Embedded newform 11.13.d.a.6.5

• Label: 11.13.d.a.6.5
• Level: $11$
• Weight: $13$
• Character: 11.6
• 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			6.5
Character	\(\chi\)	\(=\)	11.6
Dual form			11.13.d.a.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-21.1209 + 6.86260i) q^{2} +(669.622 + 486.509i) q^{3} +(-2914.74 + 2117.68i) q^{4} +(-1648.51 + 5073.61i) q^{5} +(-17481.7 - 5680.16i) q^{6} +(14639.8 + 20149.9i) q^{7} +(100496. - 138321. i) q^{8} +(47478.3 + 146123. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.1209	+	6.86260i	−0.330014	+	0.107228i	−0.469338	−	0.883019i	\(-0.655507\pi\)
							0.139323	+	0.990247i	\(0.455507\pi\)
\(3\)	669.622	+	486.509i	0.918548	+	0.667364i	0.943162	−	0.332333i	\(-0.107836\pi\)
							−0.0246140	+	0.999697i	\(0.507836\pi\)
\(5\)	−1648.51	+	5073.61i	−0.105505	+	0.324711i	−0.989849	−	0.142126i	\(-0.954606\pi\)
							0.884344	+	0.466836i	\(0.154606\pi\)
\(7\)	14639.8	+	20149.9i	0.124436	+	0.171272i	0.866690	−	0.498847i	\(-0.166243\pi\)
							−0.742254	+	0.670119i	\(0.766243\pi\)
\(11\)	−1.74309e6	+	316335.i	−0.983928	+	0.178563i
							\(13\)	−6.93819e6	+	2.25436e6i	−1.43743	+	0.467049i	−0.921095	−	0.389339i	\(-0.872704\pi\)
−0.516334	+	0.856387i	\(0.672704\pi\)
\(17\)	−1.36800e6	−	444489.i	−0.0566749	−	0.0184148i	0.280542	−	0.959842i	\(-0.409486\pi\)
							−0.337217	+	0.941427i	\(0.609486\pi\)
\(19\)	−2.41697e7	+	3.32667e7i	−0.513746	+	0.707111i	−0.984545	−	0.175129i	\(-0.943966\pi\)
							0.470799	+	0.882240i	\(0.343966\pi\)
\(23\)	1.46866e7			0.0992100			0.0496050	−	0.998769i	\(-0.484204\pi\)
							0.0496050	+	0.998769i	\(0.484204\pi\)
\(29\)	1.29527e7	+	1.78279e7i	0.0217757	+	0.0299717i	0.819766	−	0.572699i	\(-0.194103\pi\)
							−0.797990	+	0.602670i	\(0.794103\pi\)
\(31\)	1.80505e8	+	5.55537e8i	0.203385	+	0.625954i	0.999776	+	0.0211720i	\(0.00673978\pi\)
							−0.796391	+	0.604782i	\(0.793260\pi\)
\(37\)	−1.99521e8	+	1.44960e8i	−0.0777638	+	0.0564987i	−0.625988	−	0.779833i	\(-0.715304\pi\)
							0.548224	+	0.836331i	\(0.315304\pi\)
\(41\)	1.35264e8	−	1.86176e8i	0.0284761	−	0.0391940i	−0.794542	−	0.607210i	\(-0.792289\pi\)
							0.823018	+	0.568016i	\(0.192289\pi\)
\(43\)			9.67113e9i			1.52991i	0.644082	+	0.764956i	\(0.277239\pi\)
							−0.644082	+	0.764956i	\(0.722761\pi\)
\(47\)	9.85944e9	+	7.16331e9i	0.914672	+	0.664548i	0.942192	−	0.335073i	\(-0.108761\pi\)
							−0.0275202	+	0.999621i	\(0.508761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.6.5	yes	44
11.2	odd	10	inner	11.13.d.a.2.5	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.5	✓	44	11.2	odd	10	inner
11.13.d.a.6.5	yes	44	1.1	even	1	trivial