Embedded newform 91.10.bb.a.5.15

• Label: 91.10.bb.a.5.15
• Level: $91$
• Weight: $10$
• Character: 91.5
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.84691 - 29.2851i) q^{2} +(148.521 - 85.7485i) q^{3} +(-352.636 + 203.594i) q^{4} +(1538.27 - 412.177i) q^{5} +(-3676.58 - 3676.58i) q^{6} +(-6345.80 + 290.604i) q^{7} +(-2246.96 - 2246.96i) q^{8} +(4864.12 - 8424.90i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.84691	−	29.2851i	−0.346788	−	1.29423i	−0.890510	−	0.454964i	\(-0.849652\pi\)
							0.543722	−	0.839265i	\(-0.317014\pi\)
\(3\)	148.521	−	85.7485i	1.05862	−	0.611197i	0.133573	−	0.991039i	\(-0.457355\pi\)
							0.925051	+	0.379842i	\(0.124022\pi\)
\(5\)	1538.27	−	412.177i	1.10069	−	0.294930i	0.337646	−	0.941273i	\(-0.390369\pi\)
							0.763047	+	0.646343i	\(0.223703\pi\)
\(7\)	−6345.80	+	290.604i	−0.998953	+	0.0457468i
							\(11\)	6013.49	−	22442.7i	0.123840	−	0.462176i	−0.875956	−	0.482391i	\(-0.839768\pi\)
0.999796	+	0.0202152i	\(0.00643515\pi\)
\(13\)	−90480.5	−	49170.9i	−0.878638	−	0.477489i
							\(17\)	−19880.3	−	34433.6i	−0.0577300	−	0.0999913i	0.835716	−	0.549162i	\(-0.185053\pi\)
−0.893446	+	0.449170i	\(0.851720\pi\)
\(19\)	519953.	−	139321.i	0.915320	−	0.245259i	0.229736	−	0.973253i	\(-0.426214\pi\)
							0.685584	+	0.727994i	\(0.259547\pi\)
\(23\)	−240823.	−	139039.i	−0.179441	−	0.103601i	0.407589	−	0.913166i	\(-0.366370\pi\)
							−0.587030	+	0.809565i	\(0.699703\pi\)
\(29\)	−4.42256e6			−1.16114			−0.580568	−	0.814212i	\(-0.697169\pi\)
							−0.580568	+	0.814212i	\(0.697169\pi\)
\(31\)	1.86131e6	−	6.94650e6i	0.361985	−	1.35095i	−0.509477	−	0.860485i	\(-0.670161\pi\)
							0.871462	−	0.490463i	\(-0.163172\pi\)
\(37\)	−1.40923e7	+	3.77603e6i	−1.23616	+	0.331229i	−0.816976	−	0.576672i	\(-0.804351\pi\)
							−0.419186	+	0.907900i	\(0.637684\pi\)
\(41\)	−2.41120e7	−	2.41120e7i	−1.33262	−	1.33262i	−0.903018	−	0.429603i	\(-0.858653\pi\)
							−0.429603	−	0.903018i	\(-0.641347\pi\)
\(43\)			2.64623e7i			1.18037i	0.807267	+	0.590186i	\(0.200946\pi\)
							−0.807267	+	0.590186i	\(0.799054\pi\)
\(47\)	3.29356e6	+	1.22917e7i	0.0984522	+	0.367429i	0.997520	−	0.0703803i	\(-0.0224213\pi\)
							−0.899068	+	0.437809i	\(0.855755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.15	✓	328
7.3	odd	6	inner	91.10.bb.a.31.68	yes	328
13.8	odd	4	inner	91.10.bb.a.47.68	yes	328
91.73	even	12	inner	91.10.bb.a.73.15	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.15	✓	328	1.1	even	1	trivial
91.10.bb.a.31.68	yes	328	7.3	odd	6	inner
91.10.bb.a.47.68	yes	328	13.8	odd	4	inner
91.10.bb.a.73.15	yes	328	91.73	even	12	inner