Embedded newform 91.2.bb.a.73.4

• Label: 91.2.bb.a.73.4
• Level: $91$
• Weight: $2$
• Character: 91.73
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.4
Character	\(\chi\)	\(=\)	91.73
Dual form			91.2.bb.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.186083 + 0.694471i) q^{2} +(1.44134 + 0.832160i) q^{3} +(1.28439 + 0.741542i) q^{4} +(-1.87130 - 0.501414i) q^{5} +(-0.846120 + 0.846120i) q^{6} +(-0.783278 - 2.52715i) q^{7} +(-1.77076 + 1.77076i) q^{8} +(-0.115021 - 0.199221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.186083	+	0.694471i	−0.131581	+	0.491065i	−0.999989	−	0.00478511i	\(-0.998477\pi\)
							0.868408	+	0.495850i	\(0.165144\pi\)
\(3\)	1.44134	+	0.832160i	0.832160	+	0.480448i	0.854592	−	0.519301i	\(-0.173808\pi\)
							−0.0224319	+	0.999748i	\(0.507141\pi\)
\(5\)	−1.87130	−	0.501414i	−0.836871	−	0.224239i	−0.185162	−	0.982708i	\(-0.559281\pi\)
							−0.651709	+	0.758469i	\(0.725948\pi\)
\(7\)	−0.783278	−	2.52715i	−0.296051	−	0.955172i
							\(11\)	0.825667	+	3.08143i	0.248948	+	0.929087i	0.971358	+	0.237621i	\(0.0763678\pi\)
−0.722410	+	0.691465i	\(0.756966\pi\)
\(13\)	0.846120	−	3.50487i	0.234671	−	0.972075i
							\(17\)	−0.254148	+	0.440197i	−0.0616400	+	0.106764i	−0.895199	−	0.445667i	\(-0.852966\pi\)
0.833559	+	0.552431i	\(0.186300\pi\)
\(19\)	−2.65082	−	0.710286i	−0.608141	−	0.162951i	−0.0584100	−	0.998293i	\(-0.518603\pi\)
							−0.549731	+	0.835342i	\(0.685270\pi\)
\(23\)	2.49648	−	1.44134i	0.520552	−	0.300541i	−0.216609	−	0.976259i	\(-0.569500\pi\)
							0.737160	+	0.675718i	\(0.236166\pi\)
\(29\)	−2.40426			−0.446460			−0.223230	−	0.974766i	\(-0.571660\pi\)
							−0.223230	+	0.974766i	\(0.571660\pi\)
\(31\)	0.827581	+	3.08857i	0.148638	+	0.554724i	0.999566	+	0.0294427i	\(0.00937326\pi\)
							−0.850929	+	0.525281i	\(0.823960\pi\)
\(37\)	9.40142	+	2.51910i	1.54558	+	0.414138i	0.928065	−	0.372418i	\(-0.121471\pi\)
							0.617519	+	0.786556i	\(0.288138\pi\)
\(41\)	−5.34023	+	5.34023i	−0.834004	+	0.834004i	−0.988062	−	0.154058i	\(-0.950766\pi\)
							0.154058	+	0.988062i	\(0.450766\pi\)
\(43\)			12.5736i			1.91746i	0.284313	+	0.958732i	\(0.408235\pi\)
							−0.284313	+	0.958732i	\(0.591765\pi\)
\(47\)	2.88879	−	10.7811i	0.421374	−	1.57259i	−0.350342	−	0.936622i	\(-0.613935\pi\)
							0.771716	−	0.635967i	\(-0.219399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.73.4	yes	32
3.2	odd	2		819.2.fn.e.73.5		32
7.2	even	3		637.2.bc.b.411.5		32
7.3	odd	6		637.2.i.a.489.8		32
7.4	even	3		637.2.i.a.489.7		32
7.5	odd	6	inner	91.2.bb.a.47.5	yes	32
7.6	odd	2		637.2.bc.b.619.4		32
13.5	odd	4	inner	91.2.bb.a.31.5	yes	32
21.5	even	6		819.2.fn.e.775.4		32
39.5	even	4		819.2.fn.e.577.4		32
91.5	even	12	inner	91.2.bb.a.5.4	✓	32
91.18	odd	12		637.2.i.a.538.7		32
91.31	even	12		637.2.i.a.538.8		32
91.44	odd	12		637.2.bc.b.460.4		32
91.83	even	4		637.2.bc.b.31.5		32
273.5	odd	12		819.2.fn.e.460.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.4	✓	32	91.5	even	12	inner
91.2.bb.a.31.5	yes	32	13.5	odd	4	inner
91.2.bb.a.47.5	yes	32	7.5	odd	6	inner
91.2.bb.a.73.4	yes	32	1.1	even	1	trivial
637.2.i.a.489.7		32	7.4	even	3
637.2.i.a.489.8		32	7.3	odd	6
637.2.i.a.538.7		32	91.18	odd	12
637.2.i.a.538.8		32	91.31	even	12
637.2.bc.b.31.5		32	91.83	even	4
637.2.bc.b.411.5		32	7.2	even	3
637.2.bc.b.460.4		32	91.44	odd	12
637.2.bc.b.619.4		32	7.6	odd	2
819.2.fn.e.73.5		32	3.2	odd	2
819.2.fn.e.460.5		32	273.5	odd	12
819.2.fn.e.577.4		32	39.5	even	4
819.2.fn.e.775.4		32	21.5	even	6