Embedded newform 451.2.i.a.92.16

• Label: 451.2.i.a.92.16
• Level: $451$
• Weight: $2$
• Character: 451.92
• Analytic conductor: $3.601$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• 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			92.16
Character	\(\chi\)	\(=\)	451.92
Dual form			451.2.i.a.201.16

$q$-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 \(\left(\mathbb{Z}/451\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.16	✓	160	1.1	even	1	trivial
451.2.i.a.201.16	yes	160	451.201	even	5	inner
451.2.l.a.37.16	yes	160	41.37	even	5
451.2.l.a.256.16	yes	160	11.3	even	5