Embedded newform 525.2.bv.a.52.4

• Label: 525.2.bv.a.52.4
• Level: $525$
• Weight: $2$
• Character: 525.52
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bv (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(40\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			52.4
Character	\(\chi\)	\(=\)	525.52
Dual form			525.2.bv.a.313.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.872278 - 2.27236i) q^{2} +(-0.0523360 + 0.998630i) q^{3} +(-2.91647 + 2.62600i) q^{4} +(-2.11358 - 0.729921i) q^{5} +(2.31490 - 0.752156i) q^{6} +(0.892264 - 2.49076i) q^{7} +(4.17372 + 2.12662i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.872278	−	2.27236i	−0.616794	−	1.60680i	−0.783916	−	0.620866i	\(-0.786781\pi\)
							0.167123	−	0.985936i	\(-0.446552\pi\)
\(3\)	−0.0523360	+	0.998630i	−0.0302162	+	0.576559i
							\(5\)	−2.11358	−	0.729921i	−0.945221	−	0.326431i
\(7\)	0.892264	−	2.49076i	0.337244	−	0.941417i
							\(11\)	−0.394463	−	3.75306i	−0.118935	−	1.13159i	−0.877361	−	0.479831i	\(-0.840698\pi\)
0.758426	−	0.651759i	\(-0.225969\pi\)
\(13\)	0.522710	+	3.30026i	0.144974	+	0.915329i	0.947742	+	0.319037i	\(0.103360\pi\)
							−0.802768	+	0.596291i	\(0.796640\pi\)
\(17\)	−5.84393	+	3.79509i	−1.41736	+	0.920445i	−0.417431	+	0.908709i	\(0.637069\pi\)
							−0.999931	+	0.0117366i	\(0.996264\pi\)
\(19\)	0.614859	−	0.682870i	0.141058	−	0.156661i	−0.668476	−	0.743734i	\(-0.733053\pi\)
							0.809534	+	0.587073i	\(0.199720\pi\)
\(23\)	−6.52850	+	2.50606i	−1.36129	+	0.522549i	−0.925925	−	0.377706i	\(-0.876713\pi\)
							−0.435362	+	0.900256i	\(0.643380\pi\)
\(29\)	6.18154	+	2.00850i	1.14788	+	0.372970i	0.820345	−	0.571868i	\(-0.193781\pi\)
							0.327538	+	0.944838i	\(0.393781\pi\)
\(31\)	−0.952840	+	4.48276i	−0.171135	+	0.805128i	0.805900	+	0.592051i	\(0.201682\pi\)
							−0.977036	+	0.213077i	\(0.931652\pi\)
\(37\)	−2.51868	+	2.03959i	−0.414069	+	0.335306i	−0.813612	−	0.581408i	\(-0.802502\pi\)
							0.399543	+	0.916714i	\(0.369169\pi\)
\(41\)	−2.29217	+	3.15489i	−0.357976	+	0.492712i	−0.949583	−	0.313514i	\(-0.898494\pi\)
							0.591607	+	0.806226i	\(0.298494\pi\)
\(43\)	−3.78154	+	3.78154i	−0.576679	+	0.576679i	−0.933987	−	0.357308i	\(-0.883695\pi\)
							0.357308	+	0.933987i	\(0.383695\pi\)
\(47\)	−0.406212	+	0.625512i	−0.0592522	+	0.0912403i	−0.867055	−	0.498212i	\(-0.833990\pi\)
							0.807803	+	0.589453i	\(0.200657\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bv.a.52.4	✓	640
7.5	odd	6	inner	525.2.bv.a.502.4	yes	640
25.13	odd	20	inner	525.2.bv.a.388.4	yes	640
175.138	even	60	inner	525.2.bv.a.313.4	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bv.a.52.4	✓	640	1.1	even	1	trivial
525.2.bv.a.313.4	yes	640	175.138	even	60	inner
525.2.bv.a.388.4	yes	640	25.13	odd	20	inner
525.2.bv.a.502.4	yes	640	7.5	odd	6	inner