Embedded newform 91.2.ba.a.89.4

• Label: 91.2.ba.a.89.4
• Level: $91$
• Weight: $2$
• Character: 91.89
• 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.ba (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			89.4
Character	\(\chi\)	\(=\)	91.89
Dual form			91.2.ba.a.45.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.193244 - 0.193244i) q^{2} +(2.17048 + 1.25313i) q^{3} -1.92531i q^{4} +(-0.383199 - 1.43012i) q^{5} +(-0.177273 - 0.661591i) q^{6} +(-2.15474 + 1.53528i) q^{7} +(-0.758543 + 0.758543i) q^{8} +(1.64065 + 2.84169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{2} - 6 q^{3} - 6 q^{5} - 12 q^{6} - 6 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.193244	−	0.193244i	−0.136644	−	0.136644i	0.635476	−	0.772120i	\(-0.280804\pi\)
							−0.772120	+	0.635476i	\(0.780804\pi\)
\(3\)	2.17048	+	1.25313i	1.25313	+	0.723493i	0.971729	−	0.236099i	\(-0.0758689\pi\)
							0.281397	+	0.959591i	\(0.409202\pi\)
\(5\)	−0.383199	−	1.43012i	−0.171372	−	0.639568i	−0.997141	−	0.0755602i	\(-0.975925\pi\)
							0.825770	−	0.564008i	\(-0.190741\pi\)
\(7\)	−2.15474	+	1.53528i	−0.814415	+	0.580282i
							\(11\)	1.18808	+	4.43396i	0.358218	+	1.33689i	0.876386	+	0.481609i	\(0.159948\pi\)
−0.518168	+	0.855279i	\(0.673386\pi\)
\(13\)	−2.80779	−	2.26193i	−0.778741	−	0.627346i
							\(17\)	2.34163			0.567928			0.283964	−	0.958835i	\(-0.408350\pi\)
0.283964	+	0.958835i	\(0.408350\pi\)
\(19\)	−1.63673	−	0.438561i	−0.375492	−	0.100613i	0.0661367	−	0.997811i	\(-0.478933\pi\)
							−0.441629	+	0.897198i	\(0.645599\pi\)
\(23\)			4.79276i			0.999360i	0.866210	+	0.499680i	\(0.166549\pi\)
							−0.866210	+	0.499680i	\(0.833451\pi\)
\(29\)	−2.87534	−	4.98024i	−0.533938	−	0.924808i	−0.999214	−	0.0396419i	\(-0.987378\pi\)
							0.465276	−	0.885166i	\(-0.345955\pi\)
\(31\)	−5.69764	−	1.52668i	−1.02333	−	0.274199i	−0.292139	−	0.956376i	\(-0.594367\pi\)
							−0.731187	+	0.682177i	\(0.761033\pi\)
\(37\)	6.03398	−	6.03398i	0.991979	−	0.991979i	−0.00798866	−	0.999968i	\(-0.502543\pi\)
							0.999968	+	0.00798866i	\(0.00254290\pi\)
\(41\)	−0.829479	−	0.222258i	−0.129543	−	0.0347109i	0.193465	−	0.981107i	\(-0.438027\pi\)
							−0.323008	+	0.946396i	\(0.604694\pi\)
\(43\)	1.70069	+	0.981895i	0.259353	+	0.149738i	0.624039	−	0.781393i	\(-0.285490\pi\)
							−0.364686	+	0.931130i	\(0.618824\pi\)
\(47\)	7.75739	−	2.07859i	1.13153	−	0.303193i	0.355990	−	0.934490i	\(-0.384144\pi\)
							0.775542	+	0.631297i	\(0.217477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.ba.a.89.4	yes	28
3.2	odd	2		819.2.et.b.271.4		28
7.2	even	3		637.2.bd.b.440.4		28
7.3	odd	6		91.2.w.a.24.4	yes	28
7.4	even	3		637.2.x.a.570.4		28
7.5	odd	6		637.2.bd.a.440.4		28
7.6	odd	2		637.2.bb.a.362.4		28
13.6	odd	12		91.2.w.a.19.4	✓	28
21.17	even	6		819.2.gh.b.388.4		28
39.32	even	12		819.2.gh.b.19.4		28
91.6	even	12		637.2.x.a.19.4		28
91.19	even	12		637.2.bd.b.97.4		28
91.32	odd	12		637.2.bb.a.227.4		28
91.45	even	12	inner	91.2.ba.a.45.4	yes	28
91.58	odd	12		637.2.bd.a.97.4		28
273.227	odd	12		819.2.et.b.136.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.w.a.19.4	✓	28	13.6	odd	12
91.2.w.a.24.4	yes	28	7.3	odd	6
91.2.ba.a.45.4	yes	28	91.45	even	12	inner
91.2.ba.a.89.4	yes	28	1.1	even	1	trivial
637.2.x.a.19.4		28	91.6	even	12
637.2.x.a.570.4		28	7.4	even	3
637.2.bb.a.227.4		28	91.32	odd	12
637.2.bb.a.362.4		28	7.6	odd	2
637.2.bd.a.97.4		28	91.58	odd	12
637.2.bd.a.440.4		28	7.5	odd	6
637.2.bd.b.97.4		28	91.19	even	12
637.2.bd.b.440.4		28	7.2	even	3
819.2.et.b.136.4		28	273.227	odd	12
819.2.et.b.271.4		28	3.2	odd	2
819.2.gh.b.19.4		28	39.32	even	12
819.2.gh.b.388.4		28	21.17	even	6