Embedded newform 451.2.h.a.59.10

• Label: 451.2.h.a.59.10
• 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			59.10
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.527489 + 1.62344i) q^{2} +(0.911798 - 2.80623i) q^{3} +(-0.739295 - 0.537129i) q^{4} +(-0.249725 + 0.768576i) q^{5} +(4.07479 + 2.96051i) q^{6} -4.89487 q^{7} +(-1.50000 + 1.08981i) q^{8} +(-4.61648 - 3.35407i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.527489	+	1.62344i	−0.372991	+	1.14795i	0.571833	+	0.820370i	\(0.306233\pi\)
							−0.944824	+	0.327579i	\(0.893767\pi\)
\(3\)	0.911798	−	2.80623i	0.526427	−	1.62018i	−0.235049	−	0.971983i	\(-0.575525\pi\)
							0.761477	−	0.648193i	\(-0.224475\pi\)
\(5\)	−0.249725	+	0.768576i	−0.111681	+	0.343718i	−0.991240	−	0.132071i	\(-0.957837\pi\)
							0.879560	+	0.475789i	\(0.157837\pi\)
\(7\)	−4.89487			−1.85009			−0.925044	−	0.379860i	\(-0.875972\pi\)
							−0.925044	+	0.379860i	\(0.875972\pi\)
\(11\)	−2.67505	+	1.96064i	−0.806557	+	0.591156i
							\(13\)	0.153067	−	0.471092i	0.0424531	−	0.130657i	−0.927584	−	0.373616i	\(-0.878118\pi\)
0.970037	+	0.242958i	\(0.0781179\pi\)
\(17\)	−6.17958			−1.49877			−0.749384	−	0.662136i	\(-0.769650\pi\)
							−0.749384	+	0.662136i	\(0.769650\pi\)
\(19\)	4.87190	+	3.53964i	1.11769	+	0.812049i	0.983857	−	0.178954i	\(-0.0572715\pi\)
							0.133833	+	0.991004i	\(0.457271\pi\)
\(23\)	−0.947993	−	2.91762i	−0.197670	−	0.608367i	−0.999935	−	0.0113982i	\(-0.996372\pi\)
							0.802265	−	0.596968i	\(-0.203628\pi\)
\(29\)	−3.23307	−	2.34896i	−0.600366	−	0.436192i	0.245643	−	0.969360i	\(-0.421001\pi\)
							−0.846009	+	0.533169i	\(0.821001\pi\)
\(31\)	1.30563	+	4.01831i	0.234498	+	0.721710i	0.997188	+	0.0749458i	\(0.0238784\pi\)
							−0.762690	+	0.646765i	\(0.776122\pi\)
\(37\)	−5.90817	−	4.29254i	−0.971298	−	0.705689i	−0.0155507	−	0.999879i	\(-0.504950\pi\)
							−0.955747	+	0.294190i	\(0.904950\pi\)
\(41\)	−6.24665	+	1.40692i	−0.975562	+	0.219724i
							\(43\)	−1.89720	−	5.83897i	−0.289320	−	0.890435i	−0.985070	−	0.172152i	\(-0.944928\pi\)
0.695751	−	0.718283i	\(-0.255072\pi\)
\(47\)	9.84074			1.43542			0.717710	−	0.696342i	\(-0.245190\pi\)
							0.717710	+	0.696342i	\(0.245190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.10	✓	160
11.3	even	5		451.2.j.a.223.31	yes	160
41.16	even	5		451.2.j.a.180.31	yes	160
451.344	even	5	inner	451.2.h.a.344.10	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.10	✓	160	1.1	even	1	trivial
451.2.h.a.344.10	yes	160	451.344	even	5	inner
451.2.j.a.180.31	yes	160	41.16	even	5
451.2.j.a.223.31	yes	160	11.3	even	5