Embedded newform 91.2.r.a.51.7

• Label: 91.2.r.a.51.7
• 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			51.7
Root			\(-1.84073 + 1.06275i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.51
Dual form			91.2.r.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.84073 + 1.06275i) q^{2} +(0.0894272 + 0.154892i) q^{3} +(1.25885 + 2.18040i) q^{4} +(-3.12291 - 1.80301i) q^{5} +0.380153i q^{6} +(1.20931 + 2.35320i) q^{7} +1.10038i q^{8} +(1.48401 - 2.57037i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.84073	+	1.06275i	1.30159	+	0.751474i	0.980677	−	0.195634i	\(-0.0626764\pi\)
							0.320915	+	0.947108i	\(0.396010\pi\)
\(3\)	0.0894272	+	0.154892i	0.0516308	+	0.0894272i	0.890686	−	0.454620i	\(-0.150225\pi\)
							−0.839055	+	0.544047i	\(0.816891\pi\)
\(5\)	−3.12291	−	1.80301i	−1.39661	−	0.806332i	−0.402572	−	0.915388i	\(-0.631884\pi\)
							−0.994036	+	0.109056i	\(0.965217\pi\)
\(7\)	1.20931	+	2.35320i	0.457076	+	0.889428i
							\(11\)	−3.45748	+	1.99618i	−1.04247	+	0.601871i	−0.920532	−	0.390667i	\(-0.872244\pi\)
−0.121939	+	0.992538i	\(0.538911\pi\)
\(13\)	−2.51771	−	2.58092i	−0.698287	−	0.715818i
							\(17\)	2.39458	+	4.14753i	0.580771	+	1.00592i	0.995388	+	0.0959284i	\(0.0305820\pi\)
−0.414618	+	0.909996i	\(0.636085\pi\)
\(19\)	2.72850	+	1.57530i	0.625960	+	0.361398i	0.779186	−	0.626793i	\(-0.215633\pi\)
							−0.153226	+	0.988191i	\(0.548966\pi\)
\(23\)	−1.08943	+	1.88694i	−0.227161	+	0.393455i	−0.956966	−	0.290201i	\(-0.906278\pi\)
							0.729804	+	0.683656i	\(0.239611\pi\)
\(29\)	−6.57198			−1.22039			−0.610193	−	0.792253i	\(-0.708908\pi\)
							−0.610193	+	0.792253i	\(0.708908\pi\)
\(31\)	1.28753	−	0.743358i	0.231248	−	0.133511i	−0.379900	−	0.925028i	\(-0.624042\pi\)
							0.611148	+	0.791517i	\(0.290708\pi\)
\(37\)	4.29984	+	2.48252i	0.706890	+	0.408123i	0.809908	−	0.586556i	\(-0.199517\pi\)
							−0.103019	+	0.994679i	\(0.532850\pi\)
\(41\)		−	2.11931i		−	0.330981i	−0.986211	−	0.165490i	\(-0.947079\pi\)
							0.986211	−	0.165490i	\(-0.0529207\pi\)
\(43\)	−1.43145			−0.218294			−0.109147	−	0.994026i	\(-0.534812\pi\)
							−0.109147	+	0.994026i	\(0.534812\pi\)
\(47\)	0.882417	+	0.509464i	0.128714	+	0.0743129i	0.562974	−	0.826474i	\(-0.309657\pi\)
							−0.434261	+	0.900787i	\(0.642990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.r.a.51.7	yes	16
3.2	odd	2		819.2.dl.e.415.2		16
7.2	even	3		637.2.c.f.246.2		8
7.3	odd	6		637.2.r.f.116.2		16
7.4	even	3	inner	91.2.r.a.25.2	✓	16
7.5	odd	6		637.2.c.e.246.2		8
7.6	odd	2		637.2.r.f.324.7		16
13.5	odd	4		1183.2.e.i.170.7		16
13.8	odd	4		1183.2.e.i.170.2		16
13.12	even	2	inner	91.2.r.a.51.2	yes	16
21.11	odd	6		819.2.dl.e.298.7		16
39.38	odd	2		819.2.dl.e.415.7		16
91.5	even	12		8281.2.a.cj.1.2		8
91.12	odd	6		637.2.c.e.246.7		8
91.18	odd	12		1183.2.e.i.508.7		16
91.25	even	6	inner	91.2.r.a.25.7	yes	16
91.38	odd	6		637.2.r.f.116.7		16
91.44	odd	12		8281.2.a.ck.1.2		8
91.47	even	12		8281.2.a.cj.1.7		8
91.51	even	6		637.2.c.f.246.7		8
91.60	odd	12		1183.2.e.i.508.2		16
91.86	odd	12		8281.2.a.ck.1.7		8
91.90	odd	2		637.2.r.f.324.2		16
273.116	odd	6		819.2.dl.e.298.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.2	✓	16	7.4	even	3	inner
91.2.r.a.25.7	yes	16	91.25	even	6	inner
91.2.r.a.51.2	yes	16	13.12	even	2	inner
91.2.r.a.51.7	yes	16	1.1	even	1	trivial
637.2.c.e.246.2		8	7.5	odd	6
637.2.c.e.246.7		8	91.12	odd	6
637.2.c.f.246.2		8	7.2	even	3
637.2.c.f.246.7		8	91.51	even	6
637.2.r.f.116.2		16	7.3	odd	6
637.2.r.f.116.7		16	91.38	odd	6
637.2.r.f.324.2		16	91.90	odd	2
637.2.r.f.324.7		16	7.6	odd	2
819.2.dl.e.298.2		16	273.116	odd	6
819.2.dl.e.298.7		16	21.11	odd	6
819.2.dl.e.415.2		16	3.2	odd	2
819.2.dl.e.415.7		16	39.38	odd	2
1183.2.e.i.170.2		16	13.8	odd	4
1183.2.e.i.170.7		16	13.5	odd	4
1183.2.e.i.508.2		16	91.60	odd	12
1183.2.e.i.508.7		16	91.18	odd	12
8281.2.a.cj.1.2		8	91.5	even	12
8281.2.a.cj.1.7		8	91.47	even	12
8281.2.a.ck.1.2		8	91.44	odd	12
8281.2.a.ck.1.7		8	91.86	odd	12