Embedded newform 41.6.f.a.4.2

• Label: 41.6.f.a.4.2
• Level: $41$
• Weight: $6$
• Character: 41.4
• 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			4.2
Character	\(\chi\)	\(=\)	41.4
Dual form			41.6.f.a.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.55080 + 5.48598i) q^{2} -20.3153i q^{3} +(17.0301 - 52.4132i) q^{4} +(8.88512 - 27.3456i) q^{5} +(111.449 + 153.397i) q^{6} +(-62.7153 + 86.3202i) q^{7} +(66.6541 + 205.140i) q^{8} -169.711 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.55080	+	5.48598i	−1.33481	+	0.969793i	−0.335187	+	0.942152i	\(0.608800\pi\)
							−0.999618	+	0.0276412i	\(0.991200\pi\)
\(3\)		−	20.3153i		−	1.30323i	−0.758552	−	0.651613i	\(-0.774093\pi\)
							0.758552	−	0.651613i	\(-0.225907\pi\)
\(5\)	8.88512	−	27.3456i	0.158942	−	0.489172i	−0.839597	−	0.543210i	\(-0.817209\pi\)
							0.998539	+	0.0540371i	\(0.0172089\pi\)
\(7\)	−62.7153	+	86.3202i	−0.483758	+	0.665836i	−0.979222	−	0.202793i	\(-0.934998\pi\)
							0.495463	+	0.868629i	\(0.334998\pi\)
\(11\)	−79.9888	+	25.9899i	−0.199318	+	0.0647625i	−0.406975	−	0.913439i	\(-0.633416\pi\)
							0.207657	+	0.978202i	\(0.433416\pi\)
\(13\)	−254.721	−	350.593i	−0.418029	−	0.575367i	0.547125	−	0.837051i	\(-0.315722\pi\)
							−0.965154	+	0.261684i	\(0.915722\pi\)
\(17\)	−1417.95	+	460.720i	−1.18998	+	0.386647i	−0.836064	−	0.548632i	\(-0.815149\pi\)
							−0.353912	+	0.935279i	\(0.615149\pi\)
\(19\)	−1645.78	+	2265.22i	−1.04589	+	1.43955i	−0.153579	+	0.988136i	\(0.549080\pi\)
							−0.892315	+	0.451413i	\(0.850920\pi\)
\(23\)	−246.216	+	178.886i	−0.0970502	+	0.0705111i	−0.635252	−	0.772305i	\(-0.719104\pi\)
							0.538202	+	0.842816i	\(0.319104\pi\)
\(29\)	−4595.98	−	1493.32i	−1.01481	−	0.329730i	−0.246039	−	0.969260i	\(-0.579129\pi\)
							−0.768767	+	0.639529i	\(0.779129\pi\)
\(31\)	1478.27	+	4549.65i	0.276280	+	0.850303i	0.988878	+	0.148730i	\(0.0475186\pi\)
							−0.712598	+	0.701573i	\(0.752481\pi\)
\(37\)	−1974.48	+	6076.83i	−0.237109	+	0.729748i	0.759725	+	0.650244i	\(0.225333\pi\)
							−0.996835	+	0.0795034i	\(0.974667\pi\)
\(41\)	−8510.59	−	6589.85i	−0.790678	−	0.612232i
							\(43\)	14897.6	−	10823.8i	1.22870	−	0.892702i	0.231908	−	0.972738i	\(-0.425503\pi\)
0.996792	+	0.0800353i	\(0.0255033\pi\)
\(47\)	−4271.76	−	5879.58i	−0.282074	−	0.388241i	0.644346	−	0.764734i	\(-0.277130\pi\)
							−0.926419	+	0.376493i	\(0.877130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.6.f.a.4.2	✓	64
41.31	even	10	inner	41.6.f.a.31.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.4.2	✓	64	1.1	even	1	trivial
41.6.f.a.31.2	yes	64	41.31	even	10	inner