Embedded newform 91.2.r.a.25.5

• Label: 91.2.r.a.25.5
• Level: $91$
• Weight: $2$
• Character: 91.25
• 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			25.5
Root			\(-0.287846 - 0.166188i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.25
Dual form			91.2.r.a.51.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.287846 - 0.166188i) q^{2} +(-0.729919 + 1.26426i) q^{3} +(-0.944763 + 1.63638i) q^{4} +(1.25195 - 0.722811i) q^{5} +0.485214i q^{6} +(2.26391 + 1.36920i) q^{7} +1.29278i q^{8} +(0.434437 + 0.752468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.287846	−	0.166188i	0.203538	−	0.117512i	−0.394767	−	0.918781i	\(-0.629175\pi\)
							0.598304	+	0.801269i	\(0.295841\pi\)
\(3\)	−0.729919	+	1.26426i	−0.421419	+	0.729919i	−0.996079	−	0.0884737i	\(-0.971801\pi\)
							0.574660	+	0.818392i	\(0.305134\pi\)
\(5\)	1.25195	−	0.722811i	0.559887	−	0.323251i	−0.193213	−	0.981157i	\(-0.561891\pi\)
							0.753100	+	0.657906i	\(0.228558\pi\)
\(7\)	2.26391	+	1.36920i	0.855677	+	0.517510i
							\(11\)	−5.15732	−	2.97758i	−1.55499	−	0.897774i	−0.997723	−	0.0674405i	\(-0.978517\pi\)
−0.557267	−	0.830333i	\(-0.688150\pi\)
\(13\)	1.88953	−	3.07078i	0.524060	−	0.851681i
							\(17\)	2.16436	−	3.74877i	0.524933	−	0.909211i	−0.474645	−	0.880177i	\(-0.657424\pi\)
0.999578	−	0.0290341i	\(-0.00924314\pi\)
\(19\)	1.69527	−	0.978767i	0.388923	−	0.224545i	−0.292771	−	0.956183i	\(-0.594577\pi\)
							0.681693	+	0.731638i	\(0.261244\pi\)
\(23\)	−0.270081	−	0.467795i	−0.0563158	−	0.0975419i	0.836493	−	0.547977i	\(-0.184602\pi\)
							−0.892809	+	0.450436i	\(0.851269\pi\)
\(29\)	7.15857			1.32931			0.664656	−	0.747149i	\(-0.268578\pi\)
							0.664656	+	0.747149i	\(0.268578\pi\)
\(31\)	5.28968	+	3.05400i	0.950055	+	0.548514i	0.893098	−	0.449862i	\(-0.148527\pi\)
							0.0569568	+	0.998377i	\(0.481860\pi\)
\(37\)	−6.95316	+	4.01441i	−1.14309	+	0.659964i	−0.947194	−	0.320660i	\(-0.896095\pi\)
							−0.195897	+	0.980624i	\(0.562762\pi\)
\(41\)		−	7.55362i		−	1.17968i	−0.807521	−	0.589839i	\(-0.799191\pi\)
							0.807521	−	0.589839i	\(-0.200809\pi\)
\(43\)	−4.24839			−0.647873			−0.323936	−	0.946079i	\(-0.605006\pi\)
							−0.323936	+	0.946079i	\(0.605006\pi\)
\(47\)	−5.42204	+	3.13042i	−0.790886	+	0.456618i	−0.840274	−	0.542161i	\(-0.817606\pi\)
							0.0493882	+	0.998780i	\(0.484273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.r.a.25.5	yes	16
3.2	odd	2		819.2.dl.e.298.4		16
7.2	even	3	inner	91.2.r.a.51.4	yes	16
7.3	odd	6		637.2.c.e.246.5		8
7.4	even	3		637.2.c.f.246.5		8
7.5	odd	6		637.2.r.f.324.4		16
7.6	odd	2		637.2.r.f.116.5		16
13.5	odd	4		1183.2.e.i.508.4		16
13.8	odd	4		1183.2.e.i.508.5		16
13.12	even	2	inner	91.2.r.a.25.4	✓	16
21.2	odd	6		819.2.dl.e.415.5		16
39.38	odd	2		819.2.dl.e.298.5		16
91.12	odd	6		637.2.r.f.324.5		16
91.18	odd	12		8281.2.a.ck.1.5		8
91.25	even	6		637.2.c.f.246.4		8
91.31	even	12		8281.2.a.cj.1.5		8
91.38	odd	6		637.2.c.e.246.4		8
91.44	odd	12		1183.2.e.i.170.4		16
91.51	even	6	inner	91.2.r.a.51.5	yes	16
91.60	odd	12		8281.2.a.ck.1.4		8
91.73	even	12		8281.2.a.cj.1.4		8
91.86	odd	12		1183.2.e.i.170.5		16
91.90	odd	2		637.2.r.f.116.4		16
273.233	odd	6		819.2.dl.e.415.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.4	✓	16	13.12	even	2	inner
91.2.r.a.25.5	yes	16	1.1	even	1	trivial
91.2.r.a.51.4	yes	16	7.2	even	3	inner
91.2.r.a.51.5	yes	16	91.51	even	6	inner
637.2.c.e.246.4		8	91.38	odd	6
637.2.c.e.246.5		8	7.3	odd	6
637.2.c.f.246.4		8	91.25	even	6
637.2.c.f.246.5		8	7.4	even	3
637.2.r.f.116.4		16	91.90	odd	2
637.2.r.f.116.5		16	7.6	odd	2
637.2.r.f.324.4		16	7.5	odd	6
637.2.r.f.324.5		16	91.12	odd	6
819.2.dl.e.298.4		16	3.2	odd	2
819.2.dl.e.298.5		16	39.38	odd	2
819.2.dl.e.415.4		16	273.233	odd	6
819.2.dl.e.415.5		16	21.2	odd	6
1183.2.e.i.170.4		16	91.44	odd	12
1183.2.e.i.170.5		16	91.86	odd	12
1183.2.e.i.508.4		16	13.5	odd	4
1183.2.e.i.508.5		16	13.8	odd	4
8281.2.a.cj.1.4		8	91.73	even	12
8281.2.a.cj.1.5		8	91.31	even	12
8281.2.a.ck.1.4		8	91.60	odd	12
8281.2.a.ck.1.5		8	91.18	odd	12