Embedded newform 91.10.bb.a.5.9

• Label: 91.10.bb.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.88529 - 36.8924i) q^{2} +(230.291 - 132.958i) q^{3} +(-819.927 + 473.385i) q^{4} +(-1552.67 + 416.037i) q^{5} +(-7181.65 - 7181.65i) q^{6} +(-5367.23 + 3398.00i) q^{7} +(11741.9 + 11741.9i) q^{8} +(25514.4 - 44192.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.88529	−	36.8924i	−0.436872	−	1.63043i	−0.736545	−	0.676388i	\(-0.763544\pi\)
							0.299673	−	0.954042i	\(-0.403122\pi\)
\(3\)	230.291	−	132.958i	1.64146	−	0.947699i	0.661148	−	0.750256i	\(-0.270070\pi\)
							0.980314	−	0.197443i	\(-0.0632636\pi\)
\(5\)	−1552.67	+	416.037i	−1.11100	+	0.297692i	−0.767236	−	0.641364i	\(-0.778369\pi\)
							−0.343764	+	0.939056i	\(0.611702\pi\)
\(7\)	−5367.23	+	3398.00i	−0.844907	+	0.534912i
							\(11\)	−13571.8	+	50650.6i	−0.279492	+	1.04308i	0.673279	+	0.739389i	\(0.264885\pi\)
−0.952771	+	0.303690i	\(0.901781\pi\)
\(13\)	79348.4	+	65637.9i	0.770536	+	0.637396i
							\(17\)	−63168.2	−	109410.i	−0.183433	−	0.317716i	0.759614	−	0.650374i	\(-0.225388\pi\)
−0.943047	+	0.332658i	\(0.892054\pi\)
\(19\)	709631.	−	190145.i	1.24923	−	0.334729i	0.427189	−	0.904162i	\(-0.359504\pi\)
							0.822038	+	0.569433i	\(0.192837\pi\)
\(23\)	849631.	+	490535.i	0.633075	+	0.365506i	0.781942	−	0.623351i	\(-0.214229\pi\)
							−0.148867	+	0.988857i	\(0.547563\pi\)
\(29\)	−7.45831e6			−1.95817			−0.979083	−	0.203461i	\(-0.934781\pi\)
							−0.979083	+	0.203461i	\(0.934781\pi\)
\(31\)	−639054.	+	2.38498e6i	−0.124283	+	0.463829i	−0.999813	−	0.0193341i	\(-0.993845\pi\)
							0.875531	+	0.483163i	\(0.160512\pi\)
\(37\)	1.19993e6	−	321520.i	0.105256	−	0.0282033i	−0.205806	−	0.978593i	\(-0.565982\pi\)
							0.311063	+	0.950389i	\(0.399315\pi\)
\(41\)	1.34305e7	+	1.34305e7i	0.742273	+	0.742273i	0.973015	−	0.230742i	\(-0.0741153\pi\)
							−0.230742	+	0.973015i	\(0.574115\pi\)
\(43\)		−	5.02032e6i		−	0.223936i	−0.993712	−	0.111968i	\(-0.964285\pi\)
							0.993712	−	0.111968i	\(-0.0357154\pi\)
\(47\)	1.19848e7	+	4.47279e7i	0.358254	+	1.33702i	0.876340	+	0.481693i	\(0.159978\pi\)
							−0.518086	+	0.855329i	\(0.673355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.9	✓	328
7.3	odd	6	inner	91.10.bb.a.31.74	yes	328
13.8	odd	4	inner	91.10.bb.a.47.74	yes	328
91.73	even	12	inner	91.10.bb.a.73.9	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.9	✓	328	1.1	even	1	trivial
91.10.bb.a.31.74	yes	328	7.3	odd	6	inner
91.10.bb.a.47.74	yes	328	13.8	odd	4	inner
91.10.bb.a.73.9	yes	328	91.73	even	12	inner