Embedded newform 451.2.i.a.201.14

• Label: 451.2.i.a.201.14
• Level: $451$
• Weight: $2$
• Character: 451.201
• 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.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			201.14
Character	\(\chi\)	\(=\)	451.201
Dual form			451.2.i.a.92.14

$q$-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 \(\left(\mathbb{Z}/451\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.14	✓	160	451.92	even	5	inner
451.2.i.a.201.14	yes	160	1.1	even	1	trivial
451.2.l.a.37.14	yes	160	11.4	even	5
451.2.l.a.256.14	yes	160	41.10	even	5