Embedded newform 451.2.h.a.59.15

• Label: 451.2.h.a.59.15
• 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			59.15
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.304700 + 0.937771i) q^{2} +(-0.536191 + 1.65023i) q^{3} +(0.831461 + 0.604092i) q^{4} +(1.20405 - 3.70567i) q^{5} +(-1.38416 - 1.00565i) q^{6} +2.21455 q^{7} +(-2.41528 + 1.75480i) q^{8} +(-0.00869851 - 0.00631984i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.304700	+	0.937771i	−0.215456	+	0.663104i	0.783665	+	0.621183i	\(0.213348\pi\)
							−0.999121	+	0.0419210i	\(0.986652\pi\)
\(3\)	−0.536191	+	1.65023i	−0.309570	+	0.952759i	0.668362	+	0.743836i	\(0.266996\pi\)
							−0.977932	+	0.208923i	\(0.933004\pi\)
\(5\)	1.20405	−	3.70567i	0.538465	−	1.65723i	−0.197575	−	0.980288i	\(-0.563306\pi\)
							0.736040	−	0.676938i	\(-0.236694\pi\)
\(7\)	2.21455			0.837020			0.418510	−	0.908212i	\(-0.362553\pi\)
							0.418510	+	0.908212i	\(0.362553\pi\)
\(11\)	1.26780	−	3.06475i	0.382256	−	0.924056i
							\(13\)	0.320653	−	0.986869i	0.0889332	−	0.273708i	−0.896692	−	0.442655i	\(-0.854037\pi\)
0.985625	+	0.168947i	\(0.0540366\pi\)
\(17\)	4.86850			1.18078			0.590392	−	0.807116i	\(-0.298973\pi\)
							0.590392	+	0.807116i	\(0.298973\pi\)
\(19\)	1.43389	+	1.04178i	0.328957	+	0.239001i	0.739988	−	0.672620i	\(-0.234831\pi\)
							−0.411031	+	0.911621i	\(0.634831\pi\)
\(23\)	−1.11863	−	3.44278i	−0.233250	−	0.717870i	−0.997349	−	0.0727705i	\(-0.976816\pi\)
							0.764099	−	0.645099i	\(-0.223184\pi\)
\(29\)	1.83532	+	1.33344i	0.340811	+	0.247614i	0.745004	−	0.667060i	\(-0.232448\pi\)
							−0.404193	+	0.914674i	\(0.632448\pi\)
\(31\)	2.91687	+	8.97722i	0.523886	+	1.61236i	0.766508	+	0.642235i	\(0.221993\pi\)
							−0.242622	+	0.970121i	\(0.578007\pi\)
\(37\)	−7.09848	−	5.15735i	−1.16698	−	0.847863i	−0.176339	−	0.984330i	\(-0.556425\pi\)
							−0.990645	+	0.136466i	\(0.956425\pi\)
\(41\)	−5.74354	+	2.83051i	−0.896990	+	0.442051i
							\(43\)	2.68896	+	8.27576i	0.410062	+	1.26204i	0.916594	+	0.399820i	\(0.130927\pi\)
−0.506531	+	0.862222i	\(0.669073\pi\)
\(47\)	−10.1255			−1.47695			−0.738475	−	0.674281i	\(-0.764454\pi\)
							−0.738475	+	0.674281i	\(0.764454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.15	✓	160
11.3	even	5		451.2.j.a.223.26	yes	160
41.16	even	5		451.2.j.a.180.26	yes	160
451.344	even	5	inner	451.2.h.a.344.15	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.15	✓	160	1.1	even	1	trivial
451.2.h.a.344.15	yes	160	451.344	even	5	inner
451.2.j.a.180.26	yes	160	41.16	even	5
451.2.j.a.223.26	yes	160	11.3	even	5