Embedded newform 41.8.d.a.10.18

• Label: 41.8.d.a.10.18
• 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			10.18
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.69168 + 5.58834i) q^{2} +6.71135 q^{3} +(-11.6217 - 35.7678i) q^{4} +(-42.5705 - 131.018i) q^{5} +(51.6216 + 37.5053i) q^{6} +(-1081.19 + 785.532i) q^{7} +(486.551 - 1497.45i) q^{8} -2141.96 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.69168	+	5.58834i	0.679855	+	0.493944i	0.873310	−	0.487165i	\(-0.161969\pi\)
							−0.193454	+	0.981109i	\(0.561969\pi\)
\(3\)	6.71135			0.143511			0.0717555	−	0.997422i	\(-0.477140\pi\)
							0.0717555	+	0.997422i	\(0.477140\pi\)
\(5\)	−42.5705	−	131.018i	−0.152305	−	0.468746i	0.845573	−	0.533860i	\(-0.179259\pi\)
							−0.997878	+	0.0651138i	\(0.979259\pi\)
\(7\)	−1081.19	+	785.532i	−1.19141	+	0.865607i	−0.993412	−	0.114597i	\(-0.963442\pi\)
							−0.197993	+	0.980203i	\(0.563442\pi\)
\(11\)	−2318.14	+	7134.50i	−0.525128	+	1.61618i	0.238935	+	0.971035i	\(0.423202\pi\)
							−0.764063	+	0.645141i	\(0.776798\pi\)
\(13\)	−6762.23	−	4913.05i	−0.853667	−	0.620225i	0.0724877	−	0.997369i	\(-0.476906\pi\)
							−0.926155	+	0.377144i	\(0.876906\pi\)
\(17\)	3765.27	−	11588.3i	0.185877	−	0.572070i	−0.814086	−	0.580745i	\(-0.802761\pi\)
							0.999962	+	0.00867504i	\(0.00276138\pi\)
\(19\)	8793.85	−	6389.10i	0.294131	−	0.213699i	−0.430926	−	0.902387i	\(-0.641813\pi\)
							0.725058	+	0.688688i	\(0.241813\pi\)
\(23\)	−5820.96	−	4229.18i	−0.0997579	−	0.0724783i	0.536788	−	0.843717i	\(-0.319637\pi\)
							−0.636546	+	0.771239i	\(0.719637\pi\)
\(29\)	−69099.4	−	212666.i	−0.526116	−	1.61922i	−0.762099	−	0.647460i	\(-0.775831\pi\)
							0.235984	−	0.971757i	\(-0.424169\pi\)
\(31\)	−82323.5	+	253366.i	−0.496315	+	1.52750i	0.318582	+	0.947895i	\(0.396793\pi\)
							−0.814897	+	0.579606i	\(0.803207\pi\)
\(37\)	30728.2	+	94571.8i	0.0997313	+	0.306942i	0.988458	−	0.151496i	\(-0.0484092\pi\)
							−0.888726	+	0.458438i	\(0.848409\pi\)
\(41\)	−281247.	−	340080.i	−0.637302	−	0.770615i
							\(43\)	430676.	+	312905.i	0.826060	+	0.600168i	0.918442	−	0.395556i	\(-0.129448\pi\)
−0.0923820	+	0.995724i	\(0.529448\pi\)
\(47\)	754957.	+	548509.i	1.06067	+	0.770621i	0.974212	−	0.225634i	\(-0.0724455\pi\)
							0.0864570	+	0.996256i	\(0.472445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.18	✓	96
41.37	even	5	inner	41.8.d.a.37.18	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.18	✓	96	1.1	even	1	trivial
41.8.d.a.37.18	yes	96	41.37	even	5	inner