Embedded newform 41.8.d.a.37.19

• Label: 41.8.d.a.37.19
• Level: $41$
• Weight: $8$
• Character: 41.37
• Analytic conductor: $12.808$
• Analytic rank: $0$
• Dimension: $96$
• 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			37.19
Character	\(\chi\)	\(=\)	41.37
Dual form			41.8.d.a.10.19

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.19	✓	96	41.10	even	5	inner
41.8.d.a.37.19	yes	96	1.1	even	1	trivial