Embedded newform 451.2.i.a.201.16

• Label: 451.2.i.a.201.16
• Level: $451$
• Weight: $2$
• Character: 451.201
• 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.i (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			201.16
Character	\(\chi\)	\(=\)	451.201
Dual form			451.2.i.a.92.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.268311 + 0.825775i) q^{2} +(-0.984882 - 3.03116i) q^{3} +(1.00812 + 0.732442i) q^{4} +(-0.165526 + 0.120262i) q^{5} +2.76731 q^{6} +(-1.72289 + 1.25175i) q^{7} +(-2.28021 + 1.65667i) q^{8} +(-5.79086 + 4.20730i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - q^{7} + 13 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.268311	+	0.825775i	−0.189724	+	0.583911i	−0.999998	−	0.00214083i	\(-0.999319\pi\)
							0.810273	+	0.586052i	\(0.199319\pi\)
\(3\)	−0.984882	−	3.03116i	−0.568622	−	1.75004i	−0.656937	−	0.753946i	\(-0.728148\pi\)
							0.0883149	−	0.996093i	\(-0.471852\pi\)
\(5\)	−0.165526	+	0.120262i	−0.0740255	+	0.0537827i	−0.624183	−	0.781279i	\(-0.714568\pi\)
							0.550157	+	0.835061i	\(0.314568\pi\)
\(7\)	−1.72289	+	1.25175i	−0.651190	+	0.473117i	−0.863676	−	0.504047i	\(-0.831844\pi\)
							0.212486	+	0.977164i	\(0.431844\pi\)
\(11\)	−3.00347	−	1.40683i	−0.905581	−	0.424174i
							\(13\)	−1.78875	+	5.50520i	−0.496109	+	1.52687i	0.319112	+	0.947717i	\(0.396615\pi\)
−0.815221	+	0.579150i	\(0.803385\pi\)
\(17\)	0.265261	−	0.816390i	0.0643353	−	0.198004i	−0.913722	−	0.406340i	\(-0.866805\pi\)
							0.978057	+	0.208336i	\(0.0668049\pi\)
\(19\)	0.0972551	−	0.299320i	0.0223118	−	0.0686688i	−0.939281	−	0.343150i	\(-0.888506\pi\)
							0.961592	+	0.274481i	\(0.0885061\pi\)
\(23\)	2.78413	−	2.02279i	0.580531	−	0.421780i	−0.258385	−	0.966042i	\(-0.583190\pi\)
							0.838915	+	0.544262i	\(0.183190\pi\)
\(29\)	0.840416	+	0.610598i	0.156061	+	0.113385i	0.663076	−	0.748553i	\(-0.269251\pi\)
							−0.507014	+	0.861938i	\(0.669251\pi\)
\(31\)	−1.80960	−	1.31475i	−0.325014	−	0.236137i	0.413298	−	0.910596i	\(-0.364377\pi\)
							−0.738312	+	0.674459i	\(0.764377\pi\)
\(37\)	−7.08584	−	5.14817i	−1.16491	−	0.846353i	−0.174515	−	0.984654i	\(-0.555836\pi\)
							−0.990390	+	0.138301i	\(0.955836\pi\)
\(41\)	5.99045	+	2.26152i	0.935551	+	0.353190i
							\(43\)	−6.58553	+	4.78467i	−1.00428	+	0.729656i	−0.963003	−	0.269492i	\(-0.913144\pi\)
−0.0412820	+	0.999148i	\(0.513144\pi\)
\(47\)	−5.44864	−	3.95867i	−0.794766	−	0.577431i	0.114608	−	0.993411i	\(-0.463439\pi\)
							−0.909374	+	0.415979i	\(0.863439\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.16	✓	160	451.92	even	5	inner
451.2.i.a.201.16	yes	160	1.1	even	1	trivial
451.2.l.a.37.16	yes	160	11.4	even	5
451.2.l.a.256.16	yes	160	41.10	even	5