Embedded newform 451.2.h.a.59.7

• Label: 451.2.h.a.59.7
• 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			59.7
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.603793 + 1.85828i) q^{2} +(-0.444328 + 1.36750i) q^{3} +(-1.47062 - 1.06846i) q^{4} +(0.243892 - 0.750623i) q^{5} +(-2.27292 - 1.65137i) q^{6} +4.75408 q^{7} +(-0.288042 + 0.209275i) q^{8} +(0.754421 + 0.548119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.603793	+	1.85828i	−0.426946	+	1.31400i	0.474173	+	0.880432i	\(0.342747\pi\)
							−0.901119	+	0.433573i	\(0.857253\pi\)
\(3\)	−0.444328	+	1.36750i	−0.256533	+	0.789527i	0.736991	+	0.675903i	\(0.236246\pi\)
							−0.993524	+	0.113624i	\(0.963754\pi\)
\(5\)	0.243892	−	0.750623i	0.109072	−	0.335689i	−0.881593	−	0.472011i	\(-0.843528\pi\)
							0.990665	+	0.136322i	\(0.0435283\pi\)
\(7\)	4.75408			1.79687			0.898436	−	0.439104i	\(-0.144704\pi\)
							0.898436	+	0.439104i	\(0.144704\pi\)
\(11\)	0.328207	+	3.30035i	0.0989583	+	0.995092i
							\(13\)	−0.494736	+	1.52264i	−0.137215	+	0.422304i	−0.995928	−	0.0901525i	\(-0.971265\pi\)
0.858713	+	0.512457i	\(0.171265\pi\)
\(17\)	1.36580			0.331256			0.165628	−	0.986188i	\(-0.447035\pi\)
							0.165628	+	0.986188i	\(0.447035\pi\)
\(19\)	−2.94049	−	2.13639i	−0.674595	−	0.490122i	0.196965	−	0.980410i	\(-0.436891\pi\)
							−0.871560	+	0.490289i	\(0.836891\pi\)
\(23\)	0.909308	+	2.79856i	0.189604	+	0.583540i	0.999997	−	0.00234556i	\(-0.000746614\pi\)
							−0.810393	+	0.585886i	\(0.800747\pi\)
\(29\)	1.49685	+	1.08752i	0.277957	+	0.201948i	0.718026	−	0.696016i	\(-0.245046\pi\)
							−0.440069	+	0.897964i	\(0.645046\pi\)
\(31\)	−3.37883	−	10.3990i	−0.606855	−	1.86771i	−0.483499	−	0.875345i	\(-0.660634\pi\)
							−0.123356	−	0.992362i	\(-0.539366\pi\)
\(37\)	−5.24079	−	3.80766i	−0.861580	−	0.625975i	0.0667340	−	0.997771i	\(-0.478742\pi\)
							−0.928314	+	0.371796i	\(0.878742\pi\)
\(41\)	−1.21425	−	6.28694i	−0.189635	−	0.981855i
							\(43\)	1.58201	+	4.86892i	0.241254	+	0.742503i	0.996230	+	0.0867508i	\(0.0276484\pi\)
−0.754976	+	0.655752i	\(0.772352\pi\)
\(47\)	0.878882			0.128198			0.0640991	−	0.997944i	\(-0.479583\pi\)
							0.0640991	+	0.997944i	\(0.479583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.7	✓	160
11.3	even	5		451.2.j.a.223.34	yes	160
41.16	even	5		451.2.j.a.180.34	yes	160
451.344	even	5	inner	451.2.h.a.344.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.7	✓	160	1.1	even	1	trivial
451.2.h.a.344.7	yes	160	451.344	even	5	inner
451.2.j.a.180.34	yes	160	41.16	even	5
451.2.j.a.223.34	yes	160	11.3	even	5