Embedded newform 451.2.l.a.256.16

• Label: 451.2.l.a.256.16
• Level: $451$
• Weight: $2$
• Character: 451.256
• 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.l (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			256.16
Character	\(\chi\)	\(=\)	451.256
Dual form			451.2.l.a.37.16

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.868271 q^{2} +(-0.984882 - 3.03116i) q^{3} -1.24611 q^{4} +(0.0632253 - 0.194588i) q^{5} +(0.855145 + 2.63186i) q^{6} +(0.658084 + 2.02537i) q^{7} +2.81850 q^{8} +(-5.79086 + 4.20730i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 14 q^{2} - 7 q^{3} + 146 q^{4} - q^{5} - 4 q^{6} - q^{7} - 42 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.868271			−0.613960			−0.306980	−	0.951716i	\(-0.599319\pi\)
							−0.306980	+	0.951716i	\(0.599319\pi\)
\(3\)	−0.984882	−	3.03116i	−0.568622	−	1.75004i	−0.656937	−	0.753946i	\(-0.728148\pi\)
							0.0883149	−	0.996093i	\(-0.471852\pi\)
\(5\)	0.0632253	−	0.194588i	0.0282752	−	0.0870222i	−0.935923	−	0.352205i	\(-0.885432\pi\)
							0.964198	+	0.265182i	\(0.0854321\pi\)
\(7\)	0.658084	+	2.02537i	0.248732	+	0.765520i	0.995000	+	0.0998737i	\(0.0318439\pi\)
							−0.746268	+	0.665646i	\(0.768156\pi\)
\(11\)	1.60295	+	2.90354i	0.483307	+	0.875451i
							\(13\)	−5.78851			−1.60544			−0.802721	−	0.596354i	\(-0.796615\pi\)
−0.802721	+	0.596354i	\(0.796615\pi\)
\(17\)	−0.694463	+	0.504557i	−0.168432	+	0.122373i	−0.668808	−	0.743435i	\(-0.733195\pi\)
							0.500376	+	0.865808i	\(0.333195\pi\)
\(19\)	−0.254617	−	0.184990i	−0.0584132	−	0.0424396i	0.558196	−	0.829709i	\(-0.311494\pi\)
							−0.616609	+	0.787270i	\(0.711494\pi\)
\(23\)	2.78413	+	2.02279i	0.580531	+	0.421780i	0.838915	−	0.544262i	\(-0.183190\pi\)
							−0.258385	+	0.966042i	\(0.583190\pi\)
\(29\)	−1.03881			−0.192902			−0.0964512	−	0.995338i	\(-0.530749\pi\)
							−0.0964512	+	0.995338i	\(0.530749\pi\)
\(31\)	0.691206	+	2.12731i	0.124144	+	0.382077i	0.993744	−	0.111680i	\(-0.0356232\pi\)
							−0.869600	+	0.493757i	\(0.835623\pi\)
\(37\)	8.75858			1.43990			0.719951	−	0.694025i	\(-0.244164\pi\)
							0.719951	+	0.694025i	\(0.244164\pi\)
\(41\)	−3.51709	+	5.35071i	−0.549277	+	0.835640i
							\(43\)	−6.58553	−	4.78467i	−1.00428	−	0.729656i	−0.0412820	−	0.999148i	\(-0.513144\pi\)
−0.963003	+	0.269492i	\(0.913144\pi\)
\(47\)	2.08120	−	6.40526i	0.303574	−	0.934303i	−0.676632	−	0.736321i	\(-0.736561\pi\)
							0.980206	−	0.197982i	\(-0.0634388\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.l.a.256.16	yes	160
11.4	even	5		451.2.i.a.92.16	✓	160
41.37	even	5		451.2.i.a.201.16	yes	160
451.37	even	5	inner	451.2.l.a.37.16	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.16	✓	160	11.4	even	5
451.2.i.a.201.16	yes	160	41.37	even	5
451.2.l.a.37.16	yes	160	451.37	even	5	inner
451.2.l.a.256.16	yes	160	1.1	even	1	trivial