Embedded newform 525.2.bg.b.16.16

• Label: 525.2.bg.b.16.16
• 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			16.16
Character	\(\chi\)	\(=\)	525.16
Dual form			525.2.bg.b.361.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88139 - 0.837651i) q^{2} +(0.669131 - 0.743145i) q^{3} +(1.49973 - 1.66561i) q^{4} +(2.15442 - 0.598733i) q^{5} +(0.636403 - 1.95865i) q^{6} +(-0.705516 - 2.54995i) q^{7} +(0.153568 - 0.472633i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88139	−	0.837651i	1.33035	−	0.592309i	0.386377	−	0.922341i	\(-0.373726\pi\)
							0.943970	+	0.330032i	\(0.107060\pi\)
\(3\)	0.669131	−	0.743145i	0.386323	−	0.429055i
							\(5\)	2.15442	−	0.598733i	0.963485	−	0.267761i
\(7\)	−0.705516	−	2.54995i	−0.266660	−	0.963791i
							\(11\)	−0.232511	+	2.21220i	−0.0701048	+	0.667002i	0.901885	+	0.431977i	\(0.142184\pi\)
−0.971989	+	0.235025i	\(0.924483\pi\)
\(13\)	−1.57964	+	1.14768i	−0.438114	+	0.318308i	−0.784885	−	0.619642i	\(-0.787278\pi\)
							0.346771	+	0.937950i	\(0.387278\pi\)
\(17\)	−3.12802	+	0.664882i	−0.758657	+	0.161258i	−0.570969	−	0.820971i	\(-0.693433\pi\)
							−0.187687	+	0.982229i	\(0.560099\pi\)
\(19\)	1.80211	+	2.00144i	0.413432	+	0.459163i	0.913509	−	0.406818i	\(-0.133362\pi\)
							−0.500077	+	0.865981i	\(0.666695\pi\)
\(23\)	−2.65387	+	1.18158i	−0.553370	+	0.246376i	−0.664318	−	0.747450i	\(-0.731278\pi\)
							0.110948	+	0.993826i	\(0.464611\pi\)
\(29\)	−0.460534	−	1.41738i	−0.0855189	−	0.263200i	0.899148	−	0.437644i	\(-0.144187\pi\)
							−0.984667	+	0.174444i	\(0.944187\pi\)
\(31\)	2.77650	−	0.590163i	0.498674	−	0.105996i	0.0482937	−	0.998833i	\(-0.484622\pi\)
							0.450381	+	0.892837i	\(0.351288\pi\)
\(37\)	−0.0825508	−	0.785418i	−0.0135713	−	0.129122i	0.985639	−	0.168869i	\(-0.0540115\pi\)
							−0.999210	+	0.0397470i	\(0.987345\pi\)
\(41\)	6.19007	−	4.49735i	0.966726	−	0.702368i	0.0120233	−	0.999928i	\(-0.496173\pi\)
							0.954703	+	0.297560i	\(0.0961728\pi\)
\(43\)	−6.69508			−1.02099			−0.510495	−	0.859881i	\(-0.670538\pi\)
							−0.510495	+	0.859881i	\(0.670538\pi\)
\(47\)	6.94829	+	1.47690i	1.01351	+	0.215429i	0.684590	−	0.728928i	\(-0.259981\pi\)
							0.328922	+	0.944357i	\(0.393315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bg.b.16.16	✓	160
7.4	even	3	inner	525.2.bg.b.466.5	yes	160
25.11	even	5	inner	525.2.bg.b.436.5	yes	160
175.11	even	15	inner	525.2.bg.b.361.16	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bg.b.16.16	✓	160	1.1	even	1	trivial
525.2.bg.b.361.16	yes	160	175.11	even	15	inner
525.2.bg.b.436.5	yes	160	25.11	even	5	inner
525.2.bg.b.466.5	yes	160	7.4	even	3	inner