Embedded newform 91.2.g.b.81.2

• Label: 91.2.g.b.81.2
• Level: $91$
• Weight: $2$
• Character: 91.81
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.2
Root			\(-1.02197 - 1.77010i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.81
Dual form			91.2.g.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777343 + 1.34640i) q^{2} -0.489252 q^{3} +(-0.208526 - 0.361177i) q^{4} +(0.595756 + 1.03188i) q^{5} +(0.380316 - 0.658727i) q^{6} +(0.337371 + 2.62415i) q^{7} -2.46099 q^{8} -2.76063 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + q^{5} - 9 q^{6} + 9 q^{7} - 6 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.777343	+	1.34640i	−0.549665	+	0.952047i	0.448632	+	0.893716i	\(0.351911\pi\)
							−0.998297	+	0.0583310i	\(0.981422\pi\)
\(3\)	−0.489252			−0.282470			−0.141235	−	0.989976i	\(-0.545107\pi\)
							−0.141235	+	0.989976i	\(0.545107\pi\)
\(5\)	0.595756	+	1.03188i	0.266430	+	0.461470i	0.967937	−	0.251192i	\(-0.0808225\pi\)
							−0.701507	+	0.712662i	\(0.747489\pi\)
\(7\)	0.337371	+	2.62415i	0.127514	+	0.991837i
							\(11\)	2.11614			0.638040			0.319020	−	0.947748i	\(-0.396646\pi\)
0.319020	+	0.947748i	\(0.396646\pi\)
\(13\)	2.86133	−	2.19381i	0.793590	−	0.608453i
							\(17\)	0.453151	+	0.784881i	0.109905	+	0.190362i	0.915732	−	0.401790i	\(-0.131612\pi\)
−0.805826	+	0.592152i	\(0.798279\pi\)
\(19\)	6.69028			1.53486			0.767428	−	0.641135i	\(-0.221536\pi\)
							0.767428	+	0.641135i	\(0.221536\pi\)
\(23\)	−1.79866	+	3.11538i	−0.375048	+	0.649601i	−0.990334	−	0.138702i	\(-0.955707\pi\)
							0.615287	+	0.788303i	\(0.289040\pi\)
\(29\)	−4.25772	−	7.37459i	−0.790639	−	1.36943i	−0.925572	−	0.378573i	\(-0.876415\pi\)
							0.134932	−	0.990855i	\(-0.456918\pi\)
\(31\)	2.64390	−	4.57937i	0.474859	−	0.822479i	−0.524727	−	0.851271i	\(-0.675832\pi\)
							0.999585	+	0.0287913i	\(0.00916583\pi\)
\(37\)	−2.49579	+	4.32284i	−0.410306	+	0.710670i	−0.994923	−	0.100639i	\(-0.967911\pi\)
							0.584617	+	0.811309i	\(0.301245\pi\)
\(41\)	−0.768181	−	1.33053i	−0.119970	−	0.207794i	0.799786	−	0.600286i	\(-0.204946\pi\)
							−0.919755	+	0.392492i	\(0.871613\pi\)
\(43\)	−2.71636	+	4.70488i	−0.414242	+	0.717488i	−0.995349	−	0.0963397i	\(-0.969286\pi\)
							0.581107	+	0.813827i	\(0.302620\pi\)
\(47\)	1.59337	+	2.75979i	0.232416	+	0.402557i	0.958519	−	0.285030i	\(-0.0920035\pi\)
							−0.726102	+	0.687587i	\(0.758670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.g.b.81.2	yes	12
3.2	odd	2		819.2.n.d.172.5		12
7.2	even	3		91.2.h.b.16.5	yes	12
7.3	odd	6		637.2.f.j.393.2		12
7.4	even	3		637.2.f.k.393.2		12
7.5	odd	6		637.2.h.l.471.5		12
7.6	odd	2		637.2.g.l.263.2		12
13.3	even	3		1183.2.e.h.508.2		12
13.9	even	3		91.2.h.b.74.5	yes	12
13.10	even	6		1183.2.e.g.508.5		12
21.2	odd	6		819.2.s.d.289.2		12
39.35	odd	6		819.2.s.d.802.2		12
91.3	odd	6		8281.2.a.ca.1.5		6
91.9	even	3	inner	91.2.g.b.9.2	✓	12
91.10	odd	6		8281.2.a.cf.1.2		6
91.16	even	3		1183.2.e.h.170.2		12
91.23	even	6		1183.2.e.g.170.5		12
91.48	odd	6		637.2.h.l.165.5		12
91.61	odd	6		637.2.g.l.373.2		12
91.74	even	3		637.2.f.k.295.2		12
91.81	even	3		8281.2.a.bz.1.5		6
91.87	odd	6		637.2.f.j.295.2		12
91.88	even	6		8281.2.a.ce.1.2		6
273.191	odd	6		819.2.n.d.100.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.2	✓	12	91.9	even	3	inner
91.2.g.b.81.2	yes	12	1.1	even	1	trivial
91.2.h.b.16.5	yes	12	7.2	even	3
91.2.h.b.74.5	yes	12	13.9	even	3
637.2.f.j.295.2		12	91.87	odd	6
637.2.f.j.393.2		12	7.3	odd	6
637.2.f.k.295.2		12	91.74	even	3
637.2.f.k.393.2		12	7.4	even	3
637.2.g.l.263.2		12	7.6	odd	2
637.2.g.l.373.2		12	91.61	odd	6
637.2.h.l.165.5		12	91.48	odd	6
637.2.h.l.471.5		12	7.5	odd	6
819.2.n.d.100.5		12	273.191	odd	6
819.2.n.d.172.5		12	3.2	odd	2
819.2.s.d.289.2		12	21.2	odd	6
819.2.s.d.802.2		12	39.35	odd	6
1183.2.e.g.170.5		12	91.23	even	6
1183.2.e.g.508.5		12	13.10	even	6
1183.2.e.h.170.2		12	91.16	even	3
1183.2.e.h.508.2		12	13.3	even	3
8281.2.a.bz.1.5		6	91.81	even	3
8281.2.a.ca.1.5		6	91.3	odd	6
8281.2.a.ce.1.2		6	91.88	even	6
8281.2.a.cf.1.2		6	91.10	odd	6