Embedded newform 41.6.f.a.31.5

• Label: 41.6.f.a.31.5
• 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			31.5
Character	\(\chi\)	\(=\)	41.31
Dual form			41.6.f.a.4.5

$q$-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 \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.4.5	✓	64	41.4	even	10	inner
41.6.f.a.31.5	yes	64	1.1	even	1	trivial