Embedded newform 41.8.d.a.10.14

• Label: 41.8.d.a.10.14
• Level: $41$
• Weight: $8$
• Character: 41.10
• Analytic conductor: $12.808$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	41.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8077860448\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			10.14
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.94432 + 1.41263i) q^{2} -70.4343 q^{3} +(-37.7693 - 116.242i) q^{4} +(-88.4039 - 272.079i) q^{5} +(-136.947 - 99.4975i) q^{6} +(-681.343 + 495.024i) q^{7} +(185.832 - 571.933i) q^{8} +2773.99 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.94432	+	1.41263i	0.171855	+	0.124860i	0.670388	−	0.742011i	\(-0.266128\pi\)
							−0.498533	+	0.866871i	\(0.666128\pi\)
\(3\)	−70.4343			−1.50612			−0.753060	−	0.657951i	\(-0.771423\pi\)
							−0.753060	+	0.657951i	\(0.771423\pi\)
\(5\)	−88.4039	−	272.079i	−0.316283	−	0.973420i	−0.975223	−	0.221224i	\(-0.928995\pi\)
							0.658940	−	0.752196i	\(-0.271005\pi\)
\(7\)	−681.343	+	495.024i	−0.750797	+	0.545486i	−0.896074	−	0.443905i	\(-0.853593\pi\)
							0.145277	+	0.989391i	\(0.453593\pi\)
\(11\)	620.294	−	1909.07i	0.140515	−	0.432461i	−0.855892	−	0.517155i	\(-0.826991\pi\)
							0.996407	+	0.0846935i	\(0.0269911\pi\)
\(13\)	263.043	+	191.112i	0.0332066	+	0.0241260i	0.604265	−	0.796784i	\(-0.293467\pi\)
							−0.571058	+	0.820910i	\(0.693467\pi\)
\(17\)	−10128.7	+	31172.8i	−0.500013	+	1.53888i	0.308983	+	0.951068i	\(0.400011\pi\)
							−0.808996	+	0.587814i	\(0.799989\pi\)
\(19\)	18906.9	−	13736.7i	0.632387	−	0.459456i	−0.224840	−	0.974396i	\(-0.572186\pi\)
							0.857226	+	0.514940i	\(0.172186\pi\)
\(23\)	−8640.14	−	6277.43i	−0.148072	−	0.107581i	0.511283	−	0.859413i	\(-0.329171\pi\)
							−0.659355	+	0.751832i	\(0.729171\pi\)
\(29\)	13999.8	+	43086.9i	0.106593	+	0.328059i	0.990101	−	0.140357i	\(-0.0448249\pi\)
							−0.883508	+	0.468416i	\(0.844825\pi\)
\(31\)	51840.1	−	159547.i	0.312536	−	0.961886i	−0.664221	−	0.747536i	\(-0.731237\pi\)
							0.976757	−	0.214350i	\(-0.0687633\pi\)
\(37\)	92040.9	+	283273.i	0.298727	+	0.919388i	0.981944	+	0.189172i	\(0.0605804\pi\)
							−0.683217	+	0.730216i	\(0.739420\pi\)
\(41\)	439953.	−	34577.5i	0.996926	−	0.0783521i
							\(43\)	−327282.	−	237785.i	−0.627745	−	0.456083i	0.227874	−	0.973691i	\(-0.426823\pi\)
−0.855618	+	0.517608i	\(0.826823\pi\)
\(47\)	−1.07805e6	−	783251.i	−1.51460	−	1.10042i	−0.964085	−	0.265593i	\(-0.914432\pi\)
							−0.550512	−	0.834827i	\(-0.685568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.14	✓	96
41.37	even	5	inner	41.8.d.a.37.14	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.14	✓	96	1.1	even	1	trivial
41.8.d.a.37.14	yes	96	41.37	even	5	inner