Embedded newform 11.6.c.a.5.2

• Label: 11.6.c.a.5.2
• Level: $11$
• Weight: $6$
• Character: 11.5
• 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			5.2
Root			\(1.13051 + 0.821361i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.5
Dual form			11.6.c.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.740832 - 2.28005i) q^{2} +(-21.7558 - 15.8065i) q^{3} +(21.2388 - 15.4309i) q^{4} +(-5.23817 + 16.1214i) q^{5} +(-19.9222 + 61.3143i) q^{6} +(87.7560 - 63.7585i) q^{7} +(-112.982 - 82.0864i) q^{8} +(148.378 + 456.662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.740832	−	2.28005i	−0.130962	−	0.403059i	0.863978	−	0.503529i	\(-0.167965\pi\)
							−0.994940	+	0.100470i	\(0.967965\pi\)
\(3\)	−21.7558	−	15.8065i	−1.39564	−	1.01399i	−0.995220	−	0.0976567i	\(-0.968865\pi\)
							−0.400417	−	0.916333i	\(-0.631135\pi\)
\(5\)	−5.23817	+	16.1214i	−0.0937033	+	0.288389i	−0.986914	−	0.161250i	\(-0.948447\pi\)
							0.893210	+	0.449639i	\(0.148447\pi\)
\(7\)	87.7560	−	63.7585i	0.676911	−	0.491805i	−0.195420	−	0.980720i	\(-0.562607\pi\)
							0.872332	+	0.488915i	\(0.162607\pi\)
\(11\)	391.188	−	89.5726i	0.974773	−	0.223200i
							\(13\)	−235.008	−	723.279i	−0.385677	−	1.18699i	−0.935988	−	0.352032i	\(-0.885491\pi\)
0.550311	−	0.834960i	\(-0.314509\pi\)
\(17\)	−490.945	+	1510.97i	−0.412013	+	1.26805i	0.502882	+	0.864355i	\(0.332273\pi\)
							−0.914895	+	0.403691i	\(0.867727\pi\)
\(19\)	950.832	+	690.820i	0.604254	+	0.439017i	0.847386	−	0.530977i	\(-0.178175\pi\)
							−0.243132	+	0.969993i	\(0.578175\pi\)
\(23\)	1155.24			0.455356			0.227678	−	0.973736i	\(-0.426887\pi\)
							0.227678	+	0.973736i	\(0.426887\pi\)
\(29\)	−1971.40	+	1432.31i	−0.435292	+	0.316258i	−0.783761	−	0.621062i	\(-0.786702\pi\)
							0.348469	+	0.937320i	\(0.386702\pi\)
\(31\)	−577.146	−	1776.27i	−0.107865	−	0.331975i	0.882527	−	0.470262i	\(-0.155840\pi\)
							−0.990392	+	0.138287i	\(0.955840\pi\)
\(37\)	10260.8	−	7454.93i	1.23219	−	0.895240i	0.235139	−	0.971962i	\(-0.424445\pi\)
							0.997053	+	0.0767219i	\(0.0244454\pi\)
\(41\)	−2414.80	−	1754.46i	−0.224348	−	0.162998i	0.469934	−	0.882702i	\(-0.344278\pi\)
							−0.694282	+	0.719703i	\(0.744278\pi\)
\(43\)	−3678.37			−0.303378			−0.151689	−	0.988428i	\(-0.548471\pi\)
							−0.151689	+	0.988428i	\(0.548471\pi\)
\(47\)	−680.826	−	494.649i	−0.0449564	−	0.0326627i	0.565080	−	0.825036i	\(-0.308845\pi\)
							−0.610037	+	0.792373i	\(0.708845\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.5.2	✓	16
3.2	odd	2		99.6.f.a.82.3		16
11.3	even	5		121.6.a.g.1.4		8
11.8	odd	10		121.6.a.i.1.5		8
11.9	even	5	inner	11.6.c.a.9.2	yes	16
33.8	even	10		1089.6.a.bb.1.4		8
33.14	odd	10		1089.6.a.bg.1.5		8
33.20	odd	10		99.6.f.a.64.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.5.2	✓	16	1.1	even	1	trivial
11.6.c.a.9.2	yes	16	11.9	even	5	inner
99.6.f.a.64.3		16	33.20	odd	10
99.6.f.a.82.3		16	3.2	odd	2
121.6.a.g.1.4		8	11.3	even	5
121.6.a.i.1.5		8	11.8	odd	10
1089.6.a.bb.1.4		8	33.8	even	10
1089.6.a.bg.1.5		8	33.14	odd	10