Embedded newform 11.6.c.a.3.3

• Label: 11.6.c.a.3.3
• Level: $11$
• Weight: $6$
• Character: 11.3
• 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			3.3
Root			\(-1.19154 - 3.66717i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.3
Dual form			11.6.c.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.92850 - 2.85422i) q^{2} +(-7.68717 - 23.6587i) q^{3} +(-2.60202 + 8.00819i) q^{4} +(68.6567 + 49.8820i) q^{5} +(-97.7261 - 71.0022i) q^{6} +(19.5777 - 60.2541i) q^{7} +(60.6528 + 186.670i) q^{8} +(-304.049 + 220.905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.92850	−	2.85422i	0.694467	−	0.504560i	−0.183658	−	0.982990i	\(-0.558794\pi\)
							0.878126	+	0.478430i	\(0.158794\pi\)
\(3\)	−7.68717	−	23.6587i	−0.493132	−	1.51770i	−0.819848	−	0.572582i	\(-0.805942\pi\)
							0.326716	−	0.945123i	\(-0.394058\pi\)
\(5\)	68.6567	+	49.8820i	1.22817	+	0.892317i	0.996752	−	0.0805375i	\(-0.0256637\pi\)
							0.231417	+	0.972855i	\(0.425664\pi\)
\(7\)	19.5777	−	60.2541i	0.151014	−	0.464774i	−0.846721	−	0.532037i	\(-0.821427\pi\)
							0.997735	+	0.0672633i	\(0.0214268\pi\)
\(11\)	−355.035	−	187.087i	−0.884686	−	0.466188i
							\(13\)	−247.261	+	179.646i	−0.405786	+	0.294821i	−0.771894	−	0.635752i	\(-0.780690\pi\)
0.366107	+	0.930573i	\(0.380690\pi\)
\(17\)	478.069	+	347.337i	0.401207	+	0.291494i	0.770032	−	0.638005i	\(-0.220240\pi\)
							−0.368826	+	0.929499i	\(0.620240\pi\)
\(19\)	44.2308	+	136.128i	0.0281087	+	0.0865097i	0.964127	−	0.265442i	\(-0.0855180\pi\)
							−0.936018	+	0.351952i	\(0.885518\pi\)
\(23\)	−1590.28			−0.626838			−0.313419	−	0.949615i	\(-0.601474\pi\)
							−0.313419	+	0.949615i	\(0.601474\pi\)
\(29\)	2392.19	−	7362.39i	0.528202	−	1.62564i	−0.229694	−	0.973263i	\(-0.573772\pi\)
							0.757896	−	0.652376i	\(-0.226228\pi\)
\(31\)	−2328.19	+	1691.53i	−0.435125	+	0.316137i	−0.783695	−	0.621146i	\(-0.786667\pi\)
							0.348570	+	0.937283i	\(0.386667\pi\)
\(37\)	−631.520	+	1943.62i	−0.0758373	+	0.233403i	−0.981788	−	0.189980i	\(-0.939158\pi\)
							0.905951	+	0.423383i	\(0.139158\pi\)
\(41\)	−4259.43	−	13109.2i	−0.395724	−	1.21791i	−0.928396	−	0.371591i	\(-0.878812\pi\)
							0.532673	−	0.846321i	\(-0.321188\pi\)
\(43\)	19948.8			1.64530			0.822652	−	0.568546i	\(-0.192494\pi\)
							0.822652	+	0.568546i	\(0.192494\pi\)
\(47\)	281.261	+	865.632i	0.0185723	+	0.0571595i	0.959913	−	0.280298i	\(-0.0904332\pi\)
							−0.941341	+	0.337457i	\(0.890433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.3.3	✓	16
3.2	odd	2		99.6.f.a.91.2		16
11.2	odd	10		121.6.a.i.1.6		8
11.4	even	5	inner	11.6.c.a.4.3	yes	16
11.9	even	5		121.6.a.g.1.3		8
33.2	even	10		1089.6.a.bb.1.3		8
33.20	odd	10		1089.6.a.bg.1.6		8
33.26	odd	10		99.6.f.a.37.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.3.3	✓	16	1.1	even	1	trivial
11.6.c.a.4.3	yes	16	11.4	even	5	inner
99.6.f.a.37.2		16	33.26	odd	10
99.6.f.a.91.2		16	3.2	odd	2
121.6.a.g.1.3		8	11.9	even	5
121.6.a.i.1.6		8	11.2	odd	10
1089.6.a.bb.1.3		8	33.2	even	10
1089.6.a.bg.1.6		8	33.20	odd	10