Embedded newform 41.8.d.a.10.15

• Label: 41.8.d.a.10.15
• 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			10.15
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.04340 + 2.21116i) q^{2} -19.4315 q^{3} +(-35.1811 - 108.276i) q^{4} +(86.0117 + 264.717i) q^{5} +(-59.1377 - 42.9660i) q^{6} +(326.046 - 236.886i) q^{7} +(281.143 - 865.268i) q^{8} -1809.42 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.04340	+	2.21116i	0.269001	+	0.195441i	0.714106	−	0.700038i	\(-0.246834\pi\)
							−0.445105	+	0.895478i	\(0.646834\pi\)
\(3\)	−19.4315			−0.415510			−0.207755	−	0.978181i	\(-0.566616\pi\)
							−0.207755	+	0.978181i	\(0.566616\pi\)
\(5\)	86.0117	+	264.717i	0.307725	+	0.947079i	0.978646	+	0.205551i	\(0.0658986\pi\)
							−0.670922	+	0.741528i	\(0.734101\pi\)
\(7\)	326.046	−	236.886i	0.359282	−	0.261034i	−0.393470	−	0.919337i	\(-0.628726\pi\)
							0.752753	+	0.658303i	\(0.228726\pi\)
\(11\)	−1557.64	+	4793.93i	−0.352853	+	1.08597i	0.604391	+	0.796688i	\(0.293416\pi\)
							−0.957244	+	0.289281i	\(0.906584\pi\)
\(13\)	10272.9	+	7463.72i	1.29686	+	0.942223i	0.999920	−	0.0126741i	\(-0.00403440\pi\)
							0.296938	+	0.954897i	\(0.404034\pi\)
\(17\)	−6633.61	+	20416.2i	−0.327475	+	1.00787i	0.642835	+	0.766004i	\(0.277758\pi\)
							−0.970311	+	0.241861i	\(0.922242\pi\)
\(19\)	−38868.8	+	28239.8i	−1.30006	+	0.944548i	−0.999956	−	0.00935870i	\(-0.997021\pi\)
							−0.300103	+	0.953907i	\(0.597021\pi\)
\(23\)	−14140.7	−	10273.8i	−0.242339	−	0.176070i	0.459986	−	0.887926i	\(-0.347855\pi\)
							−0.702325	+	0.711857i	\(0.747855\pi\)
\(29\)	65033.5	+	200152.i	0.495158	+	1.52394i	0.816710	+	0.577048i	\(0.195795\pi\)
							−0.321552	+	0.946892i	\(0.604205\pi\)
\(31\)	4359.43	−	13416.9i	0.0262823	−	0.0808886i	−0.937055	−	0.349182i	\(-0.886460\pi\)
							0.963337	+	0.268293i	\(0.0864596\pi\)
\(37\)	−127554.	−	392571.i	−0.413989	−	1.27413i	−0.913152	−	0.407619i	\(-0.866359\pi\)
							0.499163	−	0.866508i	\(-0.333641\pi\)
\(41\)	364007.	−	249507.i	0.824832	−	0.565377i
							\(43\)	−424673.	−	308543.i	−0.814545	−	0.591801i	0.100600	−	0.994927i	\(-0.467924\pi\)
−0.915145	+	0.403126i	\(0.867924\pi\)
\(47\)	981447.	+	713063.i	1.37887	+	1.00181i	0.996984	+	0.0776022i	\(0.0247264\pi\)
							0.381889	+	0.924208i	\(0.375274\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.15	✓	96
41.37	even	5	inner	41.8.d.a.37.15	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.15	✓	96	1.1	even	1	trivial
41.8.d.a.37.15	yes	96	41.37	even	5	inner