Embedded newform 451.2.i.a.92.14

• Label: 451.2.i.a.92.14
• Level: $451$
• Weight: $2$
• Character: 451.92
• 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.i (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			92.14
Character	\(\chi\)	\(=\)	451.92
Dual form			451.2.i.a.201.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.350156 - 1.07767i) q^{2} +(0.233897 - 0.719860i) q^{3} +(0.579271 - 0.420865i) q^{4} +(1.00602 + 0.730914i) q^{5} -0.857672 q^{6} +(0.823426 + 0.598254i) q^{7} +(-2.48983 - 1.80897i) q^{8} +(1.96356 + 1.42661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - q^{7} + 13 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.350156	−	1.07767i	−0.247598	−	0.762028i	−0.995198	−	0.0978790i	\(-0.968794\pi\)
							0.747601	−	0.664149i	\(-0.231206\pi\)
\(3\)	0.233897	−	0.719860i	0.135040	−	0.415611i	−0.860556	−	0.509356i	\(-0.829884\pi\)
							0.995596	+	0.0937445i	\(0.0298837\pi\)
\(5\)	1.00602	+	0.730914i	0.449904	+	0.326875i	0.789558	−	0.613676i	\(-0.210310\pi\)
							−0.339654	+	0.940551i	\(0.610310\pi\)
\(7\)	0.823426	+	0.598254i	0.311226	+	0.226119i	0.732422	−	0.680851i	\(-0.238390\pi\)
							−0.421197	+	0.906969i	\(0.638390\pi\)
\(11\)	2.22830	−	2.45656i	0.671858	−	0.740680i
							\(13\)	−0.0428822	−	0.131978i	−0.0118934	−	0.0366040i	0.944934	−	0.327262i	\(-0.106126\pi\)
−0.956827	+	0.290658i	\(0.906126\pi\)
\(17\)	0.949645	+	2.92271i	0.230323	+	0.708860i	0.997708	+	0.0676735i	\(0.0215576\pi\)
							−0.767385	+	0.641187i	\(0.778442\pi\)
\(19\)	−1.71817	−	5.28799i	−0.394176	−	1.21315i	−0.929602	−	0.368566i	\(-0.879849\pi\)
							0.535426	−	0.844582i	\(-0.320151\pi\)
\(23\)	4.33356	+	3.14852i	0.903610	+	0.656511i	0.939391	−	0.342848i	\(-0.111392\pi\)
							−0.0357806	+	0.999360i	\(0.511392\pi\)
\(29\)	−2.48325	+	1.80419i	−0.461128	+	0.335029i	−0.793973	−	0.607952i	\(-0.791991\pi\)
							0.332846	+	0.942981i	\(0.391991\pi\)
\(31\)	−2.21336	+	1.60810i	−0.397531	+	0.288824i	−0.768535	−	0.639808i	\(-0.779014\pi\)
							0.371003	+	0.928632i	\(0.379014\pi\)
\(37\)	−2.22900	+	1.61946i	−0.366445	+	0.266238i	−0.755735	−	0.654877i	\(-0.772720\pi\)
							0.389290	+	0.921115i	\(0.372720\pi\)
\(41\)	−5.05104	+	3.93535i	−0.788840	+	0.614598i
							\(43\)	−2.61081	−	1.89687i	−0.398145	−	0.289269i	0.370640	−	0.928777i	\(-0.379138\pi\)
−0.768785	+	0.639507i	\(0.779138\pi\)
\(47\)	−0.386992	+	0.281166i	−0.0564486	+	0.0410123i	−0.615652	−	0.788018i	\(-0.711107\pi\)
							0.559203	+	0.829031i	\(0.311107\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.14	✓	160	1.1	even	1	trivial
451.2.i.a.201.14	yes	160	451.201	even	5	inner
451.2.l.a.37.14	yes	160	41.37	even	5
451.2.l.a.256.14	yes	160	11.3	even	5