Embedded newform 91.10.bc.a.6.5

• Label: 91.10.bc.a.6.5
• Level: $91$
• Weight: $10$
• Character: 91.6
• 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.bc (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			6.5
Character	\(\chi\)	\(=\)	91.6
Dual form			91.10.bc.a.76.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.4056 - 38.8341i) q^{2} +(-174.119 - 100.528i) q^{3} +(-956.405 + 552.181i) q^{4} +(1035.54 + 1035.54i) q^{5} +(-2092.09 + 7807.80i) q^{6} +(-6203.64 - 1366.90i) q^{7} +(16840.0 + 16840.0i) q^{8} +(10370.1 + 17961.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.4056	−	38.8341i	−0.459865	−	1.71624i	−0.673377	−	0.739300i	\(-0.735157\pi\)
							0.213512	−	0.976941i	\(-0.431510\pi\)
\(3\)	−174.119	−	100.528i	−1.24108	−	0.716539i	−0.271768	−	0.962363i	\(-0.587608\pi\)
							−0.969315	+	0.245824i	\(0.920942\pi\)
\(5\)	1035.54	+	1035.54i	0.740973	+	0.740973i	0.972765	−	0.231792i	\(-0.0744590\pi\)
							−0.231792	+	0.972765i	\(0.574459\pi\)
\(7\)	−6203.64	−	1366.90i	−0.976575	−	0.215177i
							\(11\)	24763.7	−	6635.41i	0.509974	−	0.136647i	0.00534905	−	0.999986i	\(-0.498297\pi\)
0.504625	+	0.863338i	\(0.331631\pi\)
\(13\)	95494.2	−	38540.4i	0.927325	−	0.374258i
							\(17\)	55167.6	+	95553.2i	0.160201	+	0.277476i	0.934941	−	0.354804i	\(-0.115453\pi\)
−0.774740	+	0.632280i	\(0.782119\pi\)
\(19\)	−21677.7	+	80902.2i	−0.0381612	+	0.142419i	−0.982378	−	0.186905i	\(-0.940154\pi\)
							0.944217	+	0.329324i	\(0.106821\pi\)
\(23\)	−1.21023e6	−	698725.i	−0.901762	−	0.520632i	−0.0239904	−	0.999712i	\(-0.507637\pi\)
							−0.877771	+	0.479080i	\(0.840970\pi\)
\(29\)	2.83305e6	−	4.90699e6i	0.743813	−	1.28832i	−0.206935	−	0.978355i	\(-0.566349\pi\)
							0.950748	−	0.309966i	\(-0.100318\pi\)
\(31\)	4.18210e6	+	4.18210e6i	0.813331	+	0.813331i	0.985132	−	0.171801i	\(-0.0549586\pi\)
							−0.171801	+	0.985132i	\(0.554959\pi\)
\(37\)	−1.59517e7	+	4.27425e6i	−1.39926	+	0.374932i	−0.878080	−	0.478513i	\(-0.841176\pi\)
							−0.521183	+	0.853445i	\(0.674509\pi\)
\(41\)	1.64467e7	−	4.40689e6i	0.908976	−	0.243559i	0.226109	−	0.974102i	\(-0.427399\pi\)
							0.682867	+	0.730543i	\(0.260733\pi\)
\(43\)	−5.51769e6	+	3.18564e6i	−0.246121	+	0.142098i	−0.617987	−	0.786188i	\(-0.712052\pi\)
							0.371866	+	0.928287i	\(0.378718\pi\)
\(47\)	2.56790e7	−	2.56790e7i	0.767606	−	0.767606i	−0.210079	−	0.977684i	\(-0.567372\pi\)
							0.977684	+	0.210079i	\(0.0673721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bc.a.6.5	✓	328
7.6	odd	2	inner	91.10.bc.a.6.6	yes	328
13.11	odd	12	inner	91.10.bc.a.76.6	yes	328
91.76	even	12	inner	91.10.bc.a.76.5	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bc.a.6.5	✓	328	1.1	even	1	trivial
91.10.bc.a.6.6	yes	328	7.6	odd	2	inner
91.10.bc.a.76.5	yes	328	91.76	even	12	inner
91.10.bc.a.76.6	yes	328	13.11	odd	12	inner