Embedded newform 11.6.c.a.4.3

• Label: 11.6.c.a.4.3
• Level: $11$
• Weight: $6$
• Character: 11.4
• 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			4.3
Root			\(-1.19154 + 3.66717i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.4
Dual form			11.6.c.a.3.3

$q$-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 \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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	0.878126	−	0.478430i	\(-0.158794\pi\)
							−0.183658	+	0.982990i	\(0.558794\pi\)
\(3\)	−7.68717	+	23.6587i	−0.493132	+	1.51770i	0.326716	+	0.945123i	\(0.394058\pi\)
							−0.819848	+	0.572582i	\(0.805942\pi\)
\(5\)	68.6567	−	49.8820i	1.22817	−	0.892317i	0.231417	−	0.972855i	\(-0.425664\pi\)
							0.996752	+	0.0805375i	\(0.0256637\pi\)
\(7\)	19.5777	+	60.2541i	0.151014	+	0.464774i	0.997735	−	0.0672633i	\(-0.0214268\pi\)
							−0.846721	+	0.532037i	\(0.821427\pi\)
\(11\)	−355.035	+	187.087i	−0.884686	+	0.466188i
							\(13\)	−247.261	−	179.646i	−0.405786	−	0.294821i	0.366107	−	0.930573i	\(-0.380690\pi\)
−0.771894	+	0.635752i	\(0.780690\pi\)
\(17\)	478.069	−	347.337i	0.401207	−	0.291494i	−0.368826	−	0.929499i	\(-0.620240\pi\)
							0.770032	+	0.638005i	\(0.220240\pi\)
\(19\)	44.2308	−	136.128i	0.0281087	−	0.0865097i	−0.936018	−	0.351952i	\(-0.885518\pi\)
							0.964127	+	0.265442i	\(0.0855180\pi\)
\(23\)	−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	0.757896	+	0.652376i	\(0.226228\pi\)
							−0.229694	+	0.973263i	\(0.573772\pi\)
\(31\)	−2328.19	−	1691.53i	−0.435125	−	0.316137i	0.348570	−	0.937283i	\(-0.386667\pi\)
							−0.783695	+	0.621146i	\(0.786667\pi\)
\(37\)	−631.520	−	1943.62i	−0.0758373	−	0.233403i	0.905951	−	0.423383i	\(-0.139158\pi\)
							−0.981788	+	0.189980i	\(0.939158\pi\)
\(41\)	−4259.43	+	13109.2i	−0.395724	+	1.21791i	0.532673	+	0.846321i	\(0.321188\pi\)
							−0.928396	+	0.371591i	\(0.878812\pi\)
\(43\)	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	−0.941341	−	0.337457i	\(-0.890433\pi\)
							0.959913	+	0.280298i	\(0.0904332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.4.3	yes	16
3.2	odd	2		99.6.f.a.37.2		16
11.3	even	5	inner	11.6.c.a.3.3	✓	16
11.5	even	5		121.6.a.g.1.3		8
11.6	odd	10		121.6.a.i.1.6		8
33.5	odd	10		1089.6.a.bg.1.6		8
33.14	odd	10		99.6.f.a.91.2		16
33.17	even	10		1089.6.a.bb.1.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.3.3	✓	16	11.3	even	5	inner
11.6.c.a.4.3	yes	16	1.1	even	1	trivial
99.6.f.a.37.2		16	3.2	odd	2
99.6.f.a.91.2		16	33.14	odd	10
121.6.a.g.1.3		8	11.5	even	5
121.6.a.i.1.6		8	11.6	odd	10
1089.6.a.bb.1.3		8	33.17	even	10
1089.6.a.bg.1.6		8	33.5	odd	10