Embedded newform 451.2.h.a.59.5

• Label: 451.2.h.a.59.5
• Level: $451$
• Weight: $2$
• Character: 451.59
• 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.h (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			59.5
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.716192 + 2.20421i) q^{2} +(0.439099 - 1.35141i) q^{3} +(-2.72758 - 1.98170i) q^{4} +(0.222886 - 0.685972i) q^{5} +(2.66431 + 1.93573i) q^{6} -0.896580 q^{7} +(2.57154 - 1.86833i) q^{8} +(0.793555 + 0.576551i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.716192	+	2.20421i	−0.506424	+	1.55861i	0.291940	+	0.956437i	\(0.405699\pi\)
							−0.798364	+	0.602176i	\(0.794301\pi\)
\(3\)	0.439099	−	1.35141i	0.253514	−	0.780236i	−0.740605	−	0.671941i	\(-0.765461\pi\)
							0.994119	−	0.108295i	\(-0.0345391\pi\)
\(5\)	0.222886	−	0.685972i	0.0996776	−	0.306776i	−0.888767	−	0.458360i	\(-0.848437\pi\)
							0.988444	+	0.151583i	\(0.0484372\pi\)
\(7\)	−0.896580			−0.338875			−0.169438	−	0.985541i	\(-0.554195\pi\)
							−0.169438	+	0.985541i	\(0.554195\pi\)
\(11\)	−1.88485	−	2.72898i	−0.568305	−	0.822818i
							\(13\)	1.01686	−	3.12958i	0.282026	−	0.867988i	−0.705248	−	0.708961i	\(-0.749164\pi\)
0.987274	−	0.159027i	\(-0.0508358\pi\)
\(17\)	5.54949			1.34595			0.672974	−	0.739666i	\(-0.265017\pi\)
							0.672974	+	0.739666i	\(0.265017\pi\)
\(19\)	−1.77991	−	1.29318i	−0.408341	−	0.296677i	0.364589	−	0.931168i	\(-0.381209\pi\)
							−0.772930	+	0.634492i	\(0.781209\pi\)
\(23\)	−2.76486	−	8.50935i	−0.576512	−	1.77432i	−0.630971	−	0.775806i	\(-0.717343\pi\)
							0.0544591	−	0.998516i	\(-0.482657\pi\)
\(29\)	8.23164	+	5.98063i	1.52858	+	1.11058i	0.957023	+	0.290011i	\(0.0936588\pi\)
							0.571553	+	0.820565i	\(0.306341\pi\)
\(31\)	−1.36191	−	4.19151i	−0.244605	−	0.752818i	−0.995701	−	0.0926248i	\(-0.970474\pi\)
							0.751096	−	0.660193i	\(-0.229526\pi\)
\(37\)	−4.06920	−	2.95645i	−0.668973	−	0.486037i	0.200708	−	0.979651i	\(-0.435676\pi\)
							−0.869681	+	0.493614i	\(0.835676\pi\)
\(41\)	0.00989249	−	6.40312i	0.00154495	−	0.999999i
							\(43\)	2.73115	+	8.40562i	0.416497	+	1.28185i	0.910905	+	0.412616i	\(0.135385\pi\)
−0.494409	+	0.869230i	\(0.664615\pi\)
\(47\)	2.28307			0.333020			0.166510	−	0.986040i	\(-0.446750\pi\)
							0.166510	+	0.986040i	\(0.446750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.5	✓	160
11.3	even	5		451.2.j.a.223.36	yes	160
41.16	even	5		451.2.j.a.180.36	yes	160
451.344	even	5	inner	451.2.h.a.344.5	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.5	✓	160	1.1	even	1	trivial
451.2.h.a.344.5	yes	160	451.344	even	5	inner
451.2.j.a.180.36	yes	160	41.16	even	5
451.2.j.a.223.36	yes	160	11.3	even	5