Embedded newform 451.2.h.a.59.4

• Label: 451.2.h.a.59.4
• 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			59.4
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.746065 + 2.29615i) q^{2} +(-0.0930617 + 0.286414i) q^{3} +(-3.09767 - 2.25059i) q^{4} +(0.524330 - 1.61372i) q^{5} +(-0.588221 - 0.427368i) q^{6} -0.287887 q^{7} +(3.57231 - 2.59543i) q^{8} +(2.35368 + 1.71005i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.746065	+	2.29615i	−0.527548	+	1.62362i	0.231675	+	0.972793i	\(0.425580\pi\)
							−0.759222	+	0.650831i	\(0.774420\pi\)
\(3\)	−0.0930617	+	0.286414i	−0.0537292	+	0.165361i	−0.974320	−	0.225166i	\(-0.927707\pi\)
							0.920591	+	0.390528i	\(0.127707\pi\)
\(5\)	0.524330	−	1.61372i	0.234487	−	0.721678i	−0.762702	−	0.646750i	\(-0.776128\pi\)
							0.997189	−	0.0749273i	\(-0.0238725\pi\)
\(7\)	−0.287887			−0.108811			−0.0544055	−	0.998519i	\(-0.517326\pi\)
							−0.0544055	+	0.998519i	\(0.517326\pi\)
\(11\)	3.29155	−	0.407040i	0.992440	−	0.122727i
							\(13\)	1.92400	−	5.92145i	0.533621	−	1.64232i	−0.212990	−	0.977054i	\(-0.568320\pi\)
0.746611	−	0.665261i	\(-0.231680\pi\)
\(17\)	−2.60643			−0.632152			−0.316076	−	0.948734i	\(-0.602366\pi\)
							−0.316076	+	0.948734i	\(0.602366\pi\)
\(19\)	4.61108	+	3.35015i	1.05785	+	0.768576i	0.973690	−	0.227875i	\(-0.0731779\pi\)
							0.0841644	+	0.996452i	\(0.473178\pi\)
\(23\)	1.70635	+	5.25161i	0.355799	+	1.09504i	0.955545	+	0.294846i	\(0.0952684\pi\)
							−0.599746	+	0.800191i	\(0.704732\pi\)
\(29\)	−4.57799	−	3.32611i	−0.850112	−	0.617643i	0.0750649	−	0.997179i	\(-0.476084\pi\)
							−0.925177	+	0.379536i	\(0.876084\pi\)
\(31\)	0.937071	+	2.88401i	0.168303	+	0.517983i	0.999264	−	0.0383468i	\(-0.0122092\pi\)
							−0.830962	+	0.556330i	\(0.812209\pi\)
\(37\)	−1.85161	−	1.34528i	−0.304403	−	0.221162i	0.425088	−	0.905152i	\(-0.360243\pi\)
							−0.729491	+	0.683990i	\(0.760243\pi\)
\(41\)	2.68773	+	5.81172i	0.419753	+	0.907639i
							\(43\)	−3.16166	−	9.73059i	−0.482149	−	1.48390i	−0.836068	−	0.548625i	\(-0.815151\pi\)
0.353920	−	0.935276i	\(-0.384849\pi\)
\(47\)	3.56683			0.520276			0.260138	−	0.965572i	\(-0.416232\pi\)
							0.260138	+	0.965572i	\(0.416232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.4	✓	160
11.3	even	5		451.2.j.a.223.37	yes	160
41.16	even	5		451.2.j.a.180.37	yes	160
451.344	even	5	inner	451.2.h.a.344.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.4	✓	160	1.1	even	1	trivial
451.2.h.a.344.4	yes	160	451.344	even	5	inner
451.2.j.a.180.37	yes	160	41.16	even	5
451.2.j.a.223.37	yes	160	11.3	even	5