Embedded newform 91.10.bb.a.5.20

• Label: 91.10.bb.a.5.20
• 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			5.20
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.27837 - 27.1633i) q^{2} +(-177.038 + 102.213i) q^{3} +(-241.463 + 139.408i) q^{4} +(171.168 - 45.8642i) q^{5} +(4064.99 + 4064.99i) q^{6} +(-5809.82 - 2568.98i) q^{7} +(-4636.82 - 4636.82i) q^{8} +(11053.5 - 19145.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.27837	−	27.1633i	−0.321662	−	1.20046i	−0.917625	−	0.397446i	\(-0.869897\pi\)
							0.595964	−	0.803011i	\(-0.296770\pi\)
\(3\)	−177.038	+	102.213i	−1.26189	+	0.728553i	−0.973440	−	0.228942i	\(-0.926473\pi\)
							−0.288450	+	0.957495i	\(0.593140\pi\)
\(5\)	171.168	−	45.8642i	0.122478	−	0.0328177i	−0.197060	−	0.980392i	\(-0.563139\pi\)
							0.319537	+	0.947574i	\(0.396473\pi\)
\(7\)	−5809.82	−	2568.98i	−0.914579	−	0.404407i
							\(11\)	−10352.1	+	38634.4i	−0.213186	+	0.795622i	0.773611	+	0.633661i	\(0.218449\pi\)
−0.986797	+	0.161961i	\(0.948218\pi\)
\(13\)	36163.8	+	96419.3i	0.351179	+	0.936308i
							\(17\)	27883.2	+	48295.2i	0.0809698	+	0.140244i	0.903667	−	0.428236i	\(-0.140865\pi\)
−0.822697	+	0.568480i	\(0.807532\pi\)
\(19\)	266623.	−	71441.4i	0.469360	−	0.125765i	−0.0163839	−	0.999866i	\(-0.505215\pi\)
							0.485744	+	0.874101i	\(0.338549\pi\)
\(23\)	1.78374e6	+	1.02984e6i	1.32909	+	0.767353i	0.985160	−	0.171641i	\(-0.0549068\pi\)
							0.343935	+	0.938994i	\(0.388240\pi\)
\(29\)	−1.72921e6			−0.454000			−0.227000	−	0.973895i	\(-0.572892\pi\)
							−0.227000	+	0.973895i	\(0.572892\pi\)
\(31\)	34303.2	−	128021.i	0.00667125	−	0.0248974i	−0.962510	−	0.271246i	\(-0.912564\pi\)
							0.969181	+	0.246349i	\(0.0792309\pi\)
\(37\)	4.61265e6	−	1.23596e6i	0.404616	−	0.108416i	−0.0507707	−	0.998710i	\(-0.516168\pi\)
							0.455386	+	0.890294i	\(0.349501\pi\)
\(41\)	−4.86772e6	−	4.86772e6i	−0.269029	−	0.269029i	0.559680	−	0.828709i	\(-0.310924\pi\)
							−0.828709	+	0.559680i	\(0.810924\pi\)
\(43\)			1.69761e7i			0.757233i	0.925554	+	0.378617i	\(0.123600\pi\)
							−0.925554	+	0.378617i	\(0.876400\pi\)
\(47\)	1.53980e6	+	5.74660e6i	0.0460281	+	0.171779i	0.985114	−	0.171904i	\(-0.0549920\pi\)
							−0.939086	+	0.343683i	\(0.888325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.20	✓	328
7.3	odd	6	inner	91.10.bb.a.31.63	yes	328
13.8	odd	4	inner	91.10.bb.a.47.63	yes	328
91.73	even	12	inner	91.10.bb.a.73.20	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.20	✓	328	1.1	even	1	trivial
91.10.bb.a.31.63	yes	328	7.3	odd	6	inner
91.10.bb.a.47.63	yes	328	13.8	odd	4	inner
91.10.bb.a.73.20	yes	328	91.73	even	12	inner