Embedded newform 525.2.bg.b.16.20

• Label: 525.2.bg.b.16.20
• 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			16.20
Character	\(\chi\)	\(=\)	525.16
Dual form			525.2.bg.b.361.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.42324 - 1.07889i) q^{2} +(0.669131 - 0.743145i) q^{3} +(3.36980 - 3.74254i) q^{4} +(-0.617899 + 2.14900i) q^{5} +(0.819687 - 2.52274i) q^{6} +(2.64524 + 0.0517719i) q^{7} +(2.48864 - 7.65924i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.42324	−	1.07889i	1.71349	−	0.762893i	0.715558	−	0.698553i	\(-0.246172\pi\)
							0.997928	−	0.0643405i	\(-0.0204944\pi\)
\(3\)	0.669131	−	0.743145i	0.386323	−	0.429055i
							\(5\)	−0.617899	+	2.14900i	−0.276333	+	0.961062i
\(7\)	2.64524	+	0.0517719i	0.999809	+	0.0195679i
							\(11\)	−0.329820	+	3.13803i	−0.0994445	+	0.946152i	0.825077	+	0.565021i	\(0.191132\pi\)
−0.924521	+	0.381131i	\(0.875535\pi\)
\(13\)	−4.92099	+	3.57531i	−1.36484	+	0.991611i	−0.366715	+	0.930333i	\(0.619518\pi\)
							−0.998121	+	0.0612779i	\(0.980482\pi\)
\(17\)	−1.87450	+	0.398438i	−0.454634	+	0.0966354i	−0.429536	−	0.903050i	\(-0.641323\pi\)
							−0.0250975	+	0.999685i	\(0.507990\pi\)
\(19\)	−4.57135	−	5.07699i	−1.04874	−	1.16474i	−0.986006	−	0.166712i	\(-0.946685\pi\)
							−0.0627330	−	0.998030i	\(-0.519982\pi\)
\(23\)	−1.60744	+	0.715677i	−0.335174	+	0.149229i	−0.567421	−	0.823428i	\(-0.692059\pi\)
							0.232247	+	0.972657i	\(0.425392\pi\)
\(29\)	−0.772429	−	2.37729i	−0.143436	−	0.441452i	0.853370	−	0.521305i	\(-0.174555\pi\)
							−0.996807	+	0.0798536i	\(0.974555\pi\)
\(31\)	−2.73759	+	0.581892i	−0.491685	+	0.104511i	−0.447080	−	0.894494i	\(-0.647536\pi\)
							−0.0446048	+	0.999005i	\(0.514203\pi\)
\(37\)	−0.742753	−	7.06683i	−0.122108	−	1.16178i	−0.868297	−	0.496045i	\(-0.834785\pi\)
							0.746189	−	0.665734i	\(-0.231882\pi\)
\(41\)	1.26026	−	0.915636i	0.196820	−	0.142998i	−0.485010	−	0.874508i	\(-0.661184\pi\)
							0.681831	+	0.731510i	\(0.261184\pi\)
\(43\)	10.4982			1.60096			0.800479	−	0.599361i	\(-0.204579\pi\)
							0.800479	+	0.599361i	\(0.204579\pi\)
\(47\)	8.85488	+	1.88216i	1.29162	+	0.274542i	0.801962	−	0.597375i	\(-0.203790\pi\)
							0.489655	+	0.871916i	\(0.337123\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bg.b.16.20	✓	160
7.4	even	3	inner	525.2.bg.b.466.1	yes	160
25.11	even	5	inner	525.2.bg.b.436.1	yes	160
175.11	even	15	inner	525.2.bg.b.361.20	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bg.b.16.20	✓	160	1.1	even	1	trivial
525.2.bg.b.361.20	yes	160	175.11	even	15	inner
525.2.bg.b.436.1	yes	160	25.11	even	5	inner
525.2.bg.b.466.1	yes	160	7.4	even	3	inner