Embedded newform 41.4.f.a.23.5

• Label: 41.4.f.a.23.5
• Level: $41$
• Weight: $4$
• Character: 41.23
• Analytic conductor: $2.419$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	41.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.41907831024\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			23.5
Character	\(\chi\)	\(=\)	41.23
Dual form			41.4.f.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.625259 - 1.92435i) q^{2} -8.16596i q^{3} +(3.15996 - 2.29585i) q^{4} +(-13.3127 + 9.67225i) q^{5} +(-15.7142 + 5.10585i) q^{6} +(17.8022 + 5.78429i) q^{7} +(-19.4894 - 14.1599i) q^{8} -39.6830 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + q^{2} - 43 q^{4} - q^{5} - 15 q^{6} - 5 q^{7} + 112 q^{8} - 370 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.625259	−	1.92435i	−0.221063	−	0.680361i	−0.998667	−	0.0516074i	\(-0.983566\pi\)
							0.777605	−	0.628753i	\(-0.216434\pi\)
\(3\)		−	8.16596i		−	1.57154i	−0.618518	−	0.785770i	\(-0.712267\pi\)
							0.618518	−	0.785770i	\(-0.287733\pi\)
\(5\)	−13.3127	+	9.67225i	−1.19073	+	0.865112i	−0.993341	−	0.115214i	\(-0.963245\pi\)
							−0.197384	+	0.980326i	\(0.563245\pi\)
\(7\)	17.8022	+	5.78429i	0.961230	+	0.312323i	0.747270	−	0.664520i	\(-0.231364\pi\)
							0.213959	+	0.976843i	\(0.431364\pi\)
\(11\)	34.2799	−	47.1822i	0.939615	−	1.29327i	−0.0163738	−	0.999866i	\(-0.505212\pi\)
							0.955989	−	0.293403i	\(-0.0947878\pi\)
\(13\)	−5.25335	+	1.70692i	−0.112078	+	0.0364164i	−0.364519	−	0.931196i	\(-0.618767\pi\)
							0.252441	+	0.967612i	\(0.418767\pi\)
\(17\)	−3.83688	+	5.28102i	−0.0547401	+	0.0753432i	−0.835509	−	0.549476i	\(-0.814827\pi\)
							0.780769	+	0.624819i	\(0.214827\pi\)
\(19\)	102.720	+	33.3758i	1.24030	+	0.402997i	0.854434	−	0.519560i	\(-0.173904\pi\)
							0.385862	+	0.922556i	\(0.373904\pi\)
\(23\)	−27.7999	−	85.5594i	−0.252030	−	0.775668i	−0.994400	−	0.105678i	\(-0.966299\pi\)
							0.742371	−	0.669990i	\(-0.233701\pi\)
\(29\)	179.022	+	246.403i	1.14633	+	1.57779i	0.752483	+	0.658612i	\(0.228856\pi\)
							0.393847	+	0.919176i	\(0.371144\pi\)
\(31\)	−7.03442	−	5.11081i	−0.0407555	−	0.0296106i	0.567221	−	0.823566i	\(-0.308019\pi\)
							−0.607976	+	0.793955i	\(0.708019\pi\)
\(37\)	70.1869	−	50.9938i	0.311856	−	0.226576i	−0.420837	−	0.907136i	\(-0.638263\pi\)
							0.732692	+	0.680560i	\(0.238263\pi\)
\(41\)	−190.472	+	180.669i	−0.725531	+	0.688190i
							\(43\)	78.2762	+	240.909i	0.277605	+	0.854380i	0.988518	+	0.151101i	\(0.0482817\pi\)
−0.710913	+	0.703279i	\(0.751718\pi\)
\(47\)	81.0242	−	26.3264i	0.251460	−	0.0817042i	−0.180575	−	0.983561i	\(-0.557796\pi\)
							0.432035	+	0.901857i	\(0.357796\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.4.f.a.23.5	✓	40
41.5	even	20		1681.4.a.l.1.24		40
41.25	even	10	inner	41.4.f.a.25.5	yes	40
41.36	even	20		1681.4.a.l.1.23		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.4.f.a.23.5	✓	40	1.1	even	1	trivial
41.4.f.a.25.5	yes	40	41.25	even	10	inner
1681.4.a.l.1.23		40	41.36	even	20
1681.4.a.l.1.24		40	41.5	even	20