Embedded newform 91.10.bb.a.31.63

• Label: 91.10.bb.a.31.63
• Level: $91$
• Weight: $10$
• Character: 91.31
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• 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			31.63
Character	\(\chi\)	\(=\)	91.31
Dual form			91.10.bb.a.47.63

$q$-expansion

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

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	27.1633	+	7.27837i	1.20046	+	0.321662i	0.803011	−	0.595964i	\(-0.203230\pi\)
							0.397446	+	0.917625i	\(0.369897\pi\)
\(3\)	−177.038	−	102.213i	−1.26189	−	0.728553i	−0.288450	−	0.957495i	\(-0.593140\pi\)
							−0.973440	+	0.228942i	\(0.926473\pi\)
\(5\)	45.8642	−	171.168i	0.0328177	−	0.122478i	−0.947574	−	0.319537i	\(-0.896473\pi\)
							0.980392	+	0.197060i	\(0.0631393\pi\)
\(7\)	−2568.98	−	5809.82i	−0.404407	−	0.914579i
							\(11\)	38634.4	−	10352.1i	0.795622	−	0.213186i	0.161961	−	0.986797i	\(-0.448218\pi\)
0.633661	+	0.773611i	\(0.281551\pi\)
\(13\)	−36163.8	−	96419.3i	−0.351179	−	0.936308i
							\(17\)	−27883.2	+	48295.2i	−0.0809698	+	0.140244i	−0.903667	−	0.428236i	\(-0.859135\pi\)
0.822697	+	0.568480i	\(0.192468\pi\)
\(19\)	71441.4	−	266623.i	0.125765	−	0.469360i	−0.874101	−	0.485744i	\(-0.838549\pi\)
							0.999866	+	0.0163839i	\(0.00521540\pi\)
\(23\)	−1.78374e6	+	1.02984e6i	−1.32909	+	0.767353i	−0.985160	−	0.171641i	\(-0.945093\pi\)
							−0.343935	+	0.938994i	\(0.611760\pi\)
\(29\)	−1.72921e6			−0.454000			−0.227000	−	0.973895i	\(-0.572892\pi\)
							−0.227000	+	0.973895i	\(0.572892\pi\)
\(31\)	128021.	−	34303.2i	0.0248974	−	0.00667125i	−0.246349	−	0.969181i	\(-0.579231\pi\)
							0.271246	+	0.962510i	\(0.412564\pi\)
\(37\)	−1.23596e6	+	4.61265e6i	−0.108416	+	0.404616i	−0.998710	−	0.0507707i	\(-0.983832\pi\)
							0.890294	+	0.455386i	\(0.150499\pi\)
\(41\)	4.86772e6	+	4.86772e6i	0.269029	+	0.269029i	0.828709	−	0.559680i	\(-0.189076\pi\)
							−0.559680	+	0.828709i	\(0.689076\pi\)
\(43\)			1.69761e7i			0.757233i	0.925554	+	0.378617i	\(0.123600\pi\)
							−0.925554	+	0.378617i	\(0.876400\pi\)
\(47\)	5.74660e6	+	1.53980e6i	0.171779	+	0.0460281i	0.343683	−	0.939086i	\(-0.388325\pi\)
							−0.171904	+	0.985114i	\(0.554992\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.20	✓	328	7.5	odd	6	inner
91.10.bb.a.31.63	yes	328	1.1	even	1	trivial
91.10.bb.a.47.63	yes	328	91.47	even	12	inner
91.10.bb.a.73.20	yes	328	13.8	odd	4	inner