Embedded newform 41.8.d.a.10.17

• Label: 41.8.d.a.10.17
• 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			10.17
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.46034 + 5.42026i) q^{2} -59.6296 q^{3} +(-13.2766 - 40.8613i) q^{4} +(149.541 + 460.239i) q^{5} +(-444.857 - 323.208i) q^{6} +(-432.561 + 314.274i) q^{7} +(487.179 - 1499.38i) q^{8} +1368.69 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.46034	+	5.42026i	0.659407	+	0.479088i	0.866463	−	0.499242i	\(-0.166388\pi\)
							−0.207055	+	0.978329i	\(0.566388\pi\)
\(3\)	−59.6296			−1.27508			−0.637540	−	0.770417i	\(-0.720048\pi\)
							−0.637540	+	0.770417i	\(0.720048\pi\)
\(5\)	149.541	+	460.239i	0.535013	+	1.64660i	0.743621	+	0.668602i	\(0.233107\pi\)
							−0.208608	+	0.977999i	\(0.566893\pi\)
\(7\)	−432.561	+	314.274i	−0.476655	+	0.346310i	−0.800029	−	0.599961i	\(-0.795183\pi\)
							0.323374	+	0.946271i	\(0.395183\pi\)
\(11\)	2287.26	−	7039.45i	0.518132	−	1.59465i	−0.259378	−	0.965776i	\(-0.583517\pi\)
							0.777510	−	0.628871i	\(-0.216483\pi\)
\(13\)	−10379.9	−	7541.42i	−1.31036	−	0.952032i	−0.999999	−	0.00141245i	\(-0.999550\pi\)
							−0.310360	−	0.950619i	\(-0.600450\pi\)
\(17\)	7538.25	−	23200.3i	0.372134	−	1.14531i	−0.573258	−	0.819375i	\(-0.694321\pi\)
							0.945392	−	0.325936i	\(-0.105679\pi\)
\(19\)	−25211.6	+	18317.3i	−0.843261	+	0.612665i	−0.923280	−	0.384128i	\(-0.874502\pi\)
							0.0800185	+	0.996793i	\(0.474502\pi\)
\(23\)	57683.9	+	41909.8i	0.988570	+	0.718238i	0.959607	−	0.281342i	\(-0.0907796\pi\)
							0.0289626	+	0.999580i	\(0.490780\pi\)
\(29\)	−39853.6	−	122657.i	−0.303441	−	0.933896i	−0.980254	−	0.197741i	\(-0.936639\pi\)
							0.676813	−	0.736155i	\(-0.263361\pi\)
\(31\)	19691.9	−	60605.3i	0.118719	−	0.365380i	−0.873985	−	0.485952i	\(-0.838473\pi\)
							0.992705	+	0.120572i	\(0.0384729\pi\)
\(37\)	−64187.6	−	197549.i	−0.208327	−	0.641164i	−0.999560	−	0.0296501i	\(-0.990561\pi\)
							0.791233	−	0.611514i	\(-0.209439\pi\)
\(41\)	−426553.	+	113169.i	−0.966561	+	0.256439i
							\(43\)	−526285.	−	382369.i	−1.00944	−	0.733403i	−0.0453506	−	0.998971i	\(-0.514441\pi\)
−0.964092	+	0.265568i	\(0.914441\pi\)
\(47\)	67754.9	+	49226.8i	0.0951916	+	0.0691607i	0.634363	−	0.773035i	\(-0.281262\pi\)
							−0.539171	+	0.842196i	\(0.681262\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.17	✓	96
41.37	even	5	inner	41.8.d.a.37.17	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.17	✓	96	1.1	even	1	trivial
41.8.d.a.37.17	yes	96	41.37	even	5	inner