Embedded newform 91.2.bb.a.73.3

• Label: 91.2.bb.a.73.3
• Level: $91$
• Weight: $2$
• Character: 91.73
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.3
Character	\(\chi\)	\(=\)	91.73
Dual form			91.2.bb.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.211401 + 0.788958i) q^{2} +(-2.60482 - 1.50389i) q^{3} +(1.15429 + 0.666428i) q^{4} +(3.03793 + 0.814012i) q^{5} +(1.73717 - 1.73717i) q^{6} +(2.28951 - 1.32595i) q^{7} +(-1.92491 + 1.92491i) q^{8} +(3.02338 + 5.23665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.211401	+	0.788958i	−0.149483	+	0.557877i	0.850032	+	0.526731i	\(0.176582\pi\)
							−0.999515	+	0.0311464i	\(0.990084\pi\)
\(3\)	−2.60482	−	1.50389i	−1.50389	−	0.868272i	−0.999990	−	0.00451177i	\(-0.998564\pi\)
							−0.503902	−	0.863761i	\(-0.668103\pi\)
\(5\)	3.03793	+	0.814012i	1.35860	+	0.364037i	0.863304	−	0.504684i	\(-0.168391\pi\)
							0.495301	+	0.868721i	\(0.335058\pi\)
\(7\)	2.28951	−	1.32595i	0.865353	−	0.501162i
							\(11\)	−0.131620	−	0.491212i	−0.0396849	−	0.148106i	0.943241	−	0.332110i	\(-0.107761\pi\)
−0.982925	+	0.184004i	\(0.941094\pi\)
\(13\)	−1.73717	−	3.15947i	−0.481804	−	0.876279i
							\(17\)	−0.606654	+	1.05076i	−0.147135	+	0.254846i	−0.930168	−	0.367135i	\(-0.880339\pi\)
0.783032	+	0.621981i	\(0.213672\pi\)
\(19\)	−1.72169	−	0.461325i	−0.394982	−	0.105835i	0.0558606	−	0.998439i	\(-0.482210\pi\)
							−0.450843	+	0.892603i	\(0.648876\pi\)
\(23\)	−4.51168	+	2.60482i	−0.940750	+	0.543142i	−0.890195	−	0.455579i	\(-0.849432\pi\)
							−0.0505543	+	0.998721i	\(0.516099\pi\)
\(29\)	1.64443			0.305362			0.152681	−	0.988276i	\(-0.451209\pi\)
							0.152681	+	0.988276i	\(0.451209\pi\)
\(31\)	−0.976210	−	3.64327i	−0.175333	−	0.654350i	−0.996495	−	0.0836552i	\(-0.973341\pi\)
							0.821162	−	0.570695i	\(-0.193326\pi\)
\(37\)	−2.66889	−	0.715128i	−0.438764	−	0.117566i	0.0326734	−	0.999466i	\(-0.489598\pi\)
							−0.471437	+	0.881900i	\(0.656265\pi\)
\(41\)	−5.55629	+	5.55629i	−0.867747	+	0.867747i	−0.992223	−	0.124476i	\(-0.960275\pi\)
							0.124476	+	0.992223i	\(0.460275\pi\)
\(43\)		−	7.46499i		−	1.13840i	−0.822199	−	0.569200i	\(-0.807253\pi\)
							0.822199	−	0.569200i	\(-0.192747\pi\)
\(47\)	−1.26875	+	4.73504i	−0.185066	+	0.690677i	0.809550	+	0.587051i	\(0.199711\pi\)
							−0.994616	+	0.103626i	\(0.966956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.73.3	yes	32
3.2	odd	2		819.2.fn.e.73.6		32
7.2	even	3		637.2.bc.b.411.6		32
7.3	odd	6		637.2.i.a.489.5		32
7.4	even	3		637.2.i.a.489.6		32
7.5	odd	6	inner	91.2.bb.a.47.6	yes	32
7.6	odd	2		637.2.bc.b.619.3		32
13.5	odd	4	inner	91.2.bb.a.31.6	yes	32
21.5	even	6		819.2.fn.e.775.3		32
39.5	even	4		819.2.fn.e.577.3		32
91.5	even	12	inner	91.2.bb.a.5.3	✓	32
91.18	odd	12		637.2.i.a.538.6		32
91.31	even	12		637.2.i.a.538.5		32
91.44	odd	12		637.2.bc.b.460.3		32
91.83	even	4		637.2.bc.b.31.6		32
273.5	odd	12		819.2.fn.e.460.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.3	✓	32	91.5	even	12	inner
91.2.bb.a.31.6	yes	32	13.5	odd	4	inner
91.2.bb.a.47.6	yes	32	7.5	odd	6	inner
91.2.bb.a.73.3	yes	32	1.1	even	1	trivial
637.2.i.a.489.5		32	7.3	odd	6
637.2.i.a.489.6		32	7.4	even	3
637.2.i.a.538.5		32	91.31	even	12
637.2.i.a.538.6		32	91.18	odd	12
637.2.bc.b.31.6		32	91.83	even	4
637.2.bc.b.411.6		32	7.2	even	3
637.2.bc.b.460.3		32	91.44	odd	12
637.2.bc.b.619.3		32	7.6	odd	2
819.2.fn.e.73.6		32	3.2	odd	2
819.2.fn.e.460.6		32	273.5	odd	12
819.2.fn.e.577.3		32	39.5	even	4
819.2.fn.e.775.3		32	21.5	even	6