Embedded newform 41.6.f.a.23.2

• Label: 41.6.f.a.23.2
• Level: $41$
• Weight: $6$
• Character: 41.23
• Analytic conductor: $6.576$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	41.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.57573661233\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			23.2
Character	\(\chi\)	\(=\)	41.23
Dual form			41.6.f.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.67523 - 8.23350i) q^{2} -18.7873i q^{3} +(-34.7451 + 25.2438i) q^{4} +(-23.8795 + 17.3495i) q^{5} +(-154.686 + 50.2604i) q^{6} +(-48.6094 - 15.7941i) q^{7} +(76.6730 + 55.7062i) q^{8} -109.964 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.67523	−	8.23350i	−0.472917	−	1.45549i	−0.848746	−	0.528801i	\(-0.822642\pi\)
							0.375828	−	0.926689i	\(-0.377358\pi\)
\(3\)		−	18.7873i		−	1.20521i	−0.798040	−	0.602604i	\(-0.794130\pi\)
							0.798040	−	0.602604i	\(-0.205870\pi\)
\(5\)	−23.8795	+	17.3495i	−0.427170	+	0.310357i	−0.780516	−	0.625135i	\(-0.785044\pi\)
							0.353346	+	0.935493i	\(0.385044\pi\)
\(7\)	−48.6094	−	15.7941i	−0.374951	−	0.121829i	0.115478	−	0.993310i	\(-0.463160\pi\)
							−0.490429	+	0.871481i	\(0.663160\pi\)
\(11\)	91.1081	−	125.400i	0.227026	−	0.312474i	−0.680274	−	0.732957i	\(-0.738139\pi\)
							0.907300	+	0.420483i	\(0.138139\pi\)
\(13\)	200.218	−	65.0549i	0.328583	−	0.106763i	−0.140080	−	0.990140i	\(-0.544736\pi\)
							0.468663	+	0.883377i	\(0.344736\pi\)
\(17\)	93.0447	−	128.065i	0.0780853	−	0.107475i	−0.768188	−	0.640225i	\(-0.778841\pi\)
							0.846273	+	0.532750i	\(0.178841\pi\)
\(19\)	−1418.02	−	460.743i	−0.901153	−	0.292802i	−0.178440	−	0.983951i	\(-0.557105\pi\)
							−0.722713	+	0.691148i	\(0.757105\pi\)
\(23\)	47.1088	+	144.986i	0.0185687	+	0.0571487i	0.959912	−	0.280303i	\(-0.0904350\pi\)
							−0.941343	+	0.337452i	\(0.890435\pi\)
\(29\)	−2583.63	−	3556.06i	−0.570472	−	0.785188i	0.422138	−	0.906532i	\(-0.361280\pi\)
							−0.992611	+	0.121344i	\(0.961280\pi\)
\(31\)	−6486.87	−	4712.98i	−1.21236	−	0.880829i	−0.216915	−	0.976191i	\(-0.569599\pi\)
							−0.995443	+	0.0953613i	\(0.969599\pi\)
\(37\)	−6327.30	+	4597.06i	−0.759826	+	0.552046i	−0.898857	−	0.438242i	\(-0.855601\pi\)
							0.139031	+	0.990288i	\(0.455601\pi\)
\(41\)	1810.79	+	10610.2i	0.168232	+	0.985747i
							\(43\)	−1086.24	−	3343.11i	−0.0895892	−	0.275727i	0.896217	−	0.443617i	\(-0.146305\pi\)
−0.985806	+	0.167889i	\(0.946305\pi\)
\(47\)	14322.7	−	4653.73i	0.945760	−	0.307296i	0.204768	−	0.978810i	\(-0.434356\pi\)
							0.740991	+	0.671514i	\(0.234356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.6.f.a.23.2	✓	64
41.25	even	10	inner	41.6.f.a.25.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.f.a.23.2	✓	64	1.1	even	1	trivial
41.6.f.a.25.2	yes	64	41.25	even	10	inner