Embedded newform 525.2.bg.b.16.2

• Label: 525.2.bg.b.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	525.16
Dual form			525.2.bg.b.361.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19023 + 0.975155i) q^{2} +(0.669131 - 0.743145i) q^{3} +(2.50794 - 2.78535i) q^{4} +(-0.992732 - 2.00362i) q^{5} +(-0.740871 + 2.28017i) q^{6} +(2.55323 + 0.693547i) q^{7} +(-1.29508 + 3.98585i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19023	+	0.975155i	−1.54873	+	0.689539i	−0.990163	−	0.139921i	\(-0.955315\pi\)
							−0.558567	+	0.829460i	\(0.688648\pi\)
\(3\)	0.669131	−	0.743145i	0.386323	−	0.429055i
							\(5\)	−0.992732	−	2.00362i	−0.443963	−	0.896045i
\(7\)	2.55323	+	0.693547i	0.965031	+	0.262136i
							\(11\)	−0.623659	+	5.93372i	−0.188040	+	1.78908i	0.340571	+	0.940219i	\(0.389380\pi\)
−0.528611	+	0.848864i	\(0.677287\pi\)
\(13\)	0.520878	−	0.378440i	0.144465	−	0.104960i	−0.513205	−	0.858266i	\(-0.671542\pi\)
							0.657670	+	0.753306i	\(0.271542\pi\)
\(17\)	0.866444	−	0.184168i	0.210143	−	0.0446674i	−0.101637	−	0.994822i	\(-0.532408\pi\)
							0.311780	+	0.950154i	\(0.399075\pi\)
\(19\)	1.93719	+	2.15146i	0.444421	+	0.493579i	0.923181	−	0.384367i	\(-0.125580\pi\)
							−0.478760	+	0.877946i	\(0.658913\pi\)
\(23\)	7.98687	−	3.55598i	1.66538	−	0.741474i	0.665391	−	0.746495i	\(-0.268265\pi\)
							0.999987	+	0.00502130i	\(0.00159834\pi\)
\(29\)	−1.28919	−	3.96772i	−0.239397	−	0.736788i	−0.996508	−	0.0835012i	\(-0.973390\pi\)
							0.757111	−	0.653286i	\(-0.226610\pi\)
\(31\)	9.17280	−	1.94974i	1.64748	−	0.350183i	0.711625	−	0.702560i	\(-0.247960\pi\)
							0.935858	+	0.352377i	\(0.114626\pi\)
\(37\)	−1.11810	−	10.6380i	−0.183815	−	1.74888i	−0.565644	−	0.824650i	\(-0.691372\pi\)
							0.381829	−	0.924233i	\(-0.375294\pi\)
\(41\)	−4.89210	+	3.55432i	−0.764017	+	0.555091i	−0.900140	−	0.435601i	\(-0.856536\pi\)
							0.136123	+	0.990692i	\(0.456536\pi\)
\(43\)	−3.17737			−0.484544			−0.242272	−	0.970208i	\(-0.577893\pi\)
							−0.242272	+	0.970208i	\(0.577893\pi\)
\(47\)	5.62689	+	1.19603i	0.820766	+	0.174459i	0.599111	−	0.800666i	\(-0.295521\pi\)
							0.221655	+	0.975125i	\(0.428854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bg.b.16.2	✓	160
7.4	even	3	inner	525.2.bg.b.466.19	yes	160
25.11	even	5	inner	525.2.bg.b.436.19	yes	160
175.11	even	15	inner	525.2.bg.b.361.2	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bg.b.16.2	✓	160	1.1	even	1	trivial
525.2.bg.b.361.2	yes	160	175.11	even	15	inner
525.2.bg.b.436.19	yes	160	25.11	even	5	inner
525.2.bg.b.466.19	yes	160	7.4	even	3	inner