Embedded newform 41.8.d.a.10.19

• Label: 41.8.d.a.10.19
• 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			10.19
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.6008 + 8.42846i) q^{2} +0.557033 q^{3} +(23.9850 + 73.8184i) q^{4} +(-118.045 - 363.305i) q^{5} +(6.46202 + 4.69493i) q^{6} +(375.838 - 273.063i) q^{7} +(223.251 - 687.097i) q^{8} -2186.69 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.6008	+	8.42846i	1.02537	+	0.744978i	0.967378	−	0.253339i	\(-0.0815287\pi\)
							0.0579966	+	0.998317i	\(0.481529\pi\)
\(3\)	0.557033			0.0119112			0.00595561	−	0.999982i	\(-0.498104\pi\)
							0.00595561	+	0.999982i	\(0.498104\pi\)
\(5\)	−118.045	−	363.305i	−0.422330	−	1.29980i	−0.905527	−	0.424288i	\(-0.860525\pi\)
							0.483197	−	0.875512i	\(-0.339475\pi\)
\(7\)	375.838	−	273.063i	0.414150	−	0.300898i	−0.361130	−	0.932516i	\(-0.617609\pi\)
							0.775280	+	0.631618i	\(0.217609\pi\)
\(11\)	1491.06	−	4589.03i	0.337771	−	1.03955i	−0.627570	−	0.778560i	\(-0.715950\pi\)
							0.965341	−	0.260992i	\(-0.0840495\pi\)
\(13\)	2069.61	+	1503.66i	0.261269	+	0.189823i	0.710706	−	0.703489i	\(-0.248376\pi\)
							−0.449438	+	0.893312i	\(0.648376\pi\)
\(17\)	−1994.50	+	6138.45i	−0.0984608	+	0.303031i	−0.988140	−	0.153555i	\(-0.950928\pi\)
							0.889679	+	0.456586i	\(0.150928\pi\)
\(19\)	−6936.65	+	5039.77i	−0.232013	+	0.168567i	−0.697718	−	0.716373i	\(-0.745801\pi\)
							0.465705	+	0.884940i	\(0.345801\pi\)
\(23\)	28688.0	+	20843.0i	0.491646	+	0.357202i	0.805817	−	0.592165i	\(-0.201727\pi\)
							−0.314171	+	0.949366i	\(0.601727\pi\)
\(29\)	16884.7	+	51965.8i	0.128559	+	0.395662i	0.994533	−	0.104426i	\(-0.0333007\pi\)
							−0.865974	+	0.500089i	\(0.833301\pi\)
\(31\)	62500.9	−	192358.i	0.376808	−	1.15970i	−0.565443	−	0.824788i	\(-0.691295\pi\)
							0.942251	−	0.334908i	\(-0.108705\pi\)
\(37\)	−116919.	−	359839.i	−0.379471	−	1.16789i	−0.940412	−	0.340036i	\(-0.889561\pi\)
							0.560941	−	0.827856i	\(-0.310439\pi\)
\(41\)	−261725.	+	355323.i	−0.593064	+	0.805156i
							\(43\)	−317160.	−	230430.i	−0.608329	−	0.441977i	0.240496	−	0.970650i	\(-0.422690\pi\)
−0.848826	+	0.528673i	\(0.822690\pi\)
\(47\)	666492.	+	484235.i	0.936381	+	0.680321i	0.947547	−	0.319617i	\(-0.103554\pi\)
							−0.0111657	+	0.999938i	\(0.503554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.19	✓	96
41.37	even	5	inner	41.8.d.a.37.19	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.19	✓	96	1.1	even	1	trivial
41.8.d.a.37.19	yes	96	41.37	even	5	inner