Embedded newform 41.8.d.a.10.7

• Label: 41.8.d.a.10.7
• 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			10.7
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.5329 - 7.65261i) q^{2} -51.6297 q^{3} +(12.8257 + 39.4733i) q^{4} +(-26.1720 - 80.5490i) q^{5} +(543.811 + 395.102i) q^{6} +(-1179.24 + 856.766i) q^{7} +(-347.989 + 1071.00i) q^{8} +478.628 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.5329	−	7.65261i	−0.930987	−	0.676401i	0.0152475	−	0.999884i	\(-0.495146\pi\)
							−0.946234	+	0.323482i	\(0.895146\pi\)
\(3\)	−51.6297			−1.10402			−0.552008	−	0.833839i	\(-0.686138\pi\)
							−0.552008	+	0.833839i	\(0.686138\pi\)
\(5\)	−26.1720	−	80.5490i	−0.0936357	−	0.288181i	0.893260	−	0.449540i	\(-0.148412\pi\)
							−0.986896	+	0.161359i	\(0.948412\pi\)
\(7\)	−1179.24	+	856.766i	−1.29945	+	0.944103i	−0.999950	−	0.00999471i	\(-0.996819\pi\)
							−0.299496	+	0.954098i	\(0.596819\pi\)
\(11\)	−609.929	+	1877.17i	−0.138167	+	0.425235i	−0.996069	−	0.0885786i	\(-0.971768\pi\)
							0.857902	+	0.513813i	\(0.171768\pi\)
\(13\)	−1948.27	−	1415.50i	−0.245950	−	0.178693i	0.457980	−	0.888963i	\(-0.348573\pi\)
							−0.703930	+	0.710269i	\(0.748573\pi\)
\(17\)	11665.6	−	35903.1i	0.575886	−	1.77239i	−0.0572573	−	0.998359i	\(-0.518236\pi\)
							0.633143	−	0.774035i	\(-0.281764\pi\)
\(19\)	−41968.3	+	30491.8i	−1.40373	+	1.01987i	−0.409534	+	0.912295i	\(0.634309\pi\)
							−0.994197	+	0.107575i	\(0.965691\pi\)
\(23\)	−79664.7	−	57879.8i	−1.36527	−	0.991927i	−0.998090	−	0.0617770i	\(-0.980323\pi\)
							−0.367180	−	0.930150i	\(-0.619677\pi\)
\(29\)	69270.4	+	213192.i	0.527417	+	1.62322i	0.759486	+	0.650524i	\(0.225451\pi\)
							−0.232068	+	0.972700i	\(0.574549\pi\)
\(31\)	26765.6	−	82375.9i	0.161365	−	0.496632i	−0.837385	−	0.546614i	\(-0.815917\pi\)
							0.998750	+	0.0499823i	\(0.0159165\pi\)
\(37\)	−51648.8	−	158959.i	−0.167631	−	0.515915i	0.831590	−	0.555391i	\(-0.187431\pi\)
							−0.999221	+	0.0394756i	\(0.987431\pi\)
\(41\)	−232393.	+	375163.i	−0.526599	+	0.850114i
							\(43\)	−168750.	−	122604.i	−0.323671	−	0.235161i	0.414069	−	0.910245i	\(-0.364107\pi\)
−0.737740	+	0.675085i	\(0.764107\pi\)
\(47\)	266226.	+	193424.i	0.374031	+	0.271749i	0.758881	−	0.651230i	\(-0.225747\pi\)
							−0.384849	+	0.922979i	\(0.625747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.7	✓	96
41.37	even	5	inner	41.8.d.a.37.7	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.7	✓	96	1.1	even	1	trivial
41.8.d.a.37.7	yes	96	41.37	even	5	inner