Embedded newform 41.8.d.a.37.9

• Label: 41.8.d.a.37.9
• 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			37.9
Character	\(\chi\)	\(=\)	41.37
Dual form			41.8.d.a.10.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.33427 + 4.60212i) q^{2} -19.9876 q^{3} +(-20.6107 + 63.4331i) q^{4} +(50.0323 - 153.984i) q^{5} +(126.607 - 91.9854i) q^{6} +(-305.258 - 221.783i) q^{7} +(-471.066 - 1449.79i) q^{8} -1787.50 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.33427	+	4.60212i	−0.559876	+	0.406774i	−0.831413	−	0.555654i	\(-0.812468\pi\)
							0.271538	+	0.962428i	\(0.412468\pi\)
\(3\)	−19.9876			−0.427402			−0.213701	−	0.976899i	\(-0.568552\pi\)
							−0.213701	+	0.976899i	\(0.568552\pi\)
\(5\)	50.0323	−	153.984i	0.179001	−	0.550909i	−0.820793	−	0.571226i	\(-0.806468\pi\)
							0.999794	+	0.0203179i	\(0.00646783\pi\)
\(7\)	−305.258	−	221.783i	−0.336375	−	0.244391i	0.406756	−	0.913537i	\(-0.366660\pi\)
							−0.743131	+	0.669146i	\(0.766660\pi\)
\(11\)	1038.12	+	3195.00i	0.235165	+	0.723763i	0.997099	+	0.0761091i	\(0.0242497\pi\)
							−0.761935	+	0.647654i	\(0.775750\pi\)
\(13\)	8408.97	−	6109.48i	1.06155	−	0.771263i	0.0871771	−	0.996193i	\(-0.472215\pi\)
							0.974375	+	0.224930i	\(0.0722154\pi\)
\(17\)	1701.90	+	5237.91i	0.0840161	+	0.258575i	0.984236	−	0.176861i	\(-0.0565943\pi\)
							−0.900220	+	0.435436i	\(0.856594\pi\)
\(19\)	31028.6	+	22543.6i	1.03783	+	0.754024i	0.969860	−	0.243664i	\(-0.0783492\pi\)
							0.0679654	+	0.997688i	\(0.478349\pi\)
\(23\)	78632.7	−	57130.0i	1.34758	−	0.979077i	0.348455	−	0.937325i	\(-0.386706\pi\)
							0.999128	−	0.0417511i	\(-0.0132937\pi\)
\(29\)	−77402.6	+	238221.i	−0.589335	+	1.81379i	−0.00822077	+	0.999966i	\(0.502617\pi\)
							−0.581115	+	0.813822i	\(0.697383\pi\)
\(31\)	−77740.2	−	239260.i	−0.468684	−	1.44246i	−0.854290	−	0.519797i	\(-0.826008\pi\)
							0.385606	−	0.922664i	\(-0.373992\pi\)
\(37\)	57752.3	−	177743.i	0.187440	−	0.576882i	−0.812542	−	0.582903i	\(-0.801917\pi\)
							0.999982	+	0.00602126i	\(0.00191664\pi\)
\(41\)	40627.6	−	439436.i	0.0920614	−	0.995753i
							\(43\)	−12504.6	+	9085.13i	−0.0239845	+	0.0174258i	−0.599713	−	0.800215i	\(-0.704719\pi\)
0.575728	+	0.817641i	\(0.304719\pi\)
\(47\)	−105122.	+	76375.8i	−0.147690	+	0.107303i	−0.659177	−	0.751988i	\(-0.729095\pi\)
							0.511486	+	0.859291i	\(0.329095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.37.9	yes	96
41.10	even	5	inner	41.8.d.a.10.9	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.9	✓	96	41.10	even	5	inner
41.8.d.a.37.9	yes	96	1.1	even	1	trivial