Embedded newform 11.6.c.a.9.1

• Label: 11.6.c.a.9.1
• Level: $11$
• Weight: $6$
• Character: 11.9
• Analytic conductor: $1.764$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.76422201794\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 5*x^15 + 86*x^14 - 146*x^13 + 7205*x^12 - 23732*x^11 + 774165*x^10 - 1228996*x^9 + 44649817*x^8 - 92720004*x^7 + 943915592*x^6 - 1892495035*x^5 + 8611100939*x^4 - 17760110698*x^3 + 18348896492*x^2 - 4224271656*x + 393784336
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			9.1
Root			\(7.81150 - 5.67539i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.9
Dual form			11.6.c.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.29275 + 10.1340i) q^{2} +(2.26410 - 1.64496i) q^{3} +(-65.9678 - 47.9284i) q^{4} +(20.4388 + 62.9043i) q^{5} +(9.21501 + 28.3609i) q^{6} +(58.3278 + 42.3777i) q^{7} +(427.066 - 310.282i) q^{8} +(-72.6709 + 223.658i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.29275	+	10.1340i	−0.582081	+	1.79146i	0.0286101	+	0.999591i	\(0.490892\pi\)
							−0.610691	+	0.791869i	\(0.709108\pi\)
\(3\)	2.26410	−	1.64496i	0.145242	−	0.105525i	−0.512791	−	0.858513i	\(-0.671389\pi\)
							0.658033	+	0.752989i	\(0.271389\pi\)
\(5\)	20.4388	+	62.9043i	0.365621	+	1.12527i	0.949591	+	0.313491i	\(0.101499\pi\)
							−0.583970	+	0.811775i	\(0.698501\pi\)
\(7\)	58.3278	+	42.3777i	0.449915	+	0.326883i	0.789562	−	0.613670i	\(-0.210308\pi\)
							−0.339647	+	0.940553i	\(0.610308\pi\)
\(11\)	369.555	−	156.461i	0.920868	−	0.389874i
							\(13\)	177.228	−	545.453i	0.290854	−	0.895156i	−0.693729	−	0.720236i	\(-0.744033\pi\)
0.984583	−	0.174920i	\(-0.0559666\pi\)
\(17\)	−84.6769	−	260.609i	−0.0710628	−	0.218709i	0.909217	−	0.416322i	\(-0.136681\pi\)
							−0.980280	+	0.197613i	\(0.936681\pi\)
\(19\)	404.202	−	293.670i	0.256871	−	0.186628i	−0.451896	−	0.892071i	\(-0.649252\pi\)
							0.708766	+	0.705443i	\(0.249252\pi\)
\(23\)	−605.239			−0.238565			−0.119283	−	0.992860i	\(-0.538059\pi\)
							−0.119283	+	0.992860i	\(0.538059\pi\)
\(29\)	43.2242	+	31.4042i	0.00954403	+	0.00693415i	0.592547	−	0.805536i	\(-0.298122\pi\)
							−0.583003	+	0.812470i	\(0.698122\pi\)
\(31\)	−1253.55	+	3858.02i	−0.234281	+	0.721042i	0.762935	+	0.646475i	\(0.223758\pi\)
							−0.997216	+	0.0745673i	\(0.976242\pi\)
\(37\)	−11517.7	−	8368.08i	−1.38312	−	1.00490i	−0.996581	−	0.0826252i	\(-0.973670\pi\)
							−0.386542	−	0.922272i	\(-0.626330\pi\)
\(41\)	6373.96	−	4630.95i	0.592174	−	0.430240i	−0.250918	−	0.968008i	\(-0.580733\pi\)
							0.843093	+	0.537769i	\(0.180733\pi\)
\(43\)	10231.1			0.843820			0.421910	−	0.906638i	\(-0.361360\pi\)
							0.421910	+	0.906638i	\(0.361360\pi\)
\(47\)	12580.0	−	9139.87i	0.830681	−	0.603525i	−0.0890707	−	0.996025i	\(-0.528390\pi\)
							0.919752	+	0.392500i	\(0.128390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.9.1	yes	16
3.2	odd	2		99.6.f.a.64.4		16
11.4	even	5		121.6.a.g.1.1		8
11.5	even	5	inner	11.6.c.a.5.1	✓	16
11.7	odd	10		121.6.a.i.1.8		8
33.5	odd	10		99.6.f.a.82.4		16
33.26	odd	10		1089.6.a.bg.1.8		8
33.29	even	10		1089.6.a.bb.1.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.5.1	✓	16	11.5	even	5	inner
11.6.c.a.9.1	yes	16	1.1	even	1	trivial
99.6.f.a.64.4		16	3.2	odd	2
99.6.f.a.82.4		16	33.5	odd	10
121.6.a.g.1.1		8	11.4	even	5
121.6.a.i.1.8		8	11.7	odd	10
1089.6.a.bb.1.1		8	33.29	even	10
1089.6.a.bg.1.8		8	33.26	odd	10