Embedded newform 11.12.c.a.3.2

• Label: 11.12.c.a.3.2
• Level: $11$
• Weight: $12$
• Character: 11.3
• Analytic conductor: $8.452$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.45177498616\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.2
Character	\(\chi\)	\(=\)	11.3
Dual form			11.12.c.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-55.1898 + 40.0977i) q^{2} +(-247.284 - 761.061i) q^{3} +(805.219 - 2478.21i) q^{4} +(-3143.04 - 2283.56i) q^{5} +(44164.4 + 32087.3i) q^{6} +(15890.9 - 48907.1i) q^{7} +(11757.6 + 36186.1i) q^{8} +(-374750. + 272272. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−55.1898	+	40.0977i	−1.21953	+	0.886043i	−0.996061	−	0.0886669i	\(-0.971739\pi\)
							−0.223473	+	0.974710i	\(0.571739\pi\)
\(3\)	−247.284	−	761.061i	−0.587528	−	1.80823i	−0.588871	−	0.808227i	\(-0.700427\pi\)
							0.00134266	−	0.999999i	\(-0.499573\pi\)
\(5\)	−3143.04	−	2283.56i	−0.449796	−	0.326796i	0.339719	−	0.940527i	\(-0.389668\pi\)
							−0.789515	+	0.613731i	\(0.789668\pi\)
\(7\)	15890.9	−	48907.1i	0.357362	−	1.09985i	−0.597265	−	0.802044i	\(-0.703746\pi\)
							0.954627	−	0.297804i	\(-0.0962542\pi\)
\(11\)	−525778.	−	94177.4i	−0.984334	−	0.176314i
							\(13\)	521612.	−	378974.i	0.389636	−	0.283087i	−0.375670	−	0.926753i	\(-0.622587\pi\)
0.765306	+	0.643666i	\(0.222587\pi\)
\(17\)	369487.	+	268448.i	0.0631147	+	0.0458555i	0.618895	−	0.785474i	\(-0.287581\pi\)
							−0.555781	+	0.831329i	\(0.687581\pi\)
\(19\)	5.09018e6	+	1.56660e7i	0.471616	+	1.45149i	0.850467	+	0.526028i	\(0.176319\pi\)
							−0.378851	+	0.925458i	\(0.623681\pi\)
\(23\)	−4.66326e6			−0.151073			−0.0755364	−	0.997143i	\(-0.524067\pi\)
							−0.0755364	+	0.997143i	\(0.524067\pi\)
\(29\)	1.21724e7	−	3.74627e7i	0.110201	−	0.339165i	−0.880715	−	0.473647i	\(-0.842937\pi\)
							0.990916	+	0.134483i	\(0.0429372\pi\)
\(31\)	3.97043e7	−	2.88469e7i	0.249086	−	0.180971i	−0.456236	−	0.889859i	\(-0.650803\pi\)
							0.705321	+	0.708888i	\(0.250803\pi\)
\(37\)	1.09690e8	−	3.37592e8i	0.260051	−	0.800355i	−0.732741	−	0.680507i	\(-0.761759\pi\)
							0.992792	−	0.119848i	\(-0.0382406\pi\)
\(41\)	−8.96832e7	−	2.76016e8i	−0.120893	−	0.372069i	0.872238	−	0.489082i	\(-0.162668\pi\)
							−0.993130	+	0.117013i	\(0.962668\pi\)
\(43\)	−3.00805e8			−0.312039			−0.156020	−	0.987754i	\(-0.549866\pi\)
							−0.156020	+	0.987754i	\(0.549866\pi\)
\(47\)	5.37252e8	+	1.65349e9i	0.341696	+	1.05163i	0.963329	+	0.268323i	\(0.0864695\pi\)
							−0.621633	+	0.783309i	\(0.713531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.3.2	✓	40
3.2	odd	2		99.12.f.a.91.9		40
11.2	odd	10		121.12.a.h.1.4		20
11.4	even	5	inner	11.12.c.a.4.2	yes	40
11.9	even	5		121.12.a.j.1.17		20
33.26	odd	10		99.12.f.a.37.9		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.2	✓	40	1.1	even	1	trivial
11.12.c.a.4.2	yes	40	11.4	even	5	inner
99.12.f.a.37.9		40	33.26	odd	10
99.12.f.a.91.9		40	3.2	odd	2
121.12.a.h.1.4		20	11.2	odd	10
121.12.a.j.1.17		20	11.9	even	5