Embedded newform 451.2.j.a.223.26

• Label: 451.2.j.a.223.26
• Level: $451$
• Weight: $2$
• Character: 451.223
• 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.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			223.26
Character	\(\chi\)	\(=\)	451.223
Dual form			451.2.j.a.180.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.797716 + 0.579574i) q^{2} +(-0.536191 - 1.65023i) q^{3} +(-0.317590 - 0.977442i) q^{4} +3.89637 q^{5} +(0.528701 - 1.62718i) q^{6} +(-1.79161 - 1.30168i) q^{7} +(0.922554 - 2.83933i) q^{8} +(-0.00869851 + 0.00631984i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - 6 q^{5} + 6 q^{6} - q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.797716	+	0.579574i	0.564070	+	0.409821i	0.832946	−	0.553354i	\(-0.186652\pi\)
							−0.268876	+	0.963175i	\(0.586652\pi\)
\(3\)	−0.536191	−	1.65023i	−0.309570	−	0.952759i	−0.977932	−	0.208923i	\(-0.933004\pi\)
							0.668362	−	0.743836i	\(-0.266996\pi\)
\(5\)	3.89637			1.74251			0.871255	−	0.490830i	\(-0.163306\pi\)
							0.871255	+	0.490830i	\(0.163306\pi\)
\(7\)	−1.79161	−	1.30168i	−0.677163	−	0.491988i	0.195252	−	0.980753i	\(-0.437447\pi\)
							−0.872415	+	0.488765i	\(0.837447\pi\)
\(11\)	−2.52298	+	2.15281i	−0.760706	+	0.649097i
							\(13\)	−0.839481	−	0.609919i	−0.232830	−	0.169161i	0.465253	−	0.885178i	\(-0.345963\pi\)
−0.698083	+	0.716017i	\(0.745963\pi\)
\(17\)	1.50445	+	4.63022i	0.364883	+	1.12299i	0.950055	+	0.312084i	\(0.101027\pi\)
							−0.585172	+	0.810909i	\(0.698973\pi\)
\(19\)	−1.77238			−0.406613			−0.203306	−	0.979115i	\(-0.565169\pi\)
							−0.203306	+	0.979115i	\(0.565169\pi\)
\(23\)	−1.11863	−	3.44278i	−0.233250	−	0.717870i	−0.997349	−	0.0727705i	\(-0.976816\pi\)
							0.764099	−	0.645099i	\(-0.223184\pi\)
\(29\)	−0.701031	−	2.15755i	−0.130178	−	0.400647i	0.864631	−	0.502408i	\(-0.167552\pi\)
							−0.994809	+	0.101761i	\(0.967552\pi\)
\(31\)	9.43920			1.69533			0.847665	−	0.530531i	\(-0.178007\pi\)
							0.847665	+	0.530531i	\(0.178007\pi\)
\(37\)	2.71138	+	8.34477i	0.445748	+	1.37187i	0.881661	+	0.471883i	\(0.156425\pi\)
							−0.435913	+	0.899989i	\(0.643575\pi\)
\(41\)	2.98289	−	5.66590i	0.465849	−	0.884864i
							\(43\)	2.68896	+	8.27576i	0.410062	+	1.26204i	0.916594	+	0.399820i	\(0.130927\pi\)
−0.506531	+	0.862222i	\(0.669073\pi\)
\(47\)	8.19167	−	5.95159i	1.19488	−	0.868129i	0.201106	−	0.979569i	\(-0.435546\pi\)
							0.993771	+	0.111440i	\(0.0355463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.j.a.223.26	yes	160
11.4	even	5		451.2.h.a.59.15	✓	160
41.16	even	5		451.2.h.a.344.15	yes	160
451.180	even	5	inner	451.2.j.a.180.26	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.15	✓	160	11.4	even	5
451.2.h.a.344.15	yes	160	41.16	even	5
451.2.j.a.180.26	yes	160	451.180	even	5	inner
451.2.j.a.223.26	yes	160	1.1	even	1	trivial