Embedded newform 525.2.bg.b.16.18

• Label: 525.2.bg.b.16.18
• 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			16.18
Character	\(\chi\)	\(=\)	525.16
Dual form			525.2.bg.b.361.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10352 - 0.936548i) q^{2} +(0.669131 - 0.743145i) q^{3} +(2.20942 - 2.45381i) q^{4} +(-0.282304 - 2.21818i) q^{5} +(0.711540 - 2.18990i) q^{6} +(0.729324 + 2.54324i) q^{7} +(0.926375 - 2.85109i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10352	−	0.936548i	1.48741	−	0.662240i	0.507499	−	0.861652i	\(-0.330570\pi\)
							0.979916	+	0.199412i	\(0.0639033\pi\)
\(3\)	0.669131	−	0.743145i	0.386323	−	0.429055i
							\(5\)	−0.282304	−	2.21818i	−0.126250	−	0.991998i
\(7\)	0.729324	+	2.54324i	0.275659	+	0.961256i
							\(11\)	0.497330	−	4.73177i	0.149950	−	1.42668i	−0.618002	−	0.786177i	\(-0.712058\pi\)
0.767952	−	0.640507i	\(-0.221276\pi\)
\(13\)	−4.46348	+	3.24291i	−1.23795	+	0.899420i	−0.997460	−	0.0712321i	\(-0.977307\pi\)
							−0.240486	+	0.970653i	\(0.577307\pi\)
\(17\)	−1.23896	+	0.263348i	−0.300491	+	0.0638714i	−0.355690	−	0.934604i	\(-0.615754\pi\)
							0.0551988	+	0.998475i	\(0.482421\pi\)
\(19\)	4.93478	+	5.48063i	1.13212	+	1.25734i	0.962336	+	0.271861i	\(0.0876393\pi\)
							0.169781	+	0.985482i	\(0.445694\pi\)
\(23\)	4.66744	−	2.07808i	0.973228	−	0.433309i	0.142382	−	0.989812i	\(-0.454524\pi\)
							0.830846	+	0.556503i	\(0.187857\pi\)
\(29\)	0.828509	+	2.54989i	0.153850	+	0.473502i	0.998043	−	0.0625381i	\(-0.0199195\pi\)
							−0.844192	+	0.536040i	\(0.819919\pi\)
\(31\)	1.77615	−	0.377533i	0.319007	−	0.0678070i	−0.0456254	−	0.998959i	\(-0.514528\pi\)
							0.364632	+	0.931152i	\(0.381195\pi\)
\(37\)	−0.519793	−	4.94550i	−0.0854535	−	0.813036i	−0.950370	−	0.311121i	\(-0.899296\pi\)
							0.864917	−	0.501915i	\(-0.167371\pi\)
\(41\)	−2.08611	+	1.51564i	−0.325795	+	0.236704i	−0.738644	−	0.674095i	\(-0.764534\pi\)
							0.412849	+	0.910799i	\(0.364534\pi\)
\(43\)	−1.05016			−0.160149			−0.0800743	−	0.996789i	\(-0.525516\pi\)
							−0.0800743	+	0.996789i	\(0.525516\pi\)
\(47\)	−10.3923	−	2.20895i	−1.51587	−	0.322209i	−0.626512	−	0.779412i	\(-0.715518\pi\)
							−0.889362	+	0.457203i	\(0.848851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bg.b.16.18	✓	160
7.4	even	3	inner	525.2.bg.b.466.3	yes	160
25.11	even	5	inner	525.2.bg.b.436.3	yes	160
175.11	even	15	inner	525.2.bg.b.361.18	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bg.b.16.18	✓	160	1.1	even	1	trivial
525.2.bg.b.361.18	yes	160	175.11	even	15	inner
525.2.bg.b.436.3	yes	160	25.11	even	5	inner
525.2.bg.b.466.3	yes	160	7.4	even	3	inner