Embedded newform 525.2.bv.a.52.11

• Label: 525.2.bv.a.52.11
• 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			52.11
Character	\(\chi\)	\(=\)	525.52
Dual form			525.2.bv.a.313.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.498488 - 1.29861i) q^{2} +(0.0523360 - 0.998630i) q^{3} +(0.0484028 - 0.0435821i) q^{4} +(-0.935202 + 2.03111i) q^{5} +(-1.32292 + 0.429841i) q^{6} +(-2.07682 - 1.63916i) q^{7} +(-2.55950 - 1.30413i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.498488	−	1.29861i	−0.352484	−	0.918253i	−0.989123	−	0.147091i	\(-0.953009\pi\)
							0.636639	−	0.771162i	\(-0.280324\pi\)
\(3\)	0.0523360	−	0.998630i	0.0302162	−	0.576559i
							\(5\)	−0.935202	+	2.03111i	−0.418235	+	0.908339i
\(7\)	−2.07682	−	1.63916i	−0.784962	−	0.619544i
							\(11\)	−0.442009	−	4.20543i	−0.133271	−	1.26799i	−0.832877	−	0.553458i	\(-0.813308\pi\)
0.699607	−	0.714528i	\(-0.253359\pi\)
\(13\)	0.637414	+	4.02447i	0.176787	+	1.11619i	0.903292	+	0.429027i	\(0.141143\pi\)
							−0.726505	+	0.687161i	\(0.758857\pi\)
\(17\)	−2.76444	+	1.79525i	−0.670475	+	0.435411i	−0.834473	−	0.551049i	\(-0.814228\pi\)
							0.163998	+	0.986461i	\(0.447561\pi\)
\(19\)	1.18180	−	1.31252i	0.271123	−	0.301112i	−0.592173	−	0.805811i	\(-0.701730\pi\)
							0.863296	+	0.504699i	\(0.168396\pi\)
\(23\)	−6.31495	+	2.42408i	−1.31676	+	0.505456i	−0.912458	−	0.409171i	\(-0.865818\pi\)
							−0.404300	+	0.914627i	\(0.632485\pi\)
\(29\)	−1.94034	−	0.630453i	−0.360311	−	0.117072i	0.123267	−	0.992374i	\(-0.460663\pi\)
							−0.483578	+	0.875301i	\(0.660663\pi\)
\(31\)	0.641149	−	3.01637i	0.115154	−	0.541756i	−0.882316	−	0.470658i	\(-0.844016\pi\)
							0.997469	−	0.0710978i	\(-0.0226502\pi\)
\(37\)	−5.43152	+	4.39836i	−0.892936	+	0.723085i	−0.961571	−	0.274556i	\(-0.911469\pi\)
							0.0686348	+	0.997642i	\(0.478136\pi\)
\(41\)	−1.24089	+	1.70794i	−0.193795	+	0.266736i	−0.894846	−	0.446376i	\(-0.852715\pi\)
							0.701051	+	0.713111i	\(0.252715\pi\)
\(43\)	5.66939	−	5.66939i	0.864573	−	0.864573i	−0.127292	−	0.991865i	\(-0.540629\pi\)
							0.991865	+	0.127292i	\(0.0406286\pi\)
\(47\)	0.0952165	−	0.146621i	0.0138888	−	0.0213868i	−0.831658	−	0.555288i	\(-0.812608\pi\)
							0.845547	+	0.533901i	\(0.179275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bv.a.52.11	✓	640
7.5	odd	6	inner	525.2.bv.a.502.11	yes	640
25.13	odd	20	inner	525.2.bv.a.388.11	yes	640
175.138	even	60	inner	525.2.bv.a.313.11	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bv.a.52.11	✓	640	1.1	even	1	trivial
525.2.bv.a.313.11	yes	640	175.138	even	60	inner
525.2.bv.a.388.11	yes	640	25.13	odd	20	inner
525.2.bv.a.502.11	yes	640	7.5	odd	6	inner