Embedded newform 41.8.d.a.10.5

• Label: 41.8.d.a.10.5
• 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			10.5
Character	\(\chi\)	\(=\)	41.10
Dual form			41.8.d.a.37.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.9001 - 9.37244i) q^{2} +24.3124 q^{3} +(39.0146 + 120.075i) q^{4} +(51.9882 + 160.003i) q^{5} +(-313.631 - 227.866i) q^{6} +(338.641 - 246.037i) q^{7} +(-8.60326 + 26.4781i) q^{8} -1595.91 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.9001	−	9.37244i	−1.14021	−	0.828414i	−0.153065	−	0.988216i	\(-0.548914\pi\)
							−0.987149	+	0.159802i	\(0.948914\pi\)
\(3\)	24.3124			0.519879			0.259940	−	0.965625i	\(-0.416297\pi\)
							0.259940	+	0.965625i	\(0.416297\pi\)
\(5\)	51.9882	+	160.003i	0.185998	+	0.572445i	0.999964	−	0.00846850i	\(-0.00269564\pi\)
							−0.813966	+	0.580913i	\(0.802696\pi\)
\(7\)	338.641	−	246.037i	0.373161	−	0.271117i	−0.385359	−	0.922767i	\(-0.625923\pi\)
							0.758521	+	0.651649i	\(0.225923\pi\)
\(11\)	2373.06	−	7303.53i	0.537569	−	1.65447i	−0.200462	−	0.979701i	\(-0.564244\pi\)
							0.738032	−	0.674766i	\(-0.235756\pi\)
\(13\)	−6224.71	−	4522.52i	−0.785810	−	0.570925i	0.120907	−	0.992664i	\(-0.461420\pi\)
							−0.906717	+	0.421739i	\(0.861420\pi\)
\(17\)	−7949.41	+	24465.8i	−0.392431	+	1.20778i	0.538513	+	0.842617i	\(0.318986\pi\)
							−0.930944	+	0.365162i	\(0.881014\pi\)
\(19\)	−45096.2	+	32764.3i	−1.50835	+	1.09588i	−0.541450	+	0.840733i	\(0.682125\pi\)
							−0.966902	+	0.255149i	\(0.917875\pi\)
\(23\)	−50034.2	−	36352.0i	−0.857471	−	0.622989i	0.0697247	−	0.997566i	\(-0.477788\pi\)
							−0.927196	+	0.374577i	\(0.877788\pi\)
\(29\)	−11058.5	−	34034.6i	−0.0841983	−	0.259136i	0.900090	−	0.435704i	\(-0.143500\pi\)
							−0.984288	+	0.176568i	\(0.943500\pi\)
\(31\)	−79173.4	+	243671.i	−0.477324	+	1.46905i	0.365474	+	0.930822i	\(0.380907\pi\)
							−0.842798	+	0.538231i	\(0.819093\pi\)
\(37\)	−25433.8	−	78277.1i	−0.0825476	−	0.254055i	0.901261	−	0.433276i	\(-0.142642\pi\)
							−0.983809	+	0.179221i	\(0.942642\pi\)
\(41\)	128535.	−	422177.i	0.291258	−	0.956644i
							\(43\)	−420211.	−	305301.i	−0.805986	−	0.585583i	0.106678	−	0.994294i	\(-0.465979\pi\)
−0.912664	+	0.408710i	\(0.865979\pi\)
\(47\)	−721989.	−	524555.i	−1.01435	−	0.736968i	−0.0492330	−	0.998787i	\(-0.515678\pi\)
							−0.965117	+	0.261819i	\(0.915678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.8.d.a.10.5	✓	96
41.37	even	5	inner	41.8.d.a.37.5	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.8.d.a.10.5	✓	96	1.1	even	1	trivial
41.8.d.a.37.5	yes	96	41.37	even	5	inner