Embedded newform 451.2.h.a.324.24

• Label: 451.2.h.a.324.24
• Level: $451$
• Weight: $2$
• Character: 451.324
• 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			324.24
Character	\(\chi\)	\(=\)	451.324
Dual form			451.2.h.a.174.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.428887 + 0.311604i) q^{2} +(-1.74772 - 1.26980i) q^{3} +(-0.531188 - 1.63483i) q^{4} +(0.377112 + 0.273988i) q^{5} +(-0.353902 - 1.08920i) q^{6} +0.230128 q^{7} +(0.609240 - 1.87505i) q^{8} +(0.515107 + 1.58534i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.428887	+	0.311604i	0.303269	+	0.220338i	0.729003	−	0.684511i	\(-0.239984\pi\)
							−0.425734	+	0.904848i	\(0.639984\pi\)
\(3\)	−1.74772	−	1.26980i	−1.00905	−	0.733117i	−0.0450397	−	0.998985i	\(-0.514341\pi\)
							−0.964009	+	0.265868i	\(0.914341\pi\)
\(5\)	0.377112	+	0.273988i	0.168650	+	0.122531i	0.668908	−	0.743345i	\(-0.266762\pi\)
							−0.500259	+	0.865876i	\(0.666762\pi\)
\(7\)	0.230128			0.0869802			0.0434901	−	0.999054i	\(-0.486152\pi\)
							0.0434901	+	0.999054i	\(0.486152\pi\)
\(11\)	−2.66989	−	1.96766i	−0.805002	−	0.593272i
							\(13\)	0.166363	+	0.120870i	0.0461408	+	0.0335233i	0.610617	−	0.791926i	\(-0.290922\pi\)
−0.564476	+	0.825450i	\(0.690922\pi\)
\(17\)	−3.51947			−0.853597			−0.426798	−	0.904347i	\(-0.640359\pi\)
							−0.426798	+	0.904347i	\(0.640359\pi\)
\(19\)	1.56415	+	4.81396i	0.358841	+	1.10440i	0.953749	+	0.300604i	\(0.0971883\pi\)
							−0.594908	+	0.803794i	\(0.702812\pi\)
\(23\)	−1.52124	+	1.10525i	−0.317201	+	0.230460i	−0.734980	−	0.678089i	\(-0.762808\pi\)
							0.417779	+	0.908548i	\(0.362808\pi\)
\(29\)	−1.70106	−	5.23531i	−0.315878	−	0.972173i	−0.975392	−	0.220480i	\(-0.929238\pi\)
							0.659514	−	0.751693i	\(-0.270762\pi\)
\(31\)	−6.59982	+	4.79505i	−1.18536	+	0.861216i	−0.992766	−	0.120062i	\(-0.961691\pi\)
							−0.192596	+	0.981278i	\(0.561691\pi\)
\(37\)	−2.84335	−	8.75094i	−0.467444	−	1.43865i	−0.855882	−	0.517170i	\(-0.826985\pi\)
							0.388438	−	0.921475i	\(-0.373015\pi\)
\(41\)	5.46864	−	3.33077i	0.854058	−	0.520179i
							\(43\)	−1.49527	+	1.08638i	−0.228027	+	0.165671i	−0.695932	−	0.718107i	\(-0.745009\pi\)
0.467906	+	0.883778i	\(0.345009\pi\)
\(47\)	3.09556			0.451533			0.225767	−	0.974181i	\(-0.427511\pi\)
							0.225767	+	0.974181i	\(0.427511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.324.24	yes	160
11.9	even	5		451.2.j.a.119.17	yes	160
41.10	even	5		451.2.j.a.379.17	yes	160
451.174	even	5	inner	451.2.h.a.174.24	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.174.24	✓	160	451.174	even	5	inner
451.2.h.a.324.24	yes	160	1.1	even	1	trivial
451.2.j.a.119.17	yes	160	11.9	even	5
451.2.j.a.379.17	yes	160	41.10	even	5