Embedded newform 525.2.bo.a.4.10

• Label: 525.2.bo.a.4.10
• Level: $525$
• Weight: $2$
• Character: 525.4
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bo (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.10
Character	\(\chi\)	\(=\)	525.4
Dual form			525.2.bo.a.394.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.383892 - 1.80607i) q^{2} +(0.406737 + 0.913545i) q^{3} +(-1.28743 + 0.573200i) q^{4} +(2.20206 - 0.388478i) q^{5} +(1.49378 - 1.08530i) q^{6} +(0.977325 - 2.45862i) q^{7} +(-0.641123 - 0.882430i) q^{8} +(-0.669131 + 0.743145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{4} - 2 q^{5} + 8 q^{6} + 60 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.383892	−	1.80607i	−0.271453	−	1.27709i	−0.876696	−	0.481045i	\(-0.840258\pi\)
							0.605243	−	0.796041i	\(-0.293076\pi\)
\(3\)	0.406737	+	0.913545i	0.234830	+	0.527436i
							\(5\)	2.20206	−	0.388478i	0.984793	−	0.173733i
\(7\)	0.977325	−	2.45862i	0.369394	−	0.929273i
							\(11\)	1.31516	+	1.46063i	0.396536	+	0.440398i	0.908040	−	0.418883i	\(-0.137578\pi\)
−0.511504	+	0.859281i	\(0.670912\pi\)
\(13\)	0.591492	−	0.192187i	0.164050	−	0.0533032i	−0.225840	−	0.974164i	\(-0.572513\pi\)
							0.389891	+	0.920861i	\(0.372513\pi\)
\(17\)	4.82198	−	0.506811i	1.16950	−	0.122920i	0.500200	−	0.865910i	\(-0.333260\pi\)
							0.669302	+	0.742990i	\(0.266593\pi\)
\(19\)	−3.46115	−	1.54101i	−0.794043	−	0.353531i	−0.0307039	−	0.999529i	\(-0.509775\pi\)
							−0.763339	+	0.645998i	\(0.776442\pi\)
\(23\)	0.481337	+	2.26451i	0.100366	+	0.472184i	0.999411	+	0.0343273i	\(0.0109289\pi\)
							−0.899045	+	0.437857i	\(0.855738\pi\)
\(29\)	−1.56272	−	1.13538i	−0.290189	−	0.210835i	0.433160	−	0.901317i	\(-0.357398\pi\)
							−0.723349	+	0.690482i	\(0.757398\pi\)
\(31\)	−0.699423	−	6.65456i	−0.125620	−	1.19520i	−0.857763	−	0.514045i	\(-0.828147\pi\)
							0.732143	−	0.681151i	\(-0.238520\pi\)
\(37\)	−1.07045	−	0.963840i	−0.175981	−	0.158454i	0.576433	−	0.817144i	\(-0.304444\pi\)
							−0.752414	+	0.658690i	\(0.771111\pi\)
\(41\)	2.49819	+	7.68864i	0.390152	+	1.20076i	0.932674	+	0.360722i	\(0.117470\pi\)
							−0.542522	+	0.840042i	\(0.682530\pi\)
\(43\)		−	7.61797i		−	1.16173i	−0.814000	−	0.580865i	\(-0.802715\pi\)
							0.814000	−	0.580865i	\(-0.197285\pi\)
\(47\)	−5.35425	−	0.562754i	−0.780997	−	0.0820861i	−0.294365	−	0.955693i	\(-0.595108\pi\)
							−0.486632	+	0.873607i	\(0.661775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bo.a.4.10	✓	320
7.2	even	3	inner	525.2.bo.a.79.10	yes	320
25.19	even	10	inner	525.2.bo.a.319.10	yes	320
175.44	even	30	inner	525.2.bo.a.394.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bo.a.4.10	✓	320	1.1	even	1	trivial
525.2.bo.a.79.10	yes	320	7.2	even	3	inner
525.2.bo.a.319.10	yes	320	25.19	even	10	inner
525.2.bo.a.394.10	yes	320	175.44	even	30	inner