Embedded newform 451.2.j.a.119.10

• Label: 451.2.j.a.119.10
• Level: $451$
• Weight: $2$
• Character: 451.119
• Analytic conductor: $3.601$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 451 = 11 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	451.j (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			119.10
Character	\(\chi\)	\(=\)	451.119
Dual form			451.2.j.a.379.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.506048 - 1.55746i) q^{2} +(-2.55608 + 1.85710i) q^{3} +(-0.551548 + 0.400723i) q^{4} -4.00839 q^{5} +(4.18585 + 3.04120i) q^{6} +(-0.366381 - 1.12760i) q^{7} +(-1.74648 - 1.26889i) q^{8} +(2.15767 - 6.64064i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - 6 q^{5} + 6 q^{6} - q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.506048	−	1.55746i	−0.357830	−	1.10129i	−0.954350	−	0.298689i	\(-0.903451\pi\)
							0.596521	−	0.802598i	\(-0.296549\pi\)
\(3\)	−2.55608	+	1.85710i	−1.47575	+	1.07220i	−0.496860	+	0.867831i	\(0.665514\pi\)
							−0.978894	+	0.204368i	\(0.934486\pi\)
\(5\)	−4.00839			−1.79261			−0.896304	−	0.443441i	\(-0.853758\pi\)
							−0.896304	+	0.443441i	\(0.853758\pi\)
\(7\)	−0.366381	−	1.12760i	−0.138479	−	0.426194i	0.857636	−	0.514257i	\(-0.171932\pi\)
							−0.996115	+	0.0880628i	\(0.971932\pi\)
\(11\)	−3.29351	+	0.390874i	−0.993031	+	0.117853i
							\(13\)	1.05979	+	3.26171i	0.293934	+	0.904635i	0.983578	+	0.180486i	\(0.0577671\pi\)
−0.689644	+	0.724149i	\(0.742233\pi\)
\(17\)	−2.86066	+	2.07839i	−0.693811	+	0.504083i	−0.877911	−	0.478824i	\(-0.841063\pi\)
							0.184100	+	0.982908i	\(0.441063\pi\)
\(19\)	3.72211			0.853911			0.426956	−	0.904273i	\(-0.359586\pi\)
							0.426956	+	0.904273i	\(0.359586\pi\)
\(23\)	6.71397	−	4.87798i	1.39996	−	1.01713i	0.405270	−	0.914197i	\(-0.367177\pi\)
							0.994688	−	0.102933i	\(-0.0328226\pi\)
\(29\)	2.29564	−	1.66788i	0.426289	−	0.309717i	−0.353874	−	0.935293i	\(-0.615136\pi\)
							0.780163	+	0.625576i	\(0.215136\pi\)
\(31\)	−4.60923			−0.827842			−0.413921	−	0.910313i	\(-0.635841\pi\)
							−0.413921	+	0.910313i	\(0.635841\pi\)
\(37\)	0.860841	−	0.625437i	0.141521	−	0.102821i	−0.514772	−	0.857327i	\(-0.672123\pi\)
							0.656293	+	0.754506i	\(0.272123\pi\)
\(41\)	5.65841	−	2.99707i	0.883695	−	0.468063i
							\(43\)	−6.73539	+	4.89355i	−1.02714	+	0.746259i	−0.967734	−	0.251975i	\(-0.918920\pi\)
−0.0594041	+	0.998234i	\(0.518920\pi\)
\(47\)	−3.79369	+	11.6758i	−0.553366	+	1.70309i	0.146853	+	0.989158i	\(0.453085\pi\)
							−0.700220	+	0.713928i	\(0.746915\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.174.31	✓	160	41.10	even	5
451.2.h.a.324.31	yes	160	11.5	even	5
451.2.j.a.119.10	yes	160	1.1	even	1	trivial
451.2.j.a.379.10	yes	160	451.379	even	5	inner