Embedded newform 451.2.h.a.59.8

• Label: 451.2.h.a.59.8
• 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			59.8
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.602884 + 1.85548i) q^{2} +(0.864312 - 2.66008i) q^{3} +(-1.46132 - 1.06171i) q^{4} +(1.15183 - 3.54497i) q^{5} +(4.41465 + 3.20743i) q^{6} +3.75276 q^{7} +(-0.305743 + 0.222135i) q^{8} +(-3.90193 - 2.83492i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.602884	+	1.85548i	−0.426303	+	1.31203i	0.475438	+	0.879749i	\(0.342290\pi\)
							−0.901741	+	0.432277i	\(0.857710\pi\)
\(3\)	0.864312	−	2.66008i	0.499011	−	1.53580i	−0.311601	−	0.950213i	\(-0.600865\pi\)
							0.810612	−	0.585584i	\(-0.199135\pi\)
\(5\)	1.15183	−	3.54497i	0.515114	−	1.58536i	−0.267961	−	0.963430i	\(-0.586350\pi\)
							0.783075	−	0.621928i	\(-0.213650\pi\)
\(7\)	3.75276			1.41841			0.709205	−	0.705003i	\(-0.249054\pi\)
							0.709205	+	0.705003i	\(0.249054\pi\)
\(11\)	−0.530290	−	3.27396i	−0.159888	−	0.987135i
							\(13\)	−0.989104	+	3.04415i	−0.274328	+	0.844295i	0.715068	+	0.699055i	\(0.246396\pi\)
−0.989396	+	0.145240i	\(0.953604\pi\)
\(17\)	−7.49261			−1.81723			−0.908613	−	0.417640i	\(-0.862857\pi\)
							−0.908613	+	0.417640i	\(0.862857\pi\)
\(19\)	2.53441	+	1.84136i	0.581433	+	0.422436i	0.839241	−	0.543760i	\(-0.183000\pi\)
							−0.257807	+	0.966196i	\(0.583000\pi\)
\(23\)	1.95494	+	6.01668i	0.407633	+	1.25457i	0.918677	+	0.395011i	\(0.129259\pi\)
							−0.511044	+	0.859555i	\(0.670741\pi\)
\(29\)	2.42503	+	1.76189i	0.450317	+	0.327174i	0.789721	−	0.613466i	\(-0.210225\pi\)
							−0.339404	+	0.940641i	\(0.610225\pi\)
\(31\)	−0.524505	−	1.61426i	−0.0942040	−	0.289930i	0.892841	−	0.450371i	\(-0.148708\pi\)
							−0.987045	+	0.160441i	\(0.948708\pi\)
\(37\)	8.14956	+	5.92100i	1.33978	+	0.973406i	0.999452	+	0.0330994i	\(0.0105378\pi\)
							0.340327	+	0.940307i	\(0.389462\pi\)
\(41\)	2.69528	−	5.80822i	0.420932	−	0.907092i
							\(43\)	−0.271161	−	0.834548i	−0.0413517	−	0.127267i	0.928249	−	0.371958i	\(-0.121314\pi\)
−0.969601	+	0.244691i	\(0.921314\pi\)
\(47\)	4.54234			0.662568			0.331284	−	0.943531i	\(-0.392518\pi\)
							0.331284	+	0.943531i	\(0.392518\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.8	✓	160
11.3	even	5		451.2.j.a.223.33	yes	160
41.16	even	5		451.2.j.a.180.33	yes	160
451.344	even	5	inner	451.2.h.a.344.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.8	✓	160	1.1	even	1	trivial
451.2.h.a.344.8	yes	160	451.344	even	5	inner
451.2.j.a.180.33	yes	160	41.16	even	5
451.2.j.a.223.33	yes	160	11.3	even	5