Embedded newform 451.2.h.a.59.6

• Label: 451.2.h.a.59.6
• Level: $451$
• Weight: $2$
• Character: 451.59
• Analytic conductor: $3.601$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 451 = 11 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	451.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.60125313116\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			59.6
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.682125 + 2.09936i) q^{2} +(0.847891 - 2.60954i) q^{3} +(-2.32400 - 1.68849i) q^{4} +(-0.661943 + 2.03725i) q^{5} +(4.90001 + 3.56007i) q^{6} +2.50637 q^{7} +(1.55836 - 1.13221i) q^{8} +(-3.66373 - 2.66186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/451\mathbb{Z}\right)^\times\).

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.682125	+	2.09936i	−0.482335	+	1.48447i	0.353468	+	0.935446i	\(0.385002\pi\)
							−0.835804	+	0.549029i	\(0.814998\pi\)
\(3\)	0.847891	−	2.60954i	0.489530	−	1.50662i	−0.335780	−	0.941940i	\(-0.609000\pi\)
							0.825311	−	0.564679i	\(-0.191000\pi\)
\(5\)	−0.661943	+	2.03725i	−0.296030	+	0.911086i	0.686844	+	0.726805i	\(0.258996\pi\)
							−0.982874	+	0.184281i	\(0.941004\pi\)
\(7\)	2.50637			0.947320			0.473660	−	0.880708i	\(-0.342933\pi\)
							0.473660	+	0.880708i	\(0.342933\pi\)
\(11\)	2.25361	+	2.43336i	0.679490	+	0.733685i
							\(13\)	0.181517	−	0.558652i	0.0503438	−	0.154942i	−0.922724	−	0.385461i	\(-0.874042\pi\)
0.973068	+	0.230519i	\(0.0740424\pi\)
\(17\)	6.29505			1.52677			0.763387	−	0.645941i	\(-0.223535\pi\)
							0.763387	+	0.645941i	\(0.223535\pi\)
\(19\)	−2.52343	−	1.83338i	−0.578915	−	0.420607i	0.259418	−	0.965765i	\(-0.416469\pi\)
							−0.838333	+	0.545159i	\(0.816469\pi\)
\(23\)	0.694351	+	2.13699i	0.144782	+	0.445594i	0.996983	−	0.0776212i	\(-0.0247325\pi\)
							−0.852201	+	0.523215i	\(0.824732\pi\)
\(29\)	0.465842	+	0.338454i	0.0865046	+	0.0628493i	0.630197	−	0.776435i	\(-0.282974\pi\)
							−0.543692	+	0.839285i	\(0.682974\pi\)
\(31\)	2.55587	+	7.86615i	0.459047	+	1.41280i	0.866317	+	0.499494i	\(0.166481\pi\)
							−0.407270	+	0.913308i	\(0.633519\pi\)
\(37\)	6.40648	+	4.65458i	1.05322	+	0.765208i	0.972822	−	0.231554i	\(-0.0743811\pi\)
							0.0803972	+	0.996763i	\(0.474381\pi\)
\(41\)	5.22059	+	3.70748i	0.815319	+	0.579012i
							\(43\)	−2.27410	−	6.99896i	−0.346797	−	1.06733i	−0.960615	−	0.277883i	\(-0.910367\pi\)
0.613818	−	0.789447i	\(-0.289633\pi\)
\(47\)	−8.17671			−1.19270			−0.596348	−	0.802726i	\(-0.703382\pi\)
							−0.596348	+	0.802726i	\(0.703382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.6	✓	160
11.3	even	5		451.2.j.a.223.35	yes	160
41.16	even	5		451.2.j.a.180.35	yes	160
451.344	even	5	inner	451.2.h.a.344.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.6	✓	160	1.1	even	1	trivial
451.2.h.a.344.6	yes	160	451.344	even	5	inner
451.2.j.a.180.35	yes	160	41.16	even	5
451.2.j.a.223.35	yes	160	11.3	even	5