Embedded newform 11.6.c.a.5.3

• Label: 11.6.c.a.5.3
• 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			5.3
Root			\(0.136182 + 0.0989419i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.5
Dual form			11.6.c.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.361034 - 1.11115i) q^{2} +(12.7716 + 9.27912i) q^{3} +(24.7842 - 18.0068i) q^{4} +(1.31139 - 4.03604i) q^{5} +(5.69949 - 17.5412i) q^{6} +(-146.551 + 106.476i) q^{7} +(-59.2025 - 43.0132i) q^{8} +(1.92087 + 5.91183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.361034	−	1.11115i	−0.0638224	−	0.196425i	0.914061	−	0.405577i	\(-0.132930\pi\)
							−0.977883	+	0.209152i	\(0.932930\pi\)
\(3\)	12.7716	+	9.27912i	0.819299	+	0.595256i	0.916512	−	0.400008i	\(-0.130993\pi\)
							−0.0972125	+	0.995264i	\(0.530993\pi\)
\(5\)	1.31139	−	4.03604i	0.0234589	−	0.0721989i	−0.938642	−	0.344894i	\(-0.887915\pi\)
							0.962101	+	0.272695i	\(0.0879149\pi\)
\(7\)	−146.551	+	106.476i	−1.13043	+	0.821307i	−0.985758	−	0.168172i	\(-0.946213\pi\)
							−0.144674	+	0.989479i	\(0.546213\pi\)
\(11\)	−240.502	−	321.262i	−0.599291	−	0.800531i
							\(13\)	185.552	+	571.070i	0.304514	+	0.937197i	0.979858	+	0.199695i	\(0.0639950\pi\)
−0.675344	+	0.737503i	\(0.736005\pi\)
\(17\)	−58.8034	+	180.978i	−0.0493492	+	0.151881i	−0.972694	−	0.232090i	\(-0.925444\pi\)
							0.923345	+	0.383971i	\(0.125444\pi\)
\(19\)	581.843	+	422.733i	0.369761	+	0.268647i	0.757112	−	0.653285i	\(-0.226610\pi\)
							−0.387351	+	0.921933i	\(0.626610\pi\)
\(23\)	−2310.03			−0.910537			−0.455269	−	0.890354i	\(-0.650457\pi\)
							−0.455269	+	0.890354i	\(0.650457\pi\)
\(29\)	6320.18	−	4591.88i	1.39552	−	1.01390i	0.400281	−	0.916392i	\(-0.368913\pi\)
							0.995235	−	0.0975090i	\(-0.0310875\pi\)
\(31\)	2092.98	+	6441.53i	0.391166	+	1.20388i	0.931908	+	0.362695i	\(0.118143\pi\)
							−0.540742	+	0.841188i	\(0.681857\pi\)
\(37\)	2492.25	−	1810.73i	0.299287	−	0.217444i	−0.427999	−	0.903779i	\(-0.640781\pi\)
							0.727286	+	0.686335i	\(0.240781\pi\)
\(41\)	−10121.7	−	7353.86i	−0.940362	−	0.683213i	0.00814615	−	0.999967i	\(-0.497407\pi\)
							−0.948508	+	0.316754i	\(0.897407\pi\)
\(43\)	−11900.8			−0.981532			−0.490766	−	0.871292i	\(-0.663283\pi\)
							−0.490766	+	0.871292i	\(0.663283\pi\)
\(47\)	22907.4	+	16643.2i	1.51262	+	1.09898i	0.964997	+	0.262259i	\(0.0844676\pi\)
							0.547624	+	0.836724i	\(0.315532\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.6.c.a.5.3	✓	16
3.2	odd	2		99.6.f.a.82.2		16
11.3	even	5		121.6.a.g.1.5		8
11.8	odd	10		121.6.a.i.1.4		8
11.9	even	5	inner	11.6.c.a.9.3	yes	16
33.8	even	10		1089.6.a.bb.1.5		8
33.14	odd	10		1089.6.a.bg.1.4		8
33.20	odd	10		99.6.f.a.64.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.5.3	✓	16	1.1	even	1	trivial
11.6.c.a.9.3	yes	16	11.9	even	5	inner
99.6.f.a.64.2		16	33.20	odd	10
99.6.f.a.82.2		16	3.2	odd	2
121.6.a.g.1.5		8	11.3	even	5
121.6.a.i.1.4		8	11.8	odd	10
1089.6.a.bb.1.5		8	33.8	even	10
1089.6.a.bg.1.4		8	33.14	odd	10