Embedded newform 41.8.d.a.10.10

• Label: 41.8.d.a.10.10
• 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			10.10
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.12065 - 3.72037i) q^{2} +62.0240 q^{3} +(-27.1743 - 83.6338i) q^{4} +(134.210 + 413.055i) q^{5} +(-317.603 - 230.752i) q^{6} +(-1138.62 + 827.255i) q^{7} +(-422.356 + 1299.88i) q^{8} +1659.98 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.12065	−	3.72037i	−0.452606	−	0.328838i	0.338018	−	0.941140i	\(-0.390244\pi\)
							−0.790624	+	0.612302i	\(0.790244\pi\)
\(3\)	62.0240			1.32628			0.663140	−	0.748495i	\(-0.269223\pi\)
							0.663140	+	0.748495i	\(0.269223\pi\)
\(5\)	134.210	+	413.055i	0.480163	+	1.47779i	0.838866	+	0.544338i	\(0.183219\pi\)
							−0.358703	+	0.933452i	\(0.616781\pi\)
\(7\)	−1138.62	+	827.255i	−1.25469	+	0.911583i	−0.998484	−	0.0550406i	\(-0.982471\pi\)
							−0.256202	+	0.966623i	\(0.582471\pi\)
\(11\)	−844.494	+	2599.08i	−0.191303	+	0.588771i	0.808697	+	0.588226i	\(0.200173\pi\)
							−1.00000		0.000544699i	\(0.999827\pi\)
\(13\)	−7560.09	−	5492.72i	−0.954388	−	0.693404i	−0.00254748	−	0.999997i	\(-0.500811\pi\)
							−0.951841	+	0.306593i	\(0.900811\pi\)
\(17\)	−2115.61	+	6511.17i	−0.104439	+	0.321431i	−0.989598	−	0.143857i	\(-0.954049\pi\)
							0.885159	+	0.465288i	\(0.154049\pi\)
\(19\)	−2814.01	+	2044.50i	−0.0941213	+	0.0683831i	−0.633851	−	0.773455i	\(-0.718527\pi\)
							0.539729	+	0.841839i	\(0.318527\pi\)
\(23\)	73994.4	+	53760.1i	1.26809	+	0.921325i	0.999125	−	0.0418242i	\(-0.0133169\pi\)
							0.268969	+	0.963149i	\(0.413317\pi\)
\(29\)	58129.1	+	178903.i	0.442589	+	1.36215i	0.885106	+	0.465389i	\(0.154085\pi\)
							−0.442517	+	0.896760i	\(0.645915\pi\)
\(31\)	62851.9	−	193438.i	0.378924	−	1.16621i	−0.561868	−	0.827227i	\(-0.689917\pi\)
							0.940793	−	0.338983i	\(-0.110083\pi\)
\(37\)	−126361.	−	388899.i	−0.410116	−	1.26221i	−0.916547	−	0.399927i	\(-0.869035\pi\)
							0.506431	−	0.862281i	\(-0.330965\pi\)
\(41\)	416728.	−	145230.i	0.944299	−	0.329089i
							\(43\)	447440.	+	325084.i	0.858213	+	0.623528i	0.927398	−	0.374076i	\(-0.122040\pi\)
−0.0691852	+	0.997604i	\(0.522040\pi\)
\(47\)	13704.9	+	9957.22i	0.0192546	+	0.0139893i	0.597371	−	0.801965i	\(-0.296212\pi\)
							−0.578116	+	0.815954i	\(0.696212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.10	✓	96
41.37	even	5	inner	41.8.d.a.37.10	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.10	✓	96	1.1	even	1	trivial
41.8.d.a.37.10	yes	96	41.37	even	5	inner