	+	0.709440i	\(0.748948\pi\)
\(37\)	−0.730221	−	0.421593i	−0.120048	−	0.0693095i	0.438774	−	0.898598i	\(-0.355413\pi\)
							−0.558821	+	0.829288i	\(0.688746\pi\)
\(41\)	10.4766	−	6.04869i	1.63618	−	0.944647i	0.654044	−	0.756457i	\(-0.273071\pi\)
							0.982133	−	0.188190i	\(-0.0602621\pi\)
\(43\)	2.41161	−	4.17704i	0.367768	−	0.636993i	−0.621448	−	0.783455i	\(-0.713455\pi\)
							0.989216	+	0.146463i	\(0.0467888\pi\)
\(47\)	3.94602	−	2.27824i	0.575587	−	0.332315i	−0.183791	−	0.982965i	\(-0.558837\pi\)
							0.759378	+	0.650650i	\(0.225504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.u.b.88.4	yes	12
3.2	odd	2		819.2.do.e.361.3		12
7.2	even	3		91.2.k.b.23.3	yes	12
7.3	odd	6		637.2.q.i.491.3		12
7.4	even	3		637.2.q.g.491.3		12
7.5	odd	6		637.2.k.i.569.3		12
7.6	odd	2		637.2.u.g.361.4		12
13.2	odd	12		1183.2.e.j.508.8		24
13.4	even	6		91.2.k.b.4.4	✓	12
13.11	odd	12		1183.2.e.j.508.5		24
21.2	odd	6		819.2.bm.f.478.4		12
39.17	odd	6		819.2.bm.f.550.3		12
91.2	odd	12		1183.2.e.j.170.8		24
91.4	even	6		637.2.q.g.589.3		12
91.11	odd	12		8281.2.a.cp.1.8		12
91.17	odd	6		637.2.q.i.589.3		12
91.24	even	12		8281.2.a.co.1.8		12
91.30	even	6	inner	91.2.u.b.30.4	yes	12
91.37	odd	12		1183.2.e.j.170.5		24
91.67	odd	12		8281.2.a.cp.1.5		12
91.69	odd	6		637.2.k.i.459.4		12
91.80	even	12		8281.2.a.co.1.5		12
91.82	odd	6		637.2.u.g.30.4		12
273.212	odd	6		819.2.do.e.667.3		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.4	✓	12	13.4	even	6
91.2.k.b.23.3	yes	12	7.2	even	3
91.2.u.b.30.4	yes	12	91.30	even	6	inner
91.2.u.b.88.4	yes	12	1.1	even	1	trivial
637.2.k.i.459.4		12	91.69	odd	6
637.2.k.i.569.3		12	7.5	odd	6
637.2.q.g.491.3		12	7.4	even	3
637.2.q.g.589.3		12	91.4	even	6
637.2.q.i.491.3		12	7.3	odd	6
637.2.q.i.589.3		12	91.17	odd	6
637.2.u.g.30.4		12	91.82	odd	6
637.2.u.g.361.4		12	7.6	odd	2
819.2.bm.f.478.4		12	21.2	odd	6
819.2.bm.f.550.3		12	39.17	odd	6
819.2.do.e.361.3		12	3.2	odd	2
819.2.do.e.667.3		12	273.212	odd	6
1183.2.e.j.170.5		24	91.37	odd	12
1183.2.e.j.170.8		24	91.2	odd	12
1183.2.e.j.508.5		24	13.11	odd	12
1183.2.e.j.508.8		24	13.2	odd	12
8281.2.a.co.1.5		12	91.80	even	12
8281.2.a.co.1.8		12	91.24	even	12
8281.2.a.cp.1.5		12	91.67	odd	12
8281.2.a.cp.1.8		12	91.11	odd	12