Embedded newform 41.6.f.a.4.10

• Label: 41.6.f.a.4.10
• Level: $41$
• Weight: $6$
• Character: 41.4
• 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			4.10
Character	\(\chi\)	\(=\)	41.4
Dual form			41.6.f.a.31.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.76846 - 1.28486i) q^{2} -11.5405i q^{3} +(-8.41197 + 25.8894i) q^{4} +(-17.5603 + 54.0450i) q^{5} +(-14.8279 - 20.4089i) q^{6} +(-61.6873 + 84.9053i) q^{7} +(40.0037 + 123.119i) q^{8} +109.817 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.76846	−	1.28486i	0.312622	−	0.227133i	−0.420399	−	0.907340i	\(-0.638110\pi\)
							0.733021	+	0.680206i	\(0.238110\pi\)
\(3\)		−	11.5405i		−	0.740324i	−0.928967	−	0.370162i	\(-0.879302\pi\)
							0.928967	−	0.370162i	\(-0.120698\pi\)
\(5\)	−17.5603	+	54.0450i	−0.314128	+	0.966786i	0.661984	+	0.749518i	\(0.269715\pi\)
							−0.976112	+	0.217268i	\(0.930285\pi\)
\(7\)	−61.6873	+	84.9053i	−0.475829	+	0.654922i	−0.977697	−	0.210022i	\(-0.932646\pi\)
							0.501868	+	0.864944i	\(0.332646\pi\)
\(11\)	17.7906	−	5.78052i	0.0443312	−	0.0144041i	−0.286767	−	0.958000i	\(-0.592581\pi\)
							0.331099	+	0.943596i	\(0.392581\pi\)
\(13\)	243.781	+	335.536i	0.400075	+	0.550656i	0.960763	−	0.277371i	\(-0.0894632\pi\)
							−0.560688	+	0.828027i	\(0.689463\pi\)
\(17\)	−904.726	+	293.963i	−0.759268	+	0.246701i	−0.662964	−	0.748651i	\(-0.730702\pi\)
							−0.0963033	+	0.995352i	\(0.530702\pi\)
\(19\)	−983.286	+	1353.38i	−0.624879	+	0.860072i	−0.997697	−	0.0678324i	\(-0.978392\pi\)
							0.372818	+	0.927905i	\(0.378392\pi\)
\(23\)	753.182	−	547.219i	0.296880	−	0.215696i	−0.429367	−	0.903130i	\(-0.641263\pi\)
							0.726246	+	0.687435i	\(0.241263\pi\)
\(29\)	2347.91	+	762.882i	0.518425	+	0.168447i	0.556531	−	0.830827i	\(-0.312132\pi\)
							−0.0381053	+	0.999274i	\(0.512132\pi\)
\(31\)	−761.532	−	2343.75i	−0.142326	−	0.438034i	0.854332	−	0.519728i	\(-0.173967\pi\)
							−0.996657	+	0.0816944i	\(0.973967\pi\)
\(37\)	1269.48	−	3907.07i	0.152448	−	0.469188i	−0.845445	−	0.534062i	\(-0.820665\pi\)
							0.997893	+	0.0648746i	\(0.0206647\pi\)
\(41\)	7474.00	−	7745.68i	0.694374	−	0.719614i
							\(43\)	−876.795	+	637.029i	−0.0723148	+	0.0525397i	−0.623355	−	0.781939i	\(-0.714231\pi\)
0.551041	+	0.834478i	\(0.314231\pi\)
\(47\)	6000.53	+	8259.02i	0.396228	+	0.545361i	0.959792	−	0.280711i	\(-0.0905703\pi\)
							−0.563564	+	0.826072i	\(0.690570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.6.f.a.4.10	✓	64
41.31	even	10	inner	41.6.f.a.31.10	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.4.10	✓	64	1.1	even	1	trivial
41.6.f.a.31.10	yes	64	41.31	even	10	inner