Embedded newform 11.12.c.a.4.10

• Label: 11.12.c.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	11.4
Dual form			11.12.c.a.3.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(66.9203 + 48.6204i) q^{2} +(-196.291 + 604.123i) q^{3} +(1481.51 + 4559.62i) q^{4} +(7813.25 - 5676.66i) q^{5} +(-42508.6 + 30884.3i) q^{6} +(-7835.02 - 24113.7i) q^{7} +(-70198.2 + 216048. i) q^{8} +(-183119. - 133044. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	66.9203	+	48.6204i	1.47874	+	1.07437i	0.977959	+	0.208798i	\(0.0669551\pi\)
							0.500785	+	0.865572i	\(0.333045\pi\)
\(3\)	−196.291	+	604.123i	−0.466374	+	1.43535i	0.390872	+	0.920445i	\(0.372173\pi\)
							−0.857246	+	0.514907i	\(0.827827\pi\)
\(5\)	7813.25	−	5676.66i	1.11814	−	0.812378i	0.134216	−	0.990952i	\(-0.457149\pi\)
							0.983926	+	0.178574i	\(0.0571485\pi\)
\(7\)	−7835.02	−	24113.7i	−0.176198	−	0.542281i	0.823488	−	0.567333i	\(-0.192025\pi\)
							−0.999686	+	0.0250520i	\(0.992025\pi\)
\(11\)	−141258.	−	515129.i	−0.264457	−	0.964398i
							\(13\)	1.48638e6	+	1.07992e6i	1.11030	+	0.806683i	0.982712	−	0.185143i	\(-0.0592750\pi\)
0.127593	+	0.991827i	\(0.459275\pi\)
\(17\)	−3.49376e6	+	2.53837e6i	−0.596794	+	0.433596i	−0.844739	−	0.535178i	\(-0.820245\pi\)
							0.247946	+	0.968774i	\(0.420245\pi\)
\(19\)	1.71393e6	−	5.27494e6i	0.158799	−	0.488734i	−0.839727	−	0.543009i	\(-0.817285\pi\)
							0.998526	+	0.0542754i	\(0.0172849\pi\)
\(23\)	1.29260e7			0.418755			0.209377	−	0.977835i	\(-0.432856\pi\)
							0.209377	+	0.977835i	\(0.432856\pi\)
\(29\)	5.54479e6	+	1.70651e7i	0.0501991	+	0.154497i	0.973014	−	0.230748i	\(-0.0741171\pi\)
							−0.922815	+	0.385244i	\(0.874117\pi\)
\(31\)	−4.11256e7	−	2.98795e7i	−0.258002	−	0.187449i	0.451264	−	0.892391i	\(-0.350973\pi\)
							−0.709266	+	0.704941i	\(0.750973\pi\)
\(37\)	−9.96056e7	−	3.06554e8i	−0.236142	−	0.726772i	−0.996968	−	0.0778151i	\(-0.975206\pi\)
							0.760825	−	0.648957i	\(-0.224794\pi\)
\(41\)	3.00281e8	−	9.24171e8i	0.404778	−	1.24578i	−0.516302	−	0.856406i	\(-0.672692\pi\)
							0.921080	−	0.389373i	\(-0.127308\pi\)
\(43\)	−8.19372e8			−0.849972			−0.424986	−	0.905200i	\(-0.639721\pi\)
							−0.424986	+	0.905200i	\(0.639721\pi\)
\(47\)	−8.83021e8	+	2.71766e9i	−0.561608	+	1.72845i	0.116214	+	0.993224i	\(0.462924\pi\)
							−0.677822	+	0.735226i	\(0.737076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.4.10	yes	40
3.2	odd	2		99.12.f.a.37.1		40
11.3	even	5	inner	11.12.c.a.3.10	✓	40
11.5	even	5		121.12.a.j.1.1		20
11.6	odd	10		121.12.a.h.1.20		20
33.14	odd	10		99.12.f.a.91.1		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.10	✓	40	11.3	even	5	inner
11.12.c.a.4.10	yes	40	1.1	even	1	trivial
99.12.f.a.37.1		40	3.2	odd	2
99.12.f.a.91.1		40	33.14	odd	10
121.12.a.h.1.20		20	11.6	odd	10
121.12.a.j.1.1		20	11.5	even	5