Embedded newform 451.2.j.a.119.3

• Label: 451.2.j.a.119.3
• 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			119.3
Character	\(\chi\)	\(=\)	451.119
Dual form			451.2.j.a.379.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.804515 - 2.47604i) q^{2} +(0.229091 - 0.166445i) q^{3} +(-3.86551 + 2.80846i) q^{4} -1.16972 q^{5} +(-0.596431 - 0.433333i) q^{6} +(-0.525573 - 1.61755i) q^{7} +(5.85124 + 4.25117i) q^{8} +(-0.902272 + 2.77691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - 6 q^{5} + 6 q^{6} - q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.804515	−	2.47604i	−0.568878	−	1.75083i	−0.656137	−	0.754642i	\(-0.727811\pi\)
							0.0872588	−	0.996186i	\(-0.472189\pi\)
\(3\)	0.229091	−	0.166445i	0.132266	−	0.0960968i	−0.519685	−	0.854358i	\(-0.673951\pi\)
							0.651951	+	0.758261i	\(0.273951\pi\)
\(5\)	−1.16972			−0.523113			−0.261556	−	0.965188i	\(-0.584236\pi\)
							−0.261556	+	0.965188i	\(0.584236\pi\)
\(7\)	−0.525573	−	1.61755i	−0.198648	−	0.611375i	−0.999915	−	0.0130678i	\(-0.995840\pi\)
							0.801267	−	0.598307i	\(-0.204160\pi\)
\(11\)	2.26648	+	2.42138i	0.683370	+	0.730072i
							\(13\)	1.00334	+	3.08797i	0.278277	+	0.856449i	0.988334	+	0.152304i	\(0.0486693\pi\)
−0.710057	+	0.704145i	\(0.751331\pi\)
\(17\)	−6.43449	+	4.67493i	−1.56059	+	1.13384i	−0.625071	+	0.780568i	\(0.714930\pi\)
							−0.935522	+	0.353269i	\(0.885070\pi\)
\(19\)	−0.642195			−0.147330			−0.0736648	−	0.997283i	\(-0.523469\pi\)
							−0.0736648	+	0.997283i	\(0.523469\pi\)
\(23\)	7.42550	−	5.39494i	1.54832	−	1.12492i	0.603491	−	0.797370i	\(-0.293776\pi\)
							0.944833	−	0.327553i	\(-0.106224\pi\)
\(29\)	−5.26729	+	3.82691i	−0.978111	+	0.710639i	−0.957286	−	0.289144i	\(-0.906629\pi\)
							−0.0208254	+	0.999783i	\(0.506629\pi\)
\(31\)	4.54358			0.816050			0.408025	−	0.912971i	\(-0.366218\pi\)
							0.408025	+	0.912971i	\(0.366218\pi\)
\(37\)	7.97040	−	5.79084i	1.31033	−	0.952008i	0.310327	−	0.950630i	\(-0.399561\pi\)
							0.999999	−	0.00137811i	\(-0.000438667\pi\)
\(41\)	−0.000300588		6.40312i	−4.69440e−5		1.00000i
							\(43\)	−0.459789	+	0.334056i	−0.0701172	+	0.0509431i	−0.622292	−	0.782785i	\(-0.713798\pi\)
0.552174	+	0.833729i	\(0.313798\pi\)
\(47\)	−3.41096	+	10.4979i	−0.497540	+	1.53127i	0.315421	+	0.948952i	\(0.397854\pi\)
							−0.812961	+	0.582319i	\(0.802146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.j.a.119.3	yes	160
11.5	even	5		451.2.h.a.324.38	yes	160
41.10	even	5		451.2.h.a.174.38	✓	160
451.379	even	5	inner	451.2.j.a.379.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.174.38	✓	160	41.10	even	5
451.2.h.a.324.38	yes	160	11.5	even	5
451.2.j.a.119.3	yes	160	1.1	even	1	trivial
451.2.j.a.379.3	yes	160	451.379	even	5	inner