Embedded newform 451.2.h.a.59.19

• Label: 451.2.h.a.59.19
• 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			59.19
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0675881 + 0.208015i) q^{2} +(-0.558994 + 1.72041i) q^{3} +(1.57933 + 1.14745i) q^{4} +(0.679232 - 2.09046i) q^{5} +(-0.320088 - 0.232558i) q^{6} -2.97826 q^{7} +(-0.699327 + 0.508091i) q^{8} +(-0.220271 - 0.160036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0675881	+	0.208015i	−0.0477920	+	0.147089i	−0.972105	−	0.234547i	\(-0.924639\pi\)
							0.924313	+	0.381636i	\(0.124639\pi\)
\(3\)	−0.558994	+	1.72041i	−0.322735	+	0.993277i	0.649717	+	0.760176i	\(0.274887\pi\)
							−0.972453	+	0.233101i	\(0.925113\pi\)
\(5\)	0.679232	−	2.09046i	0.303762	−	0.934883i	−0.676374	−	0.736558i	\(-0.736450\pi\)
							0.980136	−	0.198325i	\(-0.0635501\pi\)
\(7\)	−2.97826			−1.12568			−0.562838	−	0.826567i	\(-0.690291\pi\)
							−0.562838	+	0.826567i	\(0.690291\pi\)
\(11\)	0.406325	+	3.29164i	0.122512	+	0.992467i
							\(13\)	−1.07212	+	3.29964i	−0.297352	+	0.915156i	0.685069	+	0.728478i	\(0.259772\pi\)
−0.982421	+	0.186678i	\(0.940228\pi\)
\(17\)	−1.86266			−0.451761			−0.225880	−	0.974155i	\(-0.572526\pi\)
							−0.225880	+	0.974155i	\(0.572526\pi\)
\(19\)	3.87108	+	2.81251i	0.888088	+	0.645233i	0.935379	−	0.353648i	\(-0.115059\pi\)
							−0.0472911	+	0.998881i	\(0.515059\pi\)
\(23\)	−0.614086	−	1.88996i	−0.128046	−	0.394085i	0.866398	−	0.499354i	\(-0.166429\pi\)
							−0.994444	+	0.105270i	\(0.966429\pi\)
\(29\)	−2.57134	−	1.86819i	−0.477486	−	0.346914i	0.322866	−	0.946445i	\(-0.395354\pi\)
							−0.800352	+	0.599531i	\(0.795354\pi\)
\(31\)	−0.0890491	−	0.274065i	−0.0159937	−	0.0492235i	0.942741	−	0.333525i	\(-0.108238\pi\)
							−0.958735	+	0.284302i	\(0.908238\pi\)
\(37\)	−0.303483	−	0.220494i	−0.0498924	−	0.0362489i	0.562560	−	0.826757i	\(-0.309817\pi\)
							−0.612452	+	0.790508i	\(0.709817\pi\)
\(41\)	4.05420	−	4.95615i	0.633160	−	0.774021i
							\(43\)	0.00504419	+	0.0155244i	0.000769232	+	0.00236745i	0.951440	−	0.307833i	\(-0.0996038\pi\)
−0.950671	+	0.310200i	\(0.899604\pi\)
\(47\)	10.6305			1.55062			0.775311	−	0.631580i	\(-0.217593\pi\)
							0.775311	+	0.631580i	\(0.217593\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.19	✓	160
11.3	even	5		451.2.j.a.223.22	yes	160
41.16	even	5		451.2.j.a.180.22	yes	160
451.344	even	5	inner	451.2.h.a.344.19	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.19	✓	160	1.1	even	1	trivial
451.2.h.a.344.19	yes	160	451.344	even	5	inner
451.2.j.a.180.22	yes	160	41.16	even	5
451.2.j.a.223.22	yes	160	11.3	even	5