Embedded newform 91.10.bb.a.5.14

• Label: 91.10.bb.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.43697 - 31.4872i) q^{2} +(-61.0517 + 35.2482i) q^{3} +(-476.857 + 275.313i) q^{4} +(1307.03 - 350.218i) q^{5} +(1624.96 + 1624.96i) q^{6} +(6319.13 + 649.753i) q^{7} +(890.360 + 890.360i) q^{8} +(-7356.63 + 12742.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.43697	−	31.4872i	−0.372865	−	1.39155i	−0.856439	−	0.516248i	\(-0.827328\pi\)
							0.483574	−	0.875303i	\(-0.339338\pi\)
\(3\)	−61.0517	+	35.2482i	−0.435163	+	0.251241i	−0.701544	−	0.712626i	\(-0.747505\pi\)
							0.266381	+	0.963868i	\(0.414172\pi\)
\(5\)	1307.03	−	350.218i	0.935237	−	0.250596i	0.241150	−	0.970488i	\(-0.422475\pi\)
							0.694086	+	0.719892i	\(0.255809\pi\)
\(7\)	6319.13	+	649.753i	0.994755	+	0.102284i
							\(11\)	14844.5	−	55400.3i	0.305701	−	1.14089i	−0.626639	−	0.779310i	\(-0.715570\pi\)
0.932340	−	0.361583i	\(-0.117764\pi\)
\(13\)	−102285.	−	11925.1i	−0.993272	−	0.115803i
							\(17\)	307922.	+	533336.i	0.894170	+	1.54875i	0.834829	+	0.550509i	\(0.185566\pi\)
0.0593407	+	0.998238i	\(0.481100\pi\)
\(19\)	349471.	−	93640.3i	0.615204	−	0.164843i	0.0622576	−	0.998060i	\(-0.480170\pi\)
							0.552947	+	0.833217i	\(0.313503\pi\)
\(23\)	1.63992e6	+	946806.i	1.22193	+	0.705481i	0.965329	−	0.261036i	\(-0.0840641\pi\)
							0.256601	+	0.966517i	\(0.417397\pi\)
\(29\)	−266733.			−0.0700302			−0.0350151	−	0.999387i	\(-0.511148\pi\)
							−0.0350151	+	0.999387i	\(0.511148\pi\)
\(31\)	−1.05730e6	+	3.94588e6i	−0.205622	+	0.767391i	0.783637	+	0.621219i	\(0.213362\pi\)
							−0.989259	+	0.146172i	\(0.953305\pi\)
\(37\)	−1.07190e6	+	287216.i	−0.0940260	+	0.0251942i	−0.305525	−	0.952184i	\(-0.598832\pi\)
							0.211499	+	0.977378i	\(0.432165\pi\)
\(41\)	1.78712e7	+	1.78712e7i	0.987702	+	0.987702i	0.999925	−	0.0122236i	\(-0.00389099\pi\)
							−0.0122236	+	0.999925i	\(0.503891\pi\)
\(43\)			3.58711e7i			1.60006i	0.599960	+	0.800030i	\(0.295183\pi\)
							−0.599960	+	0.800030i	\(0.704817\pi\)
\(47\)	−1.45155e7	−	5.41727e7i	−0.433903	−	1.61935i	−0.743678	−	0.668538i	\(-0.766920\pi\)
							0.309775	−	0.950810i	\(-0.399746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.14	✓	328
7.3	odd	6	inner	91.10.bb.a.31.69	yes	328
13.8	odd	4	inner	91.10.bb.a.47.69	yes	328
91.73	even	12	inner	91.10.bb.a.73.14	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.14	✓	328	1.1	even	1	trivial
91.10.bb.a.31.69	yes	328	7.3	odd	6	inner
91.10.bb.a.47.69	yes	328	13.8	odd	4	inner
91.10.bb.a.73.14	yes	328	91.73	even	12	inner