Embedded newform 451.2.h.a.59.1

• Label: 451.2.h.a.59.1
• 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			59.1
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.865501 + 2.66374i) q^{2} +(0.258258 - 0.794835i) q^{3} +(-4.72838 - 3.43537i) q^{4} +(-0.276609 + 0.851316i) q^{5} +(1.89371 + 1.37586i) q^{6} +1.68843 q^{7} +(8.71151 - 6.32928i) q^{8} +(1.86198 + 1.35281i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.865501	+	2.66374i	−0.612002	+	1.88355i	−0.173447	+	0.984843i	\(0.555490\pi\)
							−0.438555	+	0.898704i	\(0.644510\pi\)
\(3\)	0.258258	−	0.794835i	0.149105	−	0.458898i	−0.848411	−	0.529338i	\(-0.822440\pi\)
							0.997516	+	0.0704398i	\(0.0224403\pi\)
\(5\)	−0.276609	+	0.851316i	−0.123703	+	0.380720i	−0.993663	−	0.112404i	\(-0.964145\pi\)
							0.869959	+	0.493124i	\(0.164145\pi\)
\(7\)	1.68843			0.638165			0.319082	−	0.947727i	\(-0.396625\pi\)
							0.319082	+	0.947727i	\(0.396625\pi\)
\(11\)	−2.89789	+	1.61315i	−0.873746	+	0.486383i
							\(13\)	−1.36643	+	4.20545i	−0.378981	+	1.16638i	0.561773	+	0.827292i	\(0.310120\pi\)
−0.940753	+	0.339091i	\(0.889880\pi\)
\(17\)	−6.57285			−1.59415			−0.797075	−	0.603881i	\(-0.793620\pi\)
							−0.797075	+	0.603881i	\(0.793620\pi\)
\(19\)	0.882291	+	0.641022i	0.202411	+	0.147061i	0.684374	−	0.729131i	\(-0.260076\pi\)
							−0.481962	+	0.876192i	\(0.660076\pi\)
\(23\)	0.177666	+	0.546800i	0.0370460	+	0.114016i	0.967869	−	0.251453i	\(-0.0809085\pi\)
							−0.930823	+	0.365469i	\(0.880909\pi\)
\(29\)	−3.54595	−	2.57629i	−0.658467	−	0.478404i	0.207678	−	0.978197i	\(-0.433409\pi\)
							−0.866145	+	0.499793i	\(0.833409\pi\)
\(31\)	1.35208	+	4.16127i	0.242840	+	0.747386i	0.995984	+	0.0895298i	\(0.0285364\pi\)
							−0.753144	+	0.657856i	\(0.771464\pi\)
\(37\)	2.78361	+	2.02241i	0.457623	+	0.332483i	0.792598	−	0.609744i	\(-0.208728\pi\)
							−0.334975	+	0.942227i	\(0.608728\pi\)
\(41\)	−6.21703	+	1.53250i	−0.970937	+	0.239336i
							\(43\)	3.23391	+	9.95294i	0.493166	+	1.51781i	0.819795	+	0.572657i	\(0.194087\pi\)
−0.326629	+	0.945153i	\(0.605913\pi\)
\(47\)	−7.78210			−1.13514			−0.567568	−	0.823327i	\(-0.692116\pi\)
							−0.567568	+	0.823327i	\(0.692116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.1	✓	160
11.3	even	5		451.2.j.a.223.40	yes	160
41.16	even	5		451.2.j.a.180.40	yes	160
451.344	even	5	inner	451.2.h.a.344.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.1	✓	160	1.1	even	1	trivial
451.2.h.a.344.1	yes	160	451.344	even	5	inner
451.2.j.a.180.40	yes	160	41.16	even	5
451.2.j.a.223.40	yes	160	11.3	even	5