Embedded newform 11.13.d.a.7.3

• Label: 11.13.d.a.7.3
• Level: $11$
• Weight: $13$
• Character: 11.7
• 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			7.3
Character	\(\chi\)	\(=\)	11.7
Dual form			11.13.d.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-43.8972 + 60.4193i) q^{2} +(-68.3953 + 210.499i) q^{3} +(-457.792 - 1408.94i) q^{4} +(-11528.2 + 8375.76i) q^{5} +(-9715.84 - 13372.7i) q^{6} +(-104745. + 34033.7i) q^{7} +(-185704. - 60338.9i) q^{8} +(390313. + 283579. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−43.8972	+	60.4193i	−0.685893	+	0.944051i	−0.999986	−	0.00534144i	\(-0.998300\pi\)
							0.314093	+	0.949392i	\(0.398300\pi\)
\(3\)	−68.3953	+	210.499i	−0.0938207	+	0.288750i	−0.986945	−	0.161060i	\(-0.948509\pi\)
							0.893124	+	0.449811i	\(0.148509\pi\)
\(5\)	−11528.2	+	8375.76i	−0.737807	+	0.536048i	−0.892024	−	0.451989i	\(-0.850715\pi\)
							0.154216	+	0.988037i	\(0.450715\pi\)
\(7\)	−104745.	+	34033.7i	−0.890317	+	0.289282i	−0.718235	−	0.695801i	\(-0.755050\pi\)
							−0.172082	+	0.985083i	\(0.555050\pi\)
\(11\)	988712.	−	1.46999e6i	0.558102	−	0.829772i
							\(13\)	2.00322e6	−	2.75720e6i	0.415020	−	0.571226i	−0.549414	−	0.835550i	\(-0.685149\pi\)
0.964434	+	0.264324i	\(0.0851489\pi\)
\(17\)	2.67996e6	+	3.68864e6i	0.111028	+	0.152817i	0.860915	−	0.508749i	\(-0.169892\pi\)
							−0.749886	+	0.661567i	\(0.769892\pi\)
\(19\)	−5.66366e7	−	1.84023e7i	−1.20386	−	0.391157i	−0.362680	−	0.931914i	\(-0.618138\pi\)
							−0.841179	+	0.540756i	\(0.818138\pi\)
\(23\)	5.88252e7			0.397371			0.198685	−	0.980063i	\(-0.436333\pi\)
							0.198685	+	0.980063i	\(0.436333\pi\)
\(29\)	−9.20426e7	+	2.99064e7i	−0.154739	+	0.0502779i	−0.385362	−	0.922765i	\(-0.625924\pi\)
							0.230623	+	0.973043i	\(0.425924\pi\)
\(31\)	−1.26382e9	−	9.18219e8i	−1.42402	−	1.03461i	−0.991092	−	0.133178i	\(-0.957482\pi\)
							−0.432925	−	0.901430i	\(-0.642518\pi\)
\(37\)	−5.73320e8	−	1.76450e9i	−0.223453	−	0.687719i	−0.998445	−	0.0557473i	\(-0.982246\pi\)
							0.774991	−	0.631972i	\(-0.217754\pi\)
\(41\)	−1.75691e9	−	5.70855e8i	−0.369868	−	0.120177i	0.118185	−	0.992992i	\(-0.462292\pi\)
							−0.488052	+	0.872814i	\(0.662292\pi\)
\(43\)			1.26122e9i			0.199516i	0.995012	+	0.0997582i	\(0.0318069\pi\)
							−0.995012	+	0.0997582i	\(0.968193\pi\)
\(47\)	−4.94914e9	+	1.52319e10i	−0.459137	+	1.41308i	0.407071	+	0.913397i	\(0.366550\pi\)
							−0.866208	+	0.499683i	\(0.833450\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.7.3	✓	44
11.8	odd	10	inner	11.13.d.a.8.3	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.7.3	✓	44	1.1	even	1	trivial
11.13.d.a.8.3	yes	44	11.8	odd	10	inner