Embedded newform 91.2.r.a.51.8

• Label: 91.2.r.a.51.8
• Level: $91$
• Weight: $2$
• Character: 91.51
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			51.8
Root			\(-1.97871 + 1.14241i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.51
Dual form			91.2.r.a.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.97871 + 1.14241i) q^{2} +(-1.57521 - 2.72835i) q^{3} +(1.61019 + 2.78892i) q^{4} +(1.84030 + 1.06250i) q^{5} -7.19813i q^{6} +(-2.62488 + 0.331665i) q^{7} +2.78832i q^{8} +(-3.46258 + 5.99736i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.97871	+	1.14241i	1.39916	+	0.807803i	0.994304	−	0.106579i	\(-0.0339896\pi\)
							0.404852	+	0.914382i	\(0.367323\pi\)
\(3\)	−1.57521	−	2.72835i	−0.909448	−	1.57521i	−0.814832	−	0.579697i	\(-0.803171\pi\)
							−0.0946163	−	0.995514i	\(-0.530162\pi\)
\(5\)	1.84030	+	1.06250i	0.823005	+	0.475162i	0.851452	−	0.524433i	\(-0.175723\pi\)
							−0.0284464	+	0.999595i	\(0.509056\pi\)
\(7\)	−2.62488	+	0.331665i	−0.992112	+	0.125358i
							\(11\)	0.267139	−	0.154233i	0.0805454	−	0.0465029i	−0.459186	−	0.888340i	\(-0.651859\pi\)
0.539732	+	0.841837i	\(0.318526\pi\)
\(13\)	−3.22037	+	1.62148i	−0.893170	+	0.449718i
							\(17\)	−0.887368	−	1.53697i	−0.215218	−	0.372769i	0.738122	−	0.674667i	\(-0.235713\pi\)
−0.953340	+	0.301898i	\(0.902380\pi\)
\(19\)	1.54266	+	0.890653i	0.353909	+	0.204330i	0.666406	−	0.745589i	\(-0.267832\pi\)
							−0.312496	+	0.949919i	\(0.601165\pi\)
\(23\)	0.575211	−	0.996294i	0.119940	−	0.207742i	−0.799804	−	0.600261i	\(-0.795063\pi\)
							0.919744	+	0.392520i	\(0.128397\pi\)
\(29\)	2.01052			0.373345			0.186672	−	0.982422i	\(-0.440230\pi\)
							0.186672	+	0.982422i	\(0.440230\pi\)
\(31\)	3.98791	−	2.30242i	0.716251	−	0.413527i	−0.0971205	−	0.995273i	\(-0.530963\pi\)
							0.813371	+	0.581745i	\(0.197630\pi\)
\(37\)	4.79901	+	2.77071i	0.788953	+	0.455502i	0.839594	−	0.543215i	\(-0.182793\pi\)
							−0.0506410	+	0.998717i	\(0.516126\pi\)
\(41\)			6.72984i			1.05102i	0.850786	+	0.525512i	\(0.176126\pi\)
							−0.850786	+	0.525512i	\(0.823874\pi\)
\(43\)	−1.52611			−0.232729			−0.116365	−	0.993207i	\(-0.537124\pi\)
							−0.116365	+	0.993207i	\(0.537124\pi\)
\(47\)	−8.24297	−	4.75908i	−1.20236	−	0.694183i	−0.241281	−	0.970455i	\(-0.577568\pi\)
							−0.961079	+	0.276272i	\(0.910901\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.r.a.51.8	yes	16
3.2	odd	2		819.2.dl.e.415.1		16
7.2	even	3		637.2.c.f.246.1		8
7.3	odd	6		637.2.r.f.116.1		16
7.4	even	3	inner	91.2.r.a.25.1	✓	16
7.5	odd	6		637.2.c.e.246.1		8
7.6	odd	2		637.2.r.f.324.8		16
13.5	odd	4		1183.2.e.i.170.8		16
13.8	odd	4		1183.2.e.i.170.1		16
13.12	even	2	inner	91.2.r.a.51.1	yes	16
21.11	odd	6		819.2.dl.e.298.8		16
39.38	odd	2		819.2.dl.e.415.8		16
91.5	even	12		8281.2.a.cj.1.1		8
91.12	odd	6		637.2.c.e.246.8		8
91.18	odd	12		1183.2.e.i.508.8		16
91.25	even	6	inner	91.2.r.a.25.8	yes	16
91.38	odd	6		637.2.r.f.116.8		16
91.44	odd	12		8281.2.a.ck.1.1		8
91.47	even	12		8281.2.a.cj.1.8		8
91.51	even	6		637.2.c.f.246.8		8
91.60	odd	12		1183.2.e.i.508.1		16
91.86	odd	12		8281.2.a.ck.1.8		8
91.90	odd	2		637.2.r.f.324.1		16
273.116	odd	6		819.2.dl.e.298.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.1	✓	16	7.4	even	3	inner
91.2.r.a.25.8	yes	16	91.25	even	6	inner
91.2.r.a.51.1	yes	16	13.12	even	2	inner
91.2.r.a.51.8	yes	16	1.1	even	1	trivial
637.2.c.e.246.1		8	7.5	odd	6
637.2.c.e.246.8		8	91.12	odd	6
637.2.c.f.246.1		8	7.2	even	3
637.2.c.f.246.8		8	91.51	even	6
637.2.r.f.116.1		16	7.3	odd	6
637.2.r.f.116.8		16	91.38	odd	6
637.2.r.f.324.1		16	91.90	odd	2
637.2.r.f.324.8		16	7.6	odd	2
819.2.dl.e.298.1		16	273.116	odd	6
819.2.dl.e.298.8		16	21.11	odd	6
819.2.dl.e.415.1		16	3.2	odd	2
819.2.dl.e.415.8		16	39.38	odd	2
1183.2.e.i.170.1		16	13.8	odd	4
1183.2.e.i.170.8		16	13.5	odd	4
1183.2.e.i.508.1		16	91.60	odd	12
1183.2.e.i.508.8		16	91.18	odd	12
8281.2.a.cj.1.1		8	91.5	even	12
8281.2.a.cj.1.8		8	91.47	even	12
8281.2.a.ck.1.1		8	91.44	odd	12
8281.2.a.ck.1.8		8	91.86	odd	12