Embedded newform 11.12.c.a.4.8

• Label: 11.12.c.a.4.8
• Level: $11$
• Weight: $12$
• Character: 11.4
• 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			4.8
Character	\(\chi\)	\(=\)	11.4
Dual form			11.12.c.a.3.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(44.3233 + 32.2028i) q^{2} +(122.608 - 377.349i) q^{3} +(294.671 + 906.903i) q^{4} +(3995.51 - 2902.91i) q^{5} +(17586.1 - 12777.0i) q^{6} +(-12513.3 - 38512.0i) q^{7} +(18528.6 - 57025.2i) q^{8} +(15955.6 + 11592.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	44.3233	+	32.2028i	0.979416	+	0.711587i	0.957578	−	0.288174i	\(-0.0930482\pi\)
							0.0218380	+	0.999762i	\(0.493048\pi\)
\(3\)	122.608	−	377.349i	0.291308	−	0.896553i	−0.693129	−	0.720814i	\(-0.743768\pi\)
							0.984437	−	0.175739i	\(-0.0562316\pi\)
\(5\)	3995.51	−	2902.91i	0.571790	−	0.415430i	−0.263965	−	0.964532i	\(-0.585030\pi\)
							0.835755	+	0.549102i	\(0.185030\pi\)
\(7\)	−12513.3	−	38512.0i	−0.281406	−	0.866078i	−0.987453	−	0.157913i	\(-0.949523\pi\)
							0.706047	−	0.708165i	\(-0.250477\pi\)
\(11\)	226469.	+	483760.i	0.423983	+	0.905670i
							\(13\)	252147.	+	183195.i	0.188350	+	0.136844i	0.677964	−	0.735095i	\(-0.262862\pi\)
−0.489614	+	0.871939i	\(0.662862\pi\)
\(17\)	−1.46145e6	+	1.06180e6i	−0.249640	+	0.181374i	−0.705567	−	0.708643i	\(-0.749308\pi\)
							0.455927	+	0.890017i	\(0.349308\pi\)
\(19\)	−1.83799e6	+	5.65675e6i	−0.170294	+	0.524110i	−0.999387	−	0.0349997i	\(-0.988857\pi\)
							0.829094	+	0.559110i	\(0.188857\pi\)
\(23\)	−4.04122e7			−1.30921			−0.654605	−	0.755971i	\(-0.727165\pi\)
							−0.654605	+	0.755971i	\(0.727165\pi\)
\(29\)	3.60094e6	+	1.10826e7i	0.0326007	+	0.100335i	0.966033	−	0.258419i	\(-0.0832017\pi\)
							−0.933432	+	0.358754i	\(0.883202\pi\)
\(31\)	−2.25188e7	−	1.63609e7i	−0.141272	−	0.102640i	0.514905	−	0.857247i	\(-0.327827\pi\)
							−0.656176	+	0.754607i	\(0.727827\pi\)
\(37\)	1.34028e8	+	4.12497e8i	0.317751	+	0.977937i	0.974607	+	0.223921i	\(0.0718858\pi\)
							−0.656856	+	0.754016i	\(0.728114\pi\)
\(41\)	−1.06627e8	+	3.28165e8i	−0.143733	+	0.442365i	−0.996846	−	0.0793613i	\(-0.974712\pi\)
							0.853113	+	0.521727i	\(0.174712\pi\)
\(43\)	1.72973e9			1.79432			0.897162	−	0.441702i	\(-0.145625\pi\)
							0.897162	+	0.441702i	\(0.145625\pi\)
\(47\)	−8.73444e8	+	2.68818e9i	−0.555516	+	1.70970i	0.139060	+	0.990284i	\(0.455592\pi\)
							−0.694576	+	0.719419i	\(0.744408\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.4.8	yes	40
3.2	odd	2		99.12.f.a.37.3		40
11.3	even	5	inner	11.12.c.a.3.8	✓	40
11.5	even	5		121.12.a.j.1.5		20
11.6	odd	10		121.12.a.h.1.16		20
33.14	odd	10		99.12.f.a.91.3		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.8	✓	40	11.3	even	5	inner
11.12.c.a.4.8	yes	40	1.1	even	1	trivial
99.12.f.a.37.3		40	3.2	odd	2
99.12.f.a.91.3		40	33.14	odd	10
121.12.a.h.1.16		20	11.6	odd	10
121.12.a.j.1.5		20	11.5	even	5