Embedded newform 11.13.d.a.2.4

• Label: 11.13.d.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	11.2
Dual form			11.13.d.a.6.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-28.7412 - 9.33860i) q^{2} +(-796.380 + 578.604i) q^{3} +(-2574.88 - 1870.76i) q^{4} +(-7517.68 - 23137.0i) q^{5} +(28292.3 - 9192.72i) q^{6} +(-91007.2 + 125261. i) q^{7} +(129293. + 177956. i) q^{8} +(135214. - 416146. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−28.7412	−	9.33860i	−0.449082	−	0.145916i	0.0757400	−	0.997128i	\(-0.475868\pi\)
							−0.524822	+	0.851212i	\(0.675868\pi\)
\(3\)	−796.380	+	578.604i	−1.09243	+	0.793695i	−0.979807	−	0.199943i	\(-0.935924\pi\)
							−0.112620	+	0.993638i	\(0.535924\pi\)
\(5\)	−7517.68	−	23137.0i	−0.481132	−	1.48077i	−0.837506	−	0.546428i	\(-0.815987\pi\)
							0.356374	−	0.934343i	\(-0.384013\pi\)
\(7\)	−91007.2	+	125261.i	−0.773548	+	1.06470i	0.222417	+	0.974952i	\(0.428605\pi\)
							−0.995965	+	0.0897459i	\(0.971395\pi\)
\(11\)	1.76973e6	−	80609.7i	0.998964	−	0.0455021i
							\(13\)	−3.63453e6	−	1.18093e6i	−0.752989	−	0.244661i	−0.0927222	−	0.995692i	\(-0.529557\pi\)
−0.660267	+	0.751031i	\(0.729557\pi\)
\(17\)	1.05365e7	−	3.42351e6i	0.436518	−	0.141833i	−0.0825107	−	0.996590i	\(-0.526294\pi\)
							0.519028	+	0.854757i	\(0.326294\pi\)
\(19\)	−1.48404e7	−	2.04260e7i	−0.315444	−	0.434172i	0.621625	−	0.783315i	\(-0.286473\pi\)
							−0.937069	+	0.349143i	\(0.886473\pi\)
\(23\)	1.43754e8			0.971077			0.485538	−	0.874215i	\(-0.338624\pi\)
							0.485538	+	0.874215i	\(0.338624\pi\)
\(29\)	−3.75104e8	+	5.16286e8i	−0.630614	+	0.867966i	−0.998072	−	0.0620746i	\(-0.980228\pi\)
							0.367458	+	0.930040i	\(0.380228\pi\)
\(31\)	1.54242e8	−	4.74707e8i	0.173793	−	0.534879i	−0.825783	−	0.563987i	\(-0.809267\pi\)
							0.999576	+	0.0291080i	\(0.00926668\pi\)
\(37\)	3.80317e9	+	2.76317e9i	1.48230	+	1.07695i	0.976807	+	0.214121i	\(0.0686885\pi\)
							0.505491	+	0.862832i	\(0.331311\pi\)
\(41\)	−3.92380e9	−	5.40064e9i	−0.826044	−	1.13695i	−0.988646	−	0.150261i	\(-0.951989\pi\)
							0.162602	−	0.986692i	\(-0.448011\pi\)
\(43\)			7.73686e9i			1.22392i	0.790888	+	0.611961i	\(0.209619\pi\)
							−0.790888	+	0.611961i	\(0.790381\pi\)
\(47\)	6.14444e9	−	4.46420e9i	0.570027	−	0.414149i	−0.265088	−	0.964224i	\(-0.585401\pi\)
							0.835115	+	0.550075i	\(0.185401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.2.4	✓	44
11.6	odd	10	inner	11.13.d.a.6.4	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.4	✓	44	1.1	even	1	trivial
11.13.d.a.6.4	yes	44	11.6	odd	10	inner