Embedded newform 451.2.h.a.324.31

• Label: 451.2.h.a.324.31
• 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			324.31
Character	\(\chi\)	\(=\)	451.324
Dual form			451.2.h.a.174.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32485 + 0.962560i) q^{2} +(-2.55608 - 1.85710i) q^{3} +(0.210673 + 0.648384i) q^{4} +(3.24286 + 2.35607i) q^{5} +(-1.59885 - 4.92076i) q^{6} -1.18563 q^{7} +(0.667097 - 2.05311i) q^{8} +(2.15767 + 6.64064i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32485	+	0.962560i	0.936811	+	0.680633i	0.947651	−	0.319308i	\(-0.103451\pi\)
							−0.0108401	+	0.999941i	\(0.503451\pi\)
\(3\)	−2.55608	−	1.85710i	−1.47575	−	1.07220i	−0.978894	−	0.204368i	\(-0.934486\pi\)
							−0.496860	−	0.867831i	\(-0.665514\pi\)
\(5\)	3.24286	+	2.35607i	1.45025	+	1.05367i	0.985773	+	0.168083i	\(0.0537576\pi\)
							0.464477	+	0.885585i	\(0.346242\pi\)
\(7\)	−1.18563			−0.448127			−0.224064	−	0.974575i	\(-0.571932\pi\)
							−0.224064	+	0.974575i	\(0.571932\pi\)
\(11\)	2.89426	−	1.61965i	0.872651	−	0.488344i
							\(13\)	−2.77457	−	2.01585i	−0.769528	−	0.559095i	0.132290	−	0.991211i	\(-0.457767\pi\)
−0.901818	+	0.432116i	\(0.857767\pi\)
\(17\)	3.53597			0.857598			0.428799	−	0.903400i	\(-0.358937\pi\)
							0.428799	+	0.903400i	\(0.358937\pi\)
\(19\)	1.15020	+	3.53994i	0.263873	+	0.812118i	0.991951	+	0.126623i	\(0.0404139\pi\)
							−0.728078	+	0.685494i	\(0.759586\pi\)
\(23\)	6.71397	−	4.87798i	1.39996	−	1.01713i	0.405270	−	0.914197i	\(-0.367177\pi\)
							0.994688	−	0.102933i	\(-0.0328226\pi\)
\(29\)	−0.876856	−	2.69868i	−0.162828	−	0.501133i	0.836042	−	0.548666i	\(-0.184864\pi\)
							−0.998870	+	0.0475329i	\(0.984864\pi\)
\(31\)	3.72895	−	2.70924i	0.669739	−	0.486594i	−0.200199	−	0.979755i	\(-0.564159\pi\)
							0.869938	+	0.493162i	\(0.164159\pi\)
\(37\)	−0.328812	−	1.01198i	−0.0540563	−	0.166368i	0.920384	−	0.391017i	\(-0.127877\pi\)
							−0.974440	+	0.224649i	\(0.927877\pi\)
\(41\)	4.59893	+	4.45532i	0.718231	+	0.695804i
							\(43\)	−6.73539	+	4.89355i	−1.02714	+	0.746259i	−0.967734	−	0.251975i	\(-0.918920\pi\)
−0.0594041	+	0.998234i	\(0.518920\pi\)
\(47\)	−12.2766			−1.79073			−0.895365	−	0.445333i	\(-0.853085\pi\)
							−0.895365	+	0.445333i	\(0.853085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.324.31	yes	160
11.9	even	5		451.2.j.a.119.10	yes	160
41.10	even	5		451.2.j.a.379.10	yes	160
451.174	even	5	inner	451.2.h.a.174.31	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.174.31	✓	160	451.174	even	5	inner
451.2.h.a.324.31	yes	160	1.1	even	1	trivial
451.2.j.a.119.10	yes	160	11.9	even	5
451.2.j.a.379.10	yes	160	41.10	even	5