Embedded newform 11.13.d.a.2.1

• Label: 11.13.d.a.2.1
• Level: $11$
• Weight: $13$
• Character: 11.2
• Analytic conductor: $10.054$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0539319900\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	11.2
Dual form			11.13.d.a.6.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-102.224 - 33.2144i) q^{2} +(-276.412 + 200.825i) q^{3} +(6032.72 + 4383.03i) q^{4} +(-6925.60 - 21314.8i) q^{5} +(34926.2 - 11348.2i) q^{6} +(128352. - 176662. i) q^{7} +(-212330. - 292248. i) q^{8} +(-128151. + 394409. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−102.224	−	33.2144i	−1.59724	−	0.518976i	−0.630819	−	0.775930i	\(-0.717281\pi\)
							−0.966424	+	0.256954i	\(0.917281\pi\)
\(3\)	−276.412	+	200.825i	−0.379167	+	0.275481i	−0.761002	−	0.648750i	\(-0.775292\pi\)
							0.381835	+	0.924230i	\(0.375292\pi\)
\(5\)	−6925.60	−	21314.8i	−0.443239	−	1.36415i	−0.884404	−	0.466722i	\(-0.845435\pi\)
							0.441165	−	0.897426i	\(-0.354565\pi\)
\(7\)	128352.	−	176662.i	1.09098	−	1.50160i	0.244149	−	0.969738i	\(-0.421491\pi\)
							0.846829	−	0.531865i	\(-0.178509\pi\)
\(11\)	−964895.	−	1.48573e6i	−0.544658	−	0.838658i
							\(13\)	2.85246e6	+	926820.i	0.590961	+	0.192015i	0.589205	−	0.807983i	\(-0.299441\pi\)
0.00175617	+	0.999998i	\(0.499441\pi\)
\(17\)	−2.47333e7	+	8.03634e6i	−1.02468	+	0.332939i	−0.772685	−	0.634790i	\(-0.781087\pi\)
							−0.251995	+	0.967728i	\(0.581087\pi\)
\(19\)	−1.37957e7	−	1.89881e7i	−0.293239	−	0.403608i	0.636824	−	0.771009i	\(-0.280248\pi\)
							−0.930063	+	0.367401i	\(0.880248\pi\)
\(23\)	−2.03166e7			−0.137241			−0.0686204	−	0.997643i	\(-0.521860\pi\)
							−0.0686204	+	0.997643i	\(0.521860\pi\)
\(29\)	−2.97287e8	+	4.09181e8i	−0.499791	+	0.687903i	−0.982156	−	0.188067i	\(-0.939778\pi\)
							0.482366	+	0.875970i	\(0.339778\pi\)
\(31\)	−3.09488e7	+	9.52506e7i	−0.0348717	+	0.107324i	−0.966977	−	0.254863i	\(-0.917970\pi\)
							0.932106	+	0.362187i	\(0.117970\pi\)
\(37\)	3.00640e8	+	2.18427e8i	0.117175	+	0.0851328i	0.644830	−	0.764326i	\(-0.276928\pi\)
							−0.527654	+	0.849459i	\(0.676928\pi\)
\(41\)	5.62742e7	+	7.74548e7i	0.0118469	+	0.0163059i	0.814899	−	0.579602i	\(-0.196792\pi\)
							−0.803052	+	0.595908i	\(0.796792\pi\)
\(43\)			1.17244e10i			1.85473i	0.374153	+	0.927367i	\(0.377934\pi\)
							−0.374153	+	0.927367i	\(0.622066\pi\)
\(47\)	3.70325e9	−	2.69057e9i	0.343555	−	0.249607i	−0.402605	−	0.915374i	\(-0.631895\pi\)
							0.746160	+	0.665766i	\(0.231895\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.2.1	✓	44
11.6	odd	10	inner	11.13.d.a.6.1	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.1	✓	44	1.1	even	1	trivial
11.13.d.a.6.1	yes	44	11.6	odd	10	inner