Embedded newform 91.2.g.b.9.2

• Label: 91.2.g.b.9.2
• Level: $91$
• Weight: $2$
• Character: 91.9
• 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			9.2
Root			\(-1.02197 + 1.77010i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.9
Dual form			91.2.g.b.81.2

$q$-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 \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

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

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	−0.998297	−	0.0583310i	\(-0.981422\pi\)
							0.448632	−	0.893716i	\(-0.351911\pi\)
\(3\)	−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	−0.701507	−	0.712662i	\(-0.747489\pi\)
							0.967937	+	0.251192i	\(0.0808225\pi\)
\(7\)	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	−0.805826	−	0.592152i	\(-0.798279\pi\)
0.915732	+	0.401790i	\(0.131612\pi\)
\(19\)	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	0.615287	−	0.788303i	\(-0.289040\pi\)
							−0.990334	+	0.138702i	\(0.955707\pi\)
\(29\)	−4.25772	+	7.37459i	−0.790639	+	1.36943i	0.134932	+	0.990855i	\(0.456918\pi\)
							−0.925572	+	0.378573i	\(0.876415\pi\)
\(31\)	2.64390	+	4.57937i	0.474859	+	0.822479i	0.999585	−	0.0287913i	\(-0.00916583\pi\)
							−0.524727	+	0.851271i	\(0.675832\pi\)
\(37\)	−2.49579	−	4.32284i	−0.410306	−	0.710670i	0.584617	−	0.811309i	\(-0.301245\pi\)
							−0.994923	+	0.100639i	\(0.967911\pi\)
\(41\)	−0.768181	+	1.33053i	−0.119970	+	0.207794i	−0.919755	−	0.392492i	\(-0.871613\pi\)
							0.799786	+	0.600286i	\(0.204946\pi\)
\(43\)	−2.71636	−	4.70488i	−0.414242	−	0.717488i	0.581107	−	0.813827i	\(-0.302620\pi\)
							−0.995349	+	0.0963397i	\(0.969286\pi\)
\(47\)	1.59337	−	2.75979i	0.232416	−	0.402557i	−0.726102	−	0.687587i	\(-0.758670\pi\)
							0.958519	+	0.285030i	\(0.0920035\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.2	✓	12	1.1	even	1	trivial
91.2.g.b.81.2	yes	12	91.81	even	3	inner
91.2.h.b.16.5	yes	12	13.3	even	3
91.2.h.b.74.5	yes	12	7.4	even	3
637.2.f.j.295.2		12	7.5	odd	6
637.2.f.j.393.2		12	91.68	odd	6
637.2.f.k.295.2		12	7.2	even	3
637.2.f.k.393.2		12	91.16	even	3
637.2.g.l.263.2		12	91.3	odd	6
637.2.g.l.373.2		12	7.6	odd	2
637.2.h.l.165.5		12	7.3	odd	6
637.2.h.l.471.5		12	91.55	odd	6
819.2.n.d.100.5		12	3.2	odd	2
819.2.n.d.172.5		12	273.263	odd	6
819.2.s.d.289.2		12	39.29	odd	6
819.2.s.d.802.2		12	21.11	odd	6
1183.2.e.g.170.5		12	13.4	even	6
1183.2.e.g.508.5		12	91.4	even	6
1183.2.e.h.170.2		12	13.9	even	3
1183.2.e.h.508.2		12	91.74	even	3
8281.2.a.bz.1.5		6	91.9	even	3
8281.2.a.ca.1.5		6	91.61	odd	6
8281.2.a.ce.1.2		6	91.30	even	6
8281.2.a.cf.1.2		6	91.82	odd	6