Embedded newform 451.2.l.a.37.14

• Label: 451.2.l.a.37.14
• Level: $451$
• Weight: $2$
• Character: 451.37
• 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.l (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			37.14
Character	\(\chi\)	\(=\)	451.37
Dual form			451.2.l.a.256.14

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.13313 q^{2} +(0.233897 - 0.719860i) q^{3} -0.716018 q^{4} +(-0.384264 - 1.18264i) q^{5} +(-0.265035 + 0.815694i) q^{6} +(-0.314521 + 0.967995i) q^{7} +3.07760 q^{8} +(1.96356 + 1.42661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 14 q^{2} - 7 q^{3} + 146 q^{4} - q^{5} - 4 q^{6} - q^{7} - 42 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13313			−0.801243			−0.400622	−	0.916244i	\(-0.631206\pi\)
							−0.400622	+	0.916244i	\(0.631206\pi\)
\(3\)	0.233897	−	0.719860i	0.135040	−	0.415611i	−0.860556	−	0.509356i	\(-0.829884\pi\)
							0.995596	+	0.0937445i	\(0.0298837\pi\)
\(5\)	−0.384264	−	1.18264i	−0.171848	−	0.528894i	0.827627	−	0.561278i	\(-0.189690\pi\)
							−0.999476	+	0.0323836i	\(0.989690\pi\)
\(7\)	−0.314521	+	0.967995i	−0.118878	+	0.365868i	−0.992736	−	0.120313i	\(-0.961610\pi\)
							0.873858	+	0.486181i	\(0.161610\pi\)
\(11\)	−0.358804	+	3.29716i	−0.108184	+	0.994131i
							\(13\)	−0.138770			−0.0384878			−0.0192439	−	0.999815i	\(-0.506126\pi\)
−0.0192439	+	0.999815i	\(0.506126\pi\)
\(17\)	−2.48620	−	1.80633i	−0.602993	−	0.438100i	0.243947	−	0.969789i	\(-0.421558\pi\)
							−0.846940	+	0.531689i	\(0.821558\pi\)
\(19\)	4.49823	−	3.26816i	1.03197	−	0.749767i	0.0632646	−	0.997997i	\(-0.479849\pi\)
							0.968701	+	0.248230i	\(0.0798488\pi\)
\(23\)	4.33356	−	3.14852i	0.903610	−	0.656511i	−0.0357806	−	0.999360i	\(-0.511392\pi\)
							0.939391	+	0.342848i	\(0.111392\pi\)
\(29\)	3.06946			0.569985			0.284993	−	0.958530i	\(-0.408009\pi\)
							0.284993	+	0.958530i	\(0.408009\pi\)
\(31\)	0.845429	−	2.60196i	0.151844	−	0.467326i	−0.845984	−	0.533208i	\(-0.820986\pi\)
							0.997827	+	0.0658822i	\(0.0209862\pi\)
\(37\)	2.75519			0.452950			0.226475	−	0.974017i	\(-0.427280\pi\)
							0.226475	+	0.974017i	\(0.427280\pi\)
\(41\)	1.77324	+	6.15269i	0.276933	+	0.960889i
							\(43\)	−2.61081	+	1.89687i	−0.398145	+	0.289269i	−0.768785	−	0.639507i	\(-0.779138\pi\)
0.370640	+	0.928777i	\(0.379138\pi\)
\(47\)	0.147818	+	0.454937i	0.0215615	+	0.0663594i	0.961258	−	0.275649i	\(-0.0888928\pi\)
							−0.939697	+	0.342009i	\(0.888893\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.l.a.37.14	yes	160
11.3	even	5		451.2.i.a.201.14	yes	160
41.10	even	5		451.2.i.a.92.14	✓	160
451.256	even	5	inner	451.2.l.a.256.14	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.14	✓	160	41.10	even	5
451.2.i.a.201.14	yes	160	11.3	even	5
451.2.l.a.37.14	yes	160	1.1	even	1	trivial
451.2.l.a.256.14	yes	160	451.256	even	5	inner