Embedded newform 41.6.f.a.23.4

• Label: 41.6.f.a.23.4
• Level: $41$
• Weight: $6$
• Character: 41.23
• Analytic conductor: $6.576$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	41.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.57573661233\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			23.4
Character	\(\chi\)	\(=\)	41.23
Dual form			41.6.f.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79067 - 5.51110i) q^{2} +30.5883i q^{3} +(-1.27722 + 0.927958i) q^{4} +(-15.9554 + 11.5922i) q^{5} +(168.575 - 54.7734i) q^{6} +(-104.403 - 33.9227i) q^{7} +(-142.616 - 103.616i) q^{8} -692.644 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79067	−	5.51110i	−0.316548	−	0.974235i	−0.975113	−	0.221710i	\(-0.928836\pi\)
							0.658565	−	0.752524i	\(-0.271164\pi\)
\(3\)			30.5883i			1.96224i	0.193399	+	0.981120i	\(0.438049\pi\)
							−0.193399	+	0.981120i	\(0.561951\pi\)
\(5\)	−15.9554	+	11.5922i	−0.285418	+	0.207368i	−0.721277	−	0.692647i	\(-0.756445\pi\)
							0.435859	+	0.900015i	\(0.356445\pi\)
\(7\)	−104.403	−	33.9227i	−0.805320	−	0.261664i	−0.122706	−	0.992443i	\(-0.539157\pi\)
							−0.682614	+	0.730779i	\(0.739157\pi\)
\(11\)	−82.0901	+	112.987i	−0.204555	+	0.281545i	−0.898953	−	0.438046i	\(-0.855671\pi\)
							0.694398	+	0.719591i	\(0.255671\pi\)
\(13\)	528.621	−	171.759i	0.867532	−	0.281878i	0.158761	−	0.987317i	\(-0.449250\pi\)
							0.708771	+	0.705439i	\(0.249250\pi\)
\(17\)	−551.457	+	759.015i	−0.462796	+	0.636984i	−0.975086	−	0.221829i	\(-0.928797\pi\)
							0.512290	+	0.858813i	\(0.328797\pi\)
\(19\)	−2377.75	−	772.579i	−1.51106	−	0.490974i	−0.567842	−	0.823138i	\(-0.692221\pi\)
							−0.943222	+	0.332163i	\(0.892221\pi\)
\(23\)	677.160	+	2084.09i	0.266914	+	0.821478i	0.991246	+	0.132026i	\(0.0421484\pi\)
							−0.724332	+	0.689452i	\(0.757852\pi\)
\(29\)	2227.91	+	3066.45i	0.491928	+	0.677081i	0.980742	−	0.195307i	\(-0.0625703\pi\)
							−0.488814	+	0.872388i	\(0.662570\pi\)
\(31\)	295.254	+	214.515i	0.0551813	+	0.0400916i	0.615034	−	0.788501i	\(-0.289142\pi\)
							−0.559853	+	0.828592i	\(0.689142\pi\)
\(37\)	8262.28	−	6002.90i	0.992192	−	0.720870i	0.0317922	−	0.999495i	\(-0.489879\pi\)
							0.960400	+	0.278625i	\(0.0898785\pi\)
\(41\)	10005.0	+	3969.50i	0.929514	+	0.368788i
							\(43\)	1568.65	+	4827.82i	0.129377	+	0.398181i	0.994673	−	0.103080i	\(-0.0328698\pi\)
−0.865296	+	0.501261i	\(0.832870\pi\)
\(47\)	−19211.3	+	6242.13i	−1.26856	+	0.412181i	−0.864539	−	0.502566i	\(-0.832389\pi\)
							−0.404025	+	0.914748i	\(0.632389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.6.f.a.23.4	✓	64
41.25	even	10	inner	41.6.f.a.25.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.23.4	✓	64	1.1	even	1	trivial
41.6.f.a.25.4	yes	64	41.25	even	10	inner