Embedded newform 11.6.c.a.3.4

• Label: 11.6.c.a.3.4
• 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			3.4
Root			\(-2.13151 - 6.56011i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.3
Dual form			11.6.c.a.4.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.38938 - 4.64216i) q^{2} +(3.63629 + 11.1914i) q^{3} +(9.38602 - 28.8872i) q^{4} +(-51.9930 - 37.7751i) q^{5} +(75.1856 + 54.6256i) q^{6} +(-33.5385 + 103.221i) q^{7} +(3.96879 + 12.2147i) q^{8} +(84.5674 - 61.4418i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.38938	−	4.64216i	1.12949	−	0.820625i	0.143873	−	0.989596i	\(-0.454044\pi\)
							0.985621	+	0.168971i	\(0.0540445\pi\)
\(3\)	3.63629	+	11.1914i	0.233268	+	0.717925i	0.997346	+	0.0728019i	\(0.0231941\pi\)
							−0.764078	+	0.645124i	\(0.776806\pi\)
\(5\)	−51.9930	−	37.7751i	−0.930078	−	0.675741i	0.0159338	−	0.999873i	\(-0.494928\pi\)
							−0.946012	+	0.324132i	\(0.894928\pi\)
\(7\)	−33.5385	+	103.221i	−0.258701	+	0.796200i	0.734377	+	0.678742i	\(0.237475\pi\)
							−0.993078	+	0.117458i	\(0.962525\pi\)
\(11\)	−398.317	+	48.9383i	−0.992537	+	0.121946i
							\(13\)	830.047	−	603.064i	1.36221	−	0.989704i	0.363909	−	0.931434i	\(-0.381442\pi\)
0.998301	−	0.0582693i	\(-0.0185582\pi\)
\(17\)	−1047.81	−	761.277i	−0.879346	−	0.638882i	0.0537327	−	0.998555i	\(-0.482888\pi\)
							−0.933078	+	0.359673i	\(0.882888\pi\)
\(19\)	105.257	+	323.946i	0.0668906	+	0.205868i	0.978915	−	0.204268i	\(-0.0654813\pi\)
							−0.912024	+	0.410136i	\(0.865481\pi\)
\(23\)	817.577			0.322262			0.161131	−	0.986933i	\(-0.448486\pi\)
							0.161131	+	0.986933i	\(0.448486\pi\)
\(29\)	−1222.58	+	3762.70i	−0.269948	+	0.830816i	0.720563	+	0.693389i	\(0.243883\pi\)
							−0.990512	+	0.137427i	\(0.956117\pi\)
\(31\)	−3173.87	+	2305.95i	−0.593178	+	0.430969i	−0.843451	−	0.537206i	\(-0.819480\pi\)
							0.250273	+	0.968175i	\(0.419480\pi\)
\(37\)	−3790.45	+	11665.8i	−0.455183	+	1.40091i	0.415736	+	0.909485i	\(0.363524\pi\)
							−0.870920	+	0.491425i	\(0.836476\pi\)
\(41\)	−4315.18	−	13280.8i	−0.400903	−	1.23385i	−0.924268	−	0.381745i	\(-0.875323\pi\)
							0.523364	−	0.852109i	\(-0.324677\pi\)
\(43\)	−691.044			−0.0569947			−0.0284974	−	0.999594i	\(-0.509072\pi\)
							−0.0284974	+	0.999594i	\(0.509072\pi\)
\(47\)	−4530.99	−	13944.9i	−0.299191	−	0.920814i	−0.981781	−	0.190013i	\(-0.939147\pi\)
							0.682591	−	0.730801i	\(-0.260853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.3.4	✓	16
3.2	odd	2		99.6.f.a.91.1		16
11.2	odd	10		121.6.a.i.1.7		8
11.4	even	5	inner	11.6.c.a.4.4	yes	16
11.9	even	5		121.6.a.g.1.2		8
33.2	even	10		1089.6.a.bb.1.2		8
33.20	odd	10		1089.6.a.bg.1.7		8
33.26	odd	10		99.6.f.a.37.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.3.4	✓	16	1.1	even	1	trivial
11.6.c.a.4.4	yes	16	11.4	even	5	inner
99.6.f.a.37.1		16	33.26	odd	10
99.6.f.a.91.1		16	3.2	odd	2
121.6.a.g.1.2		8	11.9	even	5
121.6.a.i.1.7		8	11.2	odd	10
1089.6.a.bb.1.2		8	33.2	even	10
1089.6.a.bg.1.7		8	33.20	odd	10