Embedded newform 11.14.c.a.5.4

• Label: 11.14.c.a.5.4
• Level: $11$
• Weight: $14$
• Character: 11.5
• Analytic conductor: $11.795$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7954021847\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	11.5
Dual form			11.14.c.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.2128 - 89.9077i) q^{2} +(-1583.89 - 1150.76i) q^{3} +(-602.533 + 437.766i) q^{4} +(-19123.6 + 58856.4i) q^{5} +(-57192.5 + 176020. i) q^{6} +(119496. - 86818.8i) q^{7} +(-569564. - 413813. i) q^{8} +(691774. + 2.12906e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.2128	−	89.9077i	−0.322759	−	0.993349i	−0.972442	−	0.233144i	\(-0.925099\pi\)
							0.649684	−	0.760205i	\(-0.274901\pi\)
\(3\)	−1583.89	−	1150.76i	−1.25440	−	0.911375i	−0.255931	−	0.966695i	\(-0.582382\pi\)
							−0.998469	+	0.0553199i	\(0.982382\pi\)
\(5\)	−19123.6	+	58856.4i	−0.547350	+	1.68457i	0.167987	+	0.985789i	\(0.446273\pi\)
							−0.715337	+	0.698780i	\(0.753727\pi\)
\(7\)	119496.	−	86818.8i	0.383898	−	0.278918i	−0.379053	−	0.925375i	\(-0.623750\pi\)
							0.762950	+	0.646457i	\(0.223750\pi\)
\(11\)	−2.88585e6	−	5.11806e6i	−0.491158	−	0.871070i
							\(13\)	3.06318e6	+	9.42750e6i	0.176011	+	0.541707i	0.999678	−	0.0253667i	\(-0.00807533\pi\)
−0.823667	+	0.567074i	\(0.808075\pi\)
\(17\)	1.20760e7	−	3.71660e7i	0.121340	−	0.373446i	−0.871877	−	0.489726i	\(-0.837097\pi\)
							0.993216	+	0.116280i	\(0.0370970\pi\)
\(19\)	2.54201e8	+	1.84688e8i	1.23959	+	0.900616i	0.997571	−	0.0696555i	\(-0.0221900\pi\)
							0.242020	+	0.970271i	\(0.422190\pi\)
\(23\)	2.30121e8			0.324134			0.162067	−	0.986780i	\(-0.448184\pi\)
							0.162067	+	0.986780i	\(0.448184\pi\)
\(29\)	1.07996e9	−	7.84640e8i	0.337149	−	0.244953i	−0.406309	−	0.913736i	\(-0.633184\pi\)
							0.743458	+	0.668782i	\(0.233184\pi\)
\(31\)	2.76527e9	+	8.51063e9i	0.559612	+	1.72231i	0.683443	+	0.730004i	\(0.260482\pi\)
							−0.123831	+	0.992303i	\(0.539518\pi\)
\(37\)	−5.21084e9	+	3.78590e9i	−0.333885	+	0.242582i	−0.742077	−	0.670314i	\(-0.766159\pi\)
							0.408192	+	0.912896i	\(0.366159\pi\)
\(41\)	2.13470e10	+	1.55095e10i	0.701847	+	0.509922i	0.880533	−	0.473984i	\(-0.157185\pi\)
							−0.178686	+	0.983906i	\(0.557185\pi\)
\(43\)	6.09429e10			1.47021			0.735103	−	0.677955i	\(-0.237134\pi\)
							0.735103	+	0.677955i	\(0.237134\pi\)
\(47\)	−4.31202e10	−	3.13287e10i	−0.583505	−	0.423941i	0.256481	−	0.966549i	\(-0.417437\pi\)
							−0.839986	+	0.542608i	\(0.817437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.5.4	✓	48
11.3	even	5		121.14.a.h.1.8		24
11.8	odd	10		121.14.a.i.1.17		24
11.9	even	5	inner	11.14.c.a.9.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.5.4	✓	48	1.1	even	1	trivial
11.14.c.a.9.4	yes	48	11.9	even	5	inner
121.14.a.h.1.8		24	11.3	even	5
121.14.a.i.1.17		24	11.8	odd	10