Embedded newform 41.8.d.a.10.6

• Label: 41.8.d.a.10.6
• Level: $41$
• Weight: $8$
• Character: 41.10
• Analytic conductor: $12.808$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	41.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8077860448\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			10.6
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.6155 - 7.71261i) q^{2} +70.2921 q^{3} +(13.6503 + 42.0112i) q^{4} +(45.2559 + 139.283i) q^{5} +(-746.185 - 542.135i) q^{6} +(1334.35 - 969.461i) q^{7} +(-339.898 + 1046.10i) q^{8} +2753.97 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.6155	−	7.71261i	−0.938286	−	0.681705i	0.00972112	−	0.999953i	\(-0.496906\pi\)
							−0.948008	+	0.318248i	\(0.896906\pi\)
\(3\)	70.2921			1.50308			0.751540	−	0.659688i	\(-0.229312\pi\)
							0.751540	+	0.659688i	\(0.229312\pi\)
\(5\)	45.2559	+	139.283i	0.161912	+	0.498315i	0.998796	−	0.0490664i	\(-0.0156246\pi\)
							−0.836883	+	0.547382i	\(0.815625\pi\)
\(7\)	1334.35	−	969.461i	1.47037	−	1.06829i	0.489862	−	0.871800i	\(-0.337047\pi\)
							0.980507	−	0.196485i	\(-0.0629529\pi\)
\(11\)	−1958.33	+	6027.11i	−0.443619	+	1.36532i	0.440371	+	0.897816i	\(0.354847\pi\)
							−0.883991	+	0.467504i	\(0.845153\pi\)
\(13\)	7760.78	+	5638.54i	0.979724	+	0.711811i	0.957647	−	0.287945i	\(-0.0929721\pi\)
							0.0220770	+	0.999756i	\(0.492972\pi\)
\(17\)	7399.36	−	22772.9i	0.365277	−	1.12421i	−0.584530	−	0.811372i	\(-0.698721\pi\)
							0.949807	−	0.312836i	\(-0.101279\pi\)
\(19\)	−2314.42	+	1681.52i	−0.0774112	+	0.0562425i	−0.625818	−	0.779969i	\(-0.715235\pi\)
							0.548407	+	0.836212i	\(0.315235\pi\)
\(23\)	−11609.0	−	8434.41i	−0.198951	−	0.144546i	0.483849	−	0.875151i	\(-0.339238\pi\)
							−0.682801	+	0.730605i	\(0.739238\pi\)
\(29\)	−38892.6	−	119699.i	−0.296124	−	0.911376i	−0.982842	−	0.184452i	\(-0.940949\pi\)
							0.686718	−	0.726924i	\(-0.259051\pi\)
\(31\)	46324.8	−	142573.i	0.279285	−	0.859552i	−0.708768	−	0.705441i	\(-0.750749\pi\)
							0.988054	−	0.154110i	\(-0.0492511\pi\)
\(37\)	51706.6	+	159136.i	0.167818	+	0.516492i	0.999233	−	0.0391607i	\(-0.0124684\pi\)
							−0.831415	+	0.555653i	\(0.812468\pi\)
\(41\)	−238739.	+	371157.i	−0.540979	+	0.841036i
							\(43\)	−362624.	−	263462.i	−0.695532	−	0.505333i	0.182942	−	0.983124i	\(-0.441438\pi\)
−0.878474	+	0.477790i	\(0.841438\pi\)
\(47\)	1.04274e6	+	757597.i	1.46499	+	1.06438i	0.982028	+	0.188736i	\(0.0604390\pi\)
							0.482962	+	0.875641i	\(0.339561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.6	✓	96
41.37	even	5	inner	41.8.d.a.37.6	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.6	✓	96	1.1	even	1	trivial
41.8.d.a.37.6	yes	96	41.37	even	5	inner