Embedded newform 91.2.w.a.24.1

• Label: 91.2.w.a.24.1
• Level: $91$
• Weight: $2$
• Character: 91.24
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			24.1
Character	\(\chi\)	\(=\)	91.24
Dual form			91.2.w.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38212 + 0.638288i) q^{2} -0.168862i q^{3} +(3.53505 - 2.04096i) q^{4} +(-2.38343 - 0.638637i) q^{5} +(0.107782 + 0.402250i) q^{6} +(0.932289 - 2.47605i) q^{7} +(-3.63054 + 3.63054i) q^{8} +2.97149 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{2} - 6 q^{4} - 6 q^{5} + 12 q^{6} + 2 q^{7} - 4 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38212	+	0.638288i	−1.68442	+	0.451338i	−0.968939	−	0.247299i	\(-0.920457\pi\)
							−0.715477	+	0.698637i	\(0.753790\pi\)
\(3\)		−	0.168862i		−	0.0974924i	−0.998811	−	0.0487462i	\(-0.984477\pi\)
							0.998811	−	0.0487462i	\(-0.0155226\pi\)
\(5\)	−2.38343	−	0.638637i	−1.06590	−	0.285607i	−0.317093	−	0.948394i	\(-0.602707\pi\)
							−0.748808	+	0.662787i	\(0.769373\pi\)
\(7\)	0.932289	−	2.47605i	0.352372	−	0.935860i
							\(11\)	3.22245	−	3.22245i	0.971606	−	0.971606i	−0.0280017	−	0.999608i	\(-0.508914\pi\)
0.999608	+	0.0280017i	\(0.00891439\pi\)
\(13\)	−1.54835	−	3.25617i	−0.429434	−	0.903098i
							\(17\)	0.0563551	+	0.0976099i	0.0136681	+	0.0236739i	0.872779	−	0.488116i	\(-0.162316\pi\)
−0.859110	+	0.511790i	\(0.828983\pi\)
\(19\)	−2.43524	+	2.43524i	−0.558683	+	0.558683i	−0.928932	−	0.370250i	\(-0.879272\pi\)
							0.370250	+	0.928932i	\(0.379272\pi\)
\(23\)	−0.565076	−	0.326247i	−0.117826	−	0.0680271i	0.439928	−	0.898033i	\(-0.355004\pi\)
							−0.557755	+	0.830006i	\(0.688337\pi\)
\(29\)	−2.82213	−	4.88807i	−0.524056	−	0.907691i	−0.999608	−	0.0280035i	\(-0.991085\pi\)
							0.475552	−	0.879688i	\(-0.342248\pi\)
\(31\)	1.43336	+	5.34937i	0.257439	+	0.960774i	0.966717	+	0.255846i	\(0.0823541\pi\)
							−0.709279	+	0.704928i	\(0.750979\pi\)
\(37\)	0.402453	+	1.50197i	0.0661628	+	0.246923i	0.991085	−	0.133235i	\(-0.0425364\pi\)
							−0.924922	+	0.380158i	\(0.875870\pi\)
\(41\)	10.6793	+	2.86152i	1.66783	+	0.446894i	0.964525	−	0.263993i	\(-0.0850396\pi\)
							0.703306	+	0.710887i	\(0.251706\pi\)
\(43\)	6.08601	+	3.51376i	0.928108	+	0.535843i	0.886213	−	0.463279i	\(-0.153327\pi\)
							0.0418951	+	0.999122i	\(0.486660\pi\)
\(47\)	−1.53354	+	5.72325i	−0.223690	+	0.834822i	0.759235	+	0.650816i	\(0.225573\pi\)
							−0.982925	+	0.184006i	\(0.941093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.w.a.24.1	yes	28
3.2	odd	2		819.2.gh.b.388.7		28
7.2	even	3		637.2.bb.a.362.7		28
7.3	odd	6		637.2.bd.b.440.7		28
7.4	even	3		637.2.bd.a.440.7		28
7.5	odd	6		91.2.ba.a.89.7	yes	28
7.6	odd	2		637.2.x.a.570.1		28
13.6	odd	12		91.2.ba.a.45.7	yes	28
21.5	even	6		819.2.et.b.271.1		28
39.32	even	12		819.2.et.b.136.1		28
91.6	even	12		637.2.bb.a.227.7		28
91.19	even	12	inner	91.2.w.a.19.1	✓	28
91.32	odd	12		637.2.bd.b.97.7		28
91.45	even	12		637.2.bd.a.97.7		28
91.58	odd	12		637.2.x.a.19.1		28
273.110	odd	12		819.2.gh.b.19.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.w.a.19.1	✓	28	91.19	even	12	inner
91.2.w.a.24.1	yes	28	1.1	even	1	trivial
91.2.ba.a.45.7	yes	28	13.6	odd	12
91.2.ba.a.89.7	yes	28	7.5	odd	6
637.2.x.a.19.1		28	91.58	odd	12
637.2.x.a.570.1		28	7.6	odd	2
637.2.bb.a.227.7		28	91.6	even	12
637.2.bb.a.362.7		28	7.2	even	3
637.2.bd.a.97.7		28	91.45	even	12
637.2.bd.a.440.7		28	7.4	even	3
637.2.bd.b.97.7		28	91.32	odd	12
637.2.bd.b.440.7		28	7.3	odd	6
819.2.et.b.136.1		28	39.32	even	12
819.2.et.b.271.1		28	21.5	even	6
819.2.gh.b.19.7		28	273.110	odd	12
819.2.gh.b.388.7		28	3.2	odd	2