Embedded newform 41.4.f.a.4.8

• Label: 41.4.f.a.4.8
• Level: $41$
• Weight: $4$
• Character: 41.4
• 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			4.8
Character	\(\chi\)	\(=\)	41.4
Dual form			41.4.f.a.31.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.80536 - 2.03822i) q^{2} -1.54098i q^{3} +(1.24361 - 3.82743i) q^{4} +(3.68903 - 11.3537i) q^{5} +(-3.14085 - 4.32301i) q^{6} +(-3.25858 + 4.48505i) q^{7} +(4.26007 + 13.1112i) q^{8} +24.6254 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + q^{2} - 43 q^{4} - q^{5} - 15 q^{6} - 5 q^{7} + 112 q^{8} - 370 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.80536	−	2.03822i	0.991846	−	0.720618i	0.0315216	−	0.999503i	\(-0.489965\pi\)
							0.960325	+	0.278885i	\(0.0899647\pi\)
\(3\)		−	1.54098i		−	0.296562i	−0.988945	−	0.148281i	\(-0.952626\pi\)
							0.988945	−	0.148281i	\(-0.0473740\pi\)
\(5\)	3.68903	−	11.3537i	0.329957	−	1.01550i	−0.639196	−	0.769044i	\(-0.720733\pi\)
							0.969153	−	0.246460i	\(-0.0792673\pi\)
\(7\)	−3.25858	+	4.48505i	−0.175947	+	0.242170i	−0.887878	−	0.460080i	\(-0.847821\pi\)
							0.711931	+	0.702249i	\(0.247821\pi\)
\(11\)	−58.9545	+	19.1555i	−1.61595	+	0.525054i	−0.970982	−	0.239154i	\(-0.923130\pi\)
							−0.644969	+	0.764208i	\(0.723130\pi\)
\(13\)	24.0572	+	33.1118i	0.513250	+	0.706429i	0.984463	−	0.175591i	\(-0.0561836\pi\)
							−0.471213	+	0.882020i	\(0.656184\pi\)
\(17\)	−28.7354	+	9.33670i	−0.409962	+	0.133205i	−0.506735	−	0.862102i	\(-0.669148\pi\)
							0.0967731	+	0.995306i	\(0.469148\pi\)
\(19\)	27.2433	−	37.4972i	0.328950	−	0.452761i	−0.612223	−	0.790685i	\(-0.709725\pi\)
							0.941173	+	0.337924i	\(0.109725\pi\)
\(23\)	−38.1092	+	27.6880i	−0.345492	+	0.251015i	−0.746976	−	0.664852i	\(-0.768495\pi\)
							0.401483	+	0.915866i	\(0.368495\pi\)
\(29\)	−201.704	−	65.5376i	−1.29157	−	0.419656i	−0.418929	−	0.908019i	\(-0.637594\pi\)
							−0.872641	+	0.488363i	\(0.837594\pi\)
\(31\)	−43.5382	−	133.997i	−0.252248	−	0.776340i	−0.994359	−	0.106063i	\(-0.966175\pi\)
							0.742111	−	0.670277i	\(-0.233825\pi\)
\(37\)	83.7230	−	257.673i	0.371999	−	1.14490i	−0.573482	−	0.819218i	\(-0.694408\pi\)
							0.945481	−	0.325677i	\(-0.105592\pi\)
\(41\)	38.9192	−	259.627i	0.148248	−	0.988950i
							\(43\)	−94.1520	+	68.4055i	−0.333908	+	0.242599i	−0.742087	−	0.670303i	\(-0.766164\pi\)
0.408179	+	0.912902i	\(0.366164\pi\)
\(47\)	−5.34141	−	7.35182i	−0.0165771	−	0.0228165i	0.800648	−	0.599135i	\(-0.204489\pi\)
							−0.817225	+	0.576319i	\(0.804489\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.4.f.a.4.8	✓	40
41.20	even	20		1681.4.a.l.1.31		40
41.21	even	20		1681.4.a.l.1.32		40
41.31	even	10	inner	41.4.f.a.31.8	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.4.f.a.4.8	✓	40	1.1	even	1	trivial
41.4.f.a.31.8	yes	40	41.31	even	10	inner
1681.4.a.l.1.31		40	41.20	even	20
1681.4.a.l.1.32		40	41.21	even	20