Embedded newform 451.2.h.a.59.14

• Label: 451.2.h.a.59.14
• 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			59.14
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.343347 + 1.05671i) q^{2} +(0.00286704 - 0.00882385i) q^{3} +(0.619281 + 0.449934i) q^{4} +(-0.934433 + 2.87589i) q^{5} +(0.00833988 + 0.00605928i) q^{6} -0.743893 q^{7} +(-2.48586 + 1.80609i) q^{8} +(2.42698 + 1.76331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.343347	+	1.05671i	−0.242783	+	0.747208i	0.753211	+	0.657779i	\(0.228504\pi\)
							−0.995993	+	0.0894287i	\(0.971496\pi\)
\(3\)	0.00286704	−	0.00882385i	0.00165529	−	0.00509445i	−0.950225	−	0.311563i	\(-0.899147\pi\)
							0.951881	+	0.306469i	\(0.0991475\pi\)
\(5\)	−0.934433	+	2.87589i	−0.417891	+	1.28614i	0.491748	+	0.870738i	\(0.336358\pi\)
							−0.909639	+	0.415399i	\(0.863642\pi\)
\(7\)	−0.743893			−0.281165			−0.140583	−	0.990069i	\(-0.544898\pi\)
							−0.140583	+	0.990069i	\(0.544898\pi\)
\(11\)	1.05100	+	3.14569i	0.316890	+	0.948462i
							\(13\)	2.06641	−	6.35976i	0.573119	−	1.76388i	−0.0693777	−	0.997590i	\(-0.522101\pi\)
0.642497	−	0.766288i	\(-0.277899\pi\)
\(17\)	−5.40748			−1.31151			−0.655753	−	0.754976i	\(-0.727649\pi\)
							−0.655753	+	0.754976i	\(0.727649\pi\)
\(19\)	−3.47796	−	2.52689i	−0.797899	−	0.579707i	0.112398	−	0.993663i	\(-0.464147\pi\)
							−0.910297	+	0.413956i	\(0.864147\pi\)
\(23\)	0.647654	+	1.99327i	0.135045	+	0.415627i	0.995597	−	0.0937364i	\(-0.0298811\pi\)
							−0.860552	+	0.509363i	\(0.829881\pi\)
\(29\)	4.34299	+	3.15537i	0.806474	+	0.585938i	0.912806	−	0.408393i	\(-0.133911\pi\)
							−0.106332	+	0.994331i	\(0.533911\pi\)
\(31\)	1.39831	+	4.30356i	0.251144	+	0.772942i	0.994565	+	0.104118i	\(0.0332020\pi\)
							−0.743421	+	0.668824i	\(0.766798\pi\)
\(37\)	1.37402	+	0.998281i	0.225887	+	0.164116i	0.694973	−	0.719036i	\(-0.255416\pi\)
							−0.469086	+	0.883153i	\(0.655416\pi\)
\(41\)	0.890291	−	6.34093i	0.139040	−	0.990287i
							\(43\)	3.20588	+	9.86668i	0.488892	+	1.50465i	0.826264	+	0.563283i	\(0.190462\pi\)
−0.337372	+	0.941371i	\(0.609538\pi\)
\(47\)	2.27868			0.332380			0.166190	−	0.986094i	\(-0.446853\pi\)
							0.166190	+	0.986094i	\(0.446853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.14	✓	160
11.3	even	5		451.2.j.a.223.27	yes	160
41.16	even	5		451.2.j.a.180.27	yes	160
451.344	even	5	inner	451.2.h.a.344.14	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.14	✓	160	1.1	even	1	trivial
451.2.h.a.344.14	yes	160	451.344	even	5	inner
451.2.j.a.180.27	yes	160	41.16	even	5
451.2.j.a.223.27	yes	160	11.3	even	5