Embedded newform 41.6.f.a.31.9

• Label: 41.6.f.a.31.9
• Level: $41$
• Weight: $6$
• Character: 41.31
• Analytic conductor: $6.576$
• Analytic rank: $0$
• Dimension: $64$
• 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			31.9
Character	\(\chi\)	\(=\)	41.31
Dual form			41.6.f.a.4.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.345180 - 0.250788i) q^{2} +28.7734i q^{3} +(-9.83229 - 30.2607i) q^{4} +(16.3294 + 50.2566i) q^{5} +(7.21603 - 9.93201i) q^{6} +(26.2129 + 36.0790i) q^{7} +(-8.41421 + 25.8963i) q^{8} -584.909 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.345180	−	0.250788i	−0.0610198	−	0.0443335i	0.556857	−	0.830608i	\(-0.312007\pi\)
							−0.617877	+	0.786275i	\(0.712007\pi\)
\(3\)			28.7734i			1.84582i	0.385021	+	0.922908i	\(0.374194\pi\)
							−0.385021	+	0.922908i	\(0.625806\pi\)
\(5\)	16.3294	+	50.2566i	0.292109	+	0.899018i	0.984177	+	0.177186i	\(0.0566994\pi\)
							−0.692069	+	0.721832i	\(0.743301\pi\)
\(7\)	26.2129	+	36.0790i	0.202195	+	0.278298i	0.898058	−	0.439877i	\(-0.144978\pi\)
							−0.695863	+	0.718175i	\(0.744978\pi\)
\(11\)	−551.496	−	179.192i	−1.37423	−	0.446516i	−0.473464	−	0.880813i	\(-0.656997\pi\)
							−0.900770	+	0.434297i	\(0.856997\pi\)
\(13\)	2.12124	−	2.91964i	0.00348122	−	0.00479149i	−0.807273	−	0.590178i	\(-0.799057\pi\)
							0.810754	+	0.585387i	\(0.199057\pi\)
\(17\)	−248.944	−	80.8869i	−0.208920	−	0.0678822i	0.202687	−	0.979243i	\(-0.435032\pi\)
							−0.411607	+	0.911361i	\(0.635032\pi\)
\(19\)	1725.75	+	2375.29i	1.09671	+	1.50950i	0.839669	+	0.543099i	\(0.182749\pi\)
							0.257046	+	0.966399i	\(0.417251\pi\)
\(23\)	111.274	+	80.8450i	0.0438604	+	0.0318665i	0.609499	−	0.792786i	\(-0.291370\pi\)
							−0.565639	+	0.824653i	\(0.691370\pi\)
\(29\)	3791.61	−	1231.97i	0.837200	−	0.272023i	0.141125	−	0.989992i	\(-0.454928\pi\)
							0.696075	+	0.717969i	\(0.254928\pi\)
\(31\)	−1709.24	+	5260.49i	−0.319447	+	0.983156i	0.654439	+	0.756115i	\(0.272905\pi\)
							−0.973885	+	0.227041i	\(0.927095\pi\)
\(37\)	4395.74	+	13528.7i	0.527871	+	1.62462i	0.758568	+	0.651593i	\(0.225899\pi\)
							−0.230698	+	0.973025i	\(0.574101\pi\)
\(41\)	−10459.4	+	2540.89i	−0.971738	+	0.236062i
							\(43\)	8151.67	+	5922.53i	0.672319	+	0.488468i	0.870801	−	0.491636i	\(-0.163601\pi\)
−0.198482	+	0.980105i	\(0.563601\pi\)
\(47\)	252.646	−	347.738i	0.0166828	−	0.0229619i	−0.800594	−	0.599207i	\(-0.795483\pi\)
							0.817277	+	0.576245i	\(0.195483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.6.f.a.31.9	yes	64
41.4	even	10	inner	41.6.f.a.4.9	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.4.9	✓	64	41.4	even	10	inner
41.6.f.a.31.9	yes	64	1.1	even	1	trivial