Embedded newform 41.6.f.a.4.5

• Label: 41.6.f.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	41.4
Dual form			41.6.f.a.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.35748 + 3.16590i) q^{2} -19.6567i q^{3} +(-0.923788 + 2.84313i) q^{4} +(-25.6787 + 79.0308i) q^{5} +(62.2312 + 85.6539i) q^{6} +(84.0596 - 115.698i) q^{7} +(-58.2368 - 179.234i) q^{8} -143.387 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.35748	+	3.16590i	−0.770301	+	0.559657i	−0.902053	−	0.431626i	\(-0.857940\pi\)
							0.131751	+	0.991283i	\(0.457940\pi\)
\(3\)		−	19.6567i		−	1.26098i	−0.776198	−	0.630490i	\(-0.782854\pi\)
							0.776198	−	0.630490i	\(-0.217146\pi\)
\(5\)	−25.6787	+	79.0308i	−0.459354	+	1.41375i	0.406592	+	0.913610i	\(0.366717\pi\)
							−0.865946	+	0.500137i	\(0.833283\pi\)
\(7\)	84.0596	−	115.698i	0.648399	−	0.892444i	−0.350630	−	0.936514i	\(-0.614032\pi\)
							0.999028	+	0.0440700i	\(0.0140325\pi\)
\(11\)	548.857	−	178.334i	1.36766	−	0.444379i	0.469065	−	0.883164i	\(-0.344591\pi\)
							0.898592	+	0.438785i	\(0.144591\pi\)
\(13\)	−423.007	−	582.220i	−0.694208	−	0.955495i	−0.999994	−	0.00339490i	\(-0.998919\pi\)
							0.305786	−	0.952100i	\(-0.401081\pi\)
\(17\)	1818.39	−	590.831i	1.52604	−	0.495839i	0.578554	−	0.815644i	\(-0.303617\pi\)
							0.947483	+	0.319805i	\(0.103617\pi\)
\(19\)	75.7349	−	104.240i	0.0481296	−	0.0662447i	−0.784274	−	0.620415i	\(-0.786964\pi\)
							0.832403	+	0.554170i	\(0.186964\pi\)
\(23\)	−134.537	+	97.7465i	−0.0530299	+	0.0385285i	−0.613984	−	0.789318i	\(-0.710434\pi\)
							0.560955	+	0.827847i	\(0.310434\pi\)
\(29\)	−829.103	−	269.392i	−0.183069	−	0.0594826i	0.216048	−	0.976383i	\(-0.430683\pi\)
							−0.399117	+	0.916900i	\(0.630683\pi\)
\(31\)	1570.85	+	4834.57i	0.293582	+	0.903554i	0.983694	+	0.179851i	\(0.0575615\pi\)
							−0.690112	+	0.723703i	\(0.742439\pi\)
\(37\)	4592.49	−	14134.2i	0.551498	−	1.69734i	−0.153520	−	0.988145i	\(-0.549061\pi\)
							0.705018	−	0.709190i	\(-0.250939\pi\)
\(41\)	4223.78	+	9900.30i	0.392412	+	0.919790i
							\(43\)	−12458.8	+	9051.85i	−1.02755	+	0.746562i	−0.967818	−	0.251652i	\(-0.919026\pi\)
−0.0597371	+	0.998214i	\(0.519026\pi\)
\(47\)	−7284.35	−	10026.0i	−0.481001	−	0.662041i	0.497696	−	0.867352i	\(-0.334180\pi\)
							−0.978697	+	0.205310i	\(0.934180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.6.f.a.4.5	✓	64
41.31	even	10	inner	41.6.f.a.31.5	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.4.5	✓	64	1.1	even	1	trivial
41.6.f.a.31.5	yes	64	41.31	even	10	inner