Embedded newform 11.6.c.a.9.4

• Label: 11.6.c.a.9.4
• 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			9.4
Root			\(-7.26917 + 5.28136i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.9
Dual form			11.6.c.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.46756 - 7.59437i) q^{2} +(0.720120 - 0.523198i) q^{3} +(-25.6970 - 18.6700i) q^{4} +(-2.24156 - 6.89880i) q^{5} +(-2.19642 - 6.75988i) q^{6} +(136.674 + 99.2995i) q^{7} +(1.52919 - 1.11102i) q^{8} +(-74.8463 + 230.353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46756	−	7.59437i	0.436207	−	1.34251i	−0.455637	−	0.890165i	\(-0.650589\pi\)
							0.891845	−	0.452342i	\(-0.149411\pi\)
\(3\)	0.720120	−	0.523198i	0.0461957	−	0.0335631i	−0.564448	−	0.825469i	\(-0.690911\pi\)
							0.610643	+	0.791906i	\(0.290911\pi\)
\(5\)	−2.24156	−	6.89880i	−0.0400982	−	0.123410i	0.929004	−	0.370071i	\(-0.120667\pi\)
							−0.969102	+	0.246661i	\(0.920667\pi\)
\(7\)	136.674	+	99.2995i	1.05424	+	0.765953i	0.973015	−	0.230743i	\(-0.0741156\pi\)
							0.0812288	+	0.996695i	\(0.474116\pi\)
\(11\)	−328.761	−	230.146i	−0.819217	−	0.573484i
							\(13\)	−82.1221	+	252.746i	−0.134773	+	0.414787i	−0.995555	−	0.0941865i	\(-0.969975\pi\)
0.860782	+	0.508974i	\(0.169975\pi\)
\(17\)	−212.954	−	655.405i	−0.178716	−	0.550031i	0.821068	−	0.570831i	\(-0.193379\pi\)
							−0.999784	+	0.0207998i	\(0.993379\pi\)
\(19\)	−2189.10	+	1590.47i	−1.39117	+	1.01075i	−0.395436	+	0.918493i	\(0.629407\pi\)
							−0.995736	+	0.0922522i	\(0.970593\pi\)
\(23\)	1409.32			0.555507			0.277754	−	0.960652i	\(-0.410410\pi\)
							0.277754	+	0.960652i	\(0.410410\pi\)
\(29\)	2168.11	+	1575.22i	0.478724	+	0.347814i	0.800832	−	0.598890i	\(-0.204391\pi\)
							−0.322107	+	0.946703i	\(0.604391\pi\)
\(31\)	1335.02	−	4108.76i	0.249507	−	0.767903i	−0.745356	−	0.666667i	\(-0.767720\pi\)
							0.994862	−	0.101236i	\(-0.0322797\pi\)
\(37\)	636.281	+	462.285i	0.0764090	+	0.0555144i	0.625334	−	0.780357i	\(-0.284963\pi\)
							−0.548925	+	0.835872i	\(0.684963\pi\)
\(41\)	10431.5	−	7578.95i	0.969144	−	0.704125i	0.0138879	−	0.999904i	\(-0.495579\pi\)
							0.955256	+	0.295779i	\(0.0955792\pi\)
\(43\)	−20070.0			−1.65530			−0.827650	−	0.561245i	\(-0.810323\pi\)
							−0.827650	+	0.561245i	\(0.810323\pi\)
\(47\)	−15223.4	+	11060.4i	−1.00523	+	0.730344i	−0.963204	−	0.268773i	\(-0.913382\pi\)
							−0.0420282	+	0.999116i	\(0.513382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.9.4	yes	16
3.2	odd	2		99.6.f.a.64.1		16
11.4	even	5		121.6.a.g.1.8		8
11.5	even	5	inner	11.6.c.a.5.4	✓	16
11.7	odd	10		121.6.a.i.1.1		8
33.5	odd	10		99.6.f.a.82.1		16
33.26	odd	10		1089.6.a.bg.1.1		8
33.29	even	10		1089.6.a.bb.1.8		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.5.4	✓	16	11.5	even	5	inner
11.6.c.a.9.4	yes	16	1.1	even	1	trivial
99.6.f.a.64.1		16	3.2	odd	2
99.6.f.a.82.1		16	33.5	odd	10
121.6.a.g.1.8		8	11.4	even	5
121.6.a.i.1.1		8	11.7	odd	10
1089.6.a.bb.1.8		8	33.29	even	10
1089.6.a.bg.1.1		8	33.26	odd	10