Embedded newform 525.2.bo.a.4.20

• Label: 525.2.bo.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	525.4
Dual form			525.2.bo.a.394.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0604608 - 0.284445i) q^{2} +(0.406737 + 0.913545i) q^{3} +(1.74984 - 0.779078i) q^{4} +(0.835291 - 2.07420i) q^{5} +(0.235262 - 0.170928i) q^{6} +(2.56191 - 0.660769i) q^{7} +(-0.669258 - 0.921154i) q^{8} +(-0.669131 + 0.743145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{4} - 2 q^{5} + 8 q^{6} + 60 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0604608	−	0.284445i	−0.0427522	−	0.201133i	0.951588	−	0.307377i	\(-0.0994511\pi\)
							−0.994340	+	0.106243i	\(0.966118\pi\)
\(3\)	0.406737	+	0.913545i	0.234830	+	0.527436i
							\(5\)	0.835291	−	2.07420i	0.373553	−	0.927609i
\(7\)	2.56191	−	0.660769i	0.968311	−	0.249747i
							\(11\)	−0.644499	−	0.715789i	−0.194324	−	0.215818i	0.638107	−	0.769948i	\(-0.279718\pi\)
−0.832430	+	0.554130i	\(0.813051\pi\)
\(13\)	1.11649	−	0.362770i	0.309659	−	0.100614i	−0.150065	−	0.988676i	\(-0.547948\pi\)
							0.459724	+	0.888062i	\(0.347948\pi\)
\(17\)	−6.23167	+	0.654975i	−1.51140	+	0.158855i	−0.823653	−	0.567093i	\(-0.808068\pi\)
							−0.687749	+	0.725948i	\(0.741401\pi\)
\(19\)	−2.32768	−	1.03635i	−0.534006	−	0.237755i	0.121974	−	0.992533i	\(-0.461078\pi\)
							−0.655980	+	0.754778i	\(0.727744\pi\)
\(23\)	1.11455	+	5.24353i	0.232399	+	1.09335i	0.927321	+	0.374266i	\(0.122105\pi\)
							−0.694922	+	0.719085i	\(0.744561\pi\)
\(29\)	0.857222	+	0.622808i	0.159182	+	0.115653i	0.664525	−	0.747266i	\(-0.268634\pi\)
							−0.505343	+	0.862919i	\(0.668634\pi\)
\(31\)	0.476065	+	4.52946i	0.0855039	+	0.813515i	0.950288	+	0.311372i	\(0.100788\pi\)
							−0.864784	+	0.502143i	\(0.832545\pi\)
\(37\)	2.57283	+	2.31658i	0.422970	+	0.380844i	0.852948	−	0.521995i	\(-0.174812\pi\)
							−0.429978	+	0.902839i	\(0.641479\pi\)
\(41\)	−1.52759	−	4.70143i	−0.238569	−	0.734240i	−0.996628	−	0.0820537i	\(-0.973852\pi\)
							0.758059	−	0.652186i	\(-0.226148\pi\)
\(43\)			9.00391i			1.37308i	0.727090	+	0.686542i	\(0.240872\pi\)
							−0.727090	+	0.686542i	\(0.759128\pi\)
\(47\)	11.0673	+	1.16322i	1.61433	+	0.169673i	0.868231	−	0.496160i	\(-0.165257\pi\)
							0.746100	+	0.665834i	\(0.231924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bo.a.4.20	✓	320
7.2	even	3	inner	525.2.bo.a.79.20	yes	320
25.19	even	10	inner	525.2.bo.a.319.20	yes	320
175.44	even	30	inner	525.2.bo.a.394.20	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bo.a.4.20	✓	320	1.1	even	1	trivial
525.2.bo.a.79.20	yes	320	7.2	even	3	inner
525.2.bo.a.319.20	yes	320	25.19	even	10	inner
525.2.bo.a.394.20	yes	320	175.44	even	30	inner