Embedded newform 525.2.bg.b.16.8

• Label: 525.2.bg.b.16.8
• Level: $525$
• Weight: $2$
• Character: 525.16
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(20\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.8
Character	\(\chi\)	\(=\)	525.16
Dual form			525.2.bg.b.361.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.504477 + 0.224608i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.13421 + 1.25967i) q^{4} +(-1.89312 + 1.19000i) q^{5} +(-0.170645 + 0.525192i) q^{6} +(0.160877 - 2.64086i) q^{7} +(0.630543 - 1.94061i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.504477	+	0.224608i	−0.356719	+	0.158822i	−0.577269	−	0.816554i	\(-0.695882\pi\)
							0.220550	+	0.975376i	\(0.429215\pi\)
\(3\)	0.669131	−	0.743145i	0.386323	−	0.429055i
							\(5\)	−1.89312	+	1.19000i	−0.846629	+	0.532183i
\(7\)	0.160877	−	2.64086i	0.0608057	−	0.998150i
							\(11\)	−0.183637	+	1.74719i	−0.0553687	+	0.526798i	0.931323	+	0.364195i	\(0.118656\pi\)
−0.986692	+	0.162603i	\(0.948011\pi\)
\(13\)	1.27740	−	0.928087i	0.354288	−	0.257405i	−0.396378	−	0.918087i	\(-0.629733\pi\)
							0.750666	+	0.660682i	\(0.229733\pi\)
\(17\)	4.55514	−	0.968226i	1.10478	−	0.234829i	0.380808	−	0.924654i	\(-0.375646\pi\)
							0.723976	+	0.689825i	\(0.242312\pi\)
\(19\)	−5.09751	−	5.66136i	−1.16945	−	1.29880i	−0.946029	−	0.324083i	\(-0.894944\pi\)
							−0.223420	−	0.974722i	\(-0.571722\pi\)
\(23\)	3.55945	−	1.58477i	0.742197	−	0.330447i	−0.000567794	−	1.00000i	\(-0.500181\pi\)
							0.742765	+	0.669552i	\(0.233514\pi\)
\(29\)	−0.469160	−	1.44393i	−0.0871209	−	0.268131i	0.897999	−	0.439997i	\(-0.145020\pi\)
							−0.985120	+	0.171866i	\(0.945020\pi\)
\(31\)	6.90701	−	1.46813i	1.24053	−	0.263684i	0.459504	−	0.888176i	\(-0.348027\pi\)
							0.781031	+	0.624492i	\(0.214694\pi\)
\(37\)	−0.509204	−	4.84475i	−0.0837127	−	0.796473i	−0.953164	−	0.302454i	\(-0.902194\pi\)
							0.869451	−	0.494019i	\(-0.164473\pi\)
\(41\)	4.55176	−	3.30705i	0.710866	−	0.516474i	−0.172587	−	0.984994i	\(-0.555213\pi\)
							0.883453	+	0.468520i	\(0.155213\pi\)
\(43\)	−2.07322			−0.316163			−0.158082	−	0.987426i	\(-0.550531\pi\)
							−0.158082	+	0.987426i	\(0.550531\pi\)
\(47\)	−7.61539	−	1.61870i	−1.11082	−	0.236112i	−0.384266	−	0.923223i	\(-0.625545\pi\)
							−0.726553	+	0.687111i	\(0.758879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bg.b.16.8	✓	160
7.4	even	3	inner	525.2.bg.b.466.13	yes	160
25.11	even	5	inner	525.2.bg.b.436.13	yes	160
175.11	even	15	inner	525.2.bg.b.361.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bg.b.16.8	✓	160	1.1	even	1	trivial
525.2.bg.b.361.8	yes	160	175.11	even	15	inner
525.2.bg.b.436.13	yes	160	25.11	even	5	inner
525.2.bg.b.466.13	yes	160	7.4	even	3	inner