Embedded newform 525.2.bv.a.52.18

• Label: 525.2.bv.a.52.18
• 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			52.18
Character	\(\chi\)	\(=\)	525.52
Dual form			525.2.bv.a.313.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.107591 - 0.280285i) q^{2} +(0.0523360 - 0.998630i) q^{3} +(1.41931 - 1.27795i) q^{4} +(1.68997 + 1.46424i) q^{5} +(-0.285532 + 0.0927749i) q^{6} +(-1.90734 - 1.83359i) q^{7} +(-1.04590 - 0.532913i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.107591	−	0.280285i	−0.0760786	−	0.198191i	0.890267	−	0.455439i	\(-0.150518\pi\)
							−0.966346	+	0.257248i	\(0.917184\pi\)
\(3\)	0.0523360	−	0.998630i	0.0302162	−	0.576559i
							\(5\)	1.68997	+	1.46424i	0.755780	+	0.654826i
\(7\)	−1.90734	−	1.83359i	−0.720909	−	0.693030i
							\(11\)	0.0760837	+	0.723888i	0.0229401	+	0.218261i	0.999986	+	0.00523303i	\(0.00166573\pi\)
−0.977046	+	0.213028i	\(0.931668\pi\)
\(13\)	−0.793179	−	5.00794i	−0.219988	−	1.38895i	−0.812298	−	0.583242i	\(-0.801784\pi\)
							0.592310	−	0.805710i	\(-0.298216\pi\)
\(17\)	3.42364	−	2.22334i	0.830354	−	0.539238i	−0.0580428	−	0.998314i	\(-0.518486\pi\)
							0.888397	+	0.459076i	\(0.151819\pi\)
\(19\)	−2.10571	+	2.33862i	−0.483082	+	0.536517i	−0.934579	−	0.355756i	\(-0.884224\pi\)
							0.451497	+	0.892273i	\(0.350890\pi\)
\(23\)	6.06988	−	2.33001i	1.26566	−	0.485840i	0.369357	−	0.929288i	\(-0.379578\pi\)
							0.896300	+	0.443447i	\(0.146245\pi\)
\(29\)	−4.57645	−	1.48698i	−0.849825	−	0.276125i	−0.148452	−	0.988920i	\(-0.547429\pi\)
							−0.701373	+	0.712795i	\(0.747429\pi\)
\(31\)	−1.89607	+	8.92033i	−0.340545	+	1.60214i	0.391030	+	0.920378i	\(0.372119\pi\)
							−0.731575	+	0.681761i	\(0.761214\pi\)
\(37\)	3.18134	−	2.57620i	0.523010	−	0.423525i	−0.331117	−	0.943590i	\(-0.607425\pi\)
							0.854127	+	0.520065i	\(0.174092\pi\)
\(41\)	5.40693	−	7.44200i	0.844421	−	1.16225i	−0.140644	−	0.990060i	\(-0.544917\pi\)
							0.985065	−	0.172185i	\(-0.0550827\pi\)
\(43\)	4.26746	−	4.26746i	0.650781	−	0.650781i	−0.302400	−	0.953181i	\(-0.597788\pi\)
							0.953181	+	0.302400i	\(0.0977878\pi\)
\(47\)	−5.28005	+	8.13056i	−0.770174	+	1.18596i	0.207991	+	0.978131i	\(0.433308\pi\)
							−0.978164	+	0.207833i	\(0.933359\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bv.a.52.18	✓	640
7.5	odd	6	inner	525.2.bv.a.502.18	yes	640
25.13	odd	20	inner	525.2.bv.a.388.18	yes	640
175.138	even	60	inner	525.2.bv.a.313.18	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bv.a.52.18	✓	640	1.1	even	1	trivial
525.2.bv.a.313.18	yes	640	175.138	even	60	inner
525.2.bv.a.388.18	yes	640	25.13	odd	20	inner
525.2.bv.a.502.18	yes	640	7.5	odd	6	inner