Embedded newform 225.4.u.a.4.20

• Label: 225.4.u.a.4.20
• Level: $225$
• Weight: $4$
• Character: 225.4
• Analytic conductor: $13.275$
• Analytic rank: $0$
• Dimension: $704$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(704\)
Relative dimension:	\(88\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.20
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.728639 - 3.42798i) q^{2} +(-4.82414 + 1.93071i) q^{3} +(-3.91175 + 1.74162i) q^{4} +(-5.26207 + 9.86462i) q^{5} +(10.1335 + 15.1303i) q^{6} +(23.0240 - 13.2929i) q^{7} +(-7.65893 - 10.5416i) q^{8} +(19.5447 - 18.6280i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.728639	−	3.42798i	−0.257613	−	1.21197i	−0.896630	−	0.442781i	\(-0.853992\pi\)
							0.639017	−	0.769192i	\(-0.279341\pi\)
\(3\)	−4.82414	+	1.93071i	−0.928407	+	0.371565i
							\(5\)	−5.26207	+	9.86462i	−0.470653	+	0.882318i
\(7\)	23.0240	−	13.2929i	1.24318	−	0.717751i								0.273441	−	0.961889i	\(-0.411838\pi\)
							0.969741	+	0.244138i	\(0.0785049\pi\)
\(11\)	−21.1063	+	4.48628i	−0.578526	+	0.122970i	−0.487875	−	0.872913i	\(-0.662228\pi\)
							−0.0906506	+	0.995883i	\(0.528895\pi\)
\(13\)	8.68684	−	40.8684i	0.185331	−	0.871912i	−0.782958	−	0.622075i	\(-0.786290\pi\)
							0.968288	−	0.249836i	\(-0.0803768\pi\)
\(17\)	11.2228	+	15.4469i	0.160114	+	0.220377i	0.881535	−	0.472119i	\(-0.156511\pi\)
							−0.721421	+	0.692497i	\(0.756511\pi\)
\(19\)	−54.4316	+	39.5469i	−0.657235	+	0.477509i	−0.865728	−	0.500515i	\(-0.833144\pi\)
							0.208493	+	0.978024i	\(0.433144\pi\)
\(23\)	−62.9364	+	56.6682i	−0.570572	+	0.513745i	−0.903147	−	0.429332i	\(-0.858749\pi\)
							0.332575	+	0.943077i	\(0.392082\pi\)
\(29\)	−11.3622	+	108.104i	−0.0727552	+	0.692219i	0.895975	+	0.444104i	\(0.146478\pi\)
							−0.968731	+	0.248115i	\(0.920189\pi\)
\(31\)	−26.1215	−	248.530i	−0.151341	−	1.43991i	−0.761774	−	0.647843i	\(-0.775671\pi\)
							0.610433	−	0.792068i	\(-0.290995\pi\)
\(37\)	−178.573	+	58.0218i	−0.793437	+	0.257803i	−0.677567	−	0.735461i	\(-0.736966\pi\)
							−0.115870	+	0.993264i	\(0.536966\pi\)
\(41\)	−460.124	−	97.8023i	−1.75266	−	0.372540i	−0.783971	−	0.620798i	\(-0.786809\pi\)
							−0.968694	+	0.248257i	\(0.920142\pi\)
\(43\)	−130.968	+	75.6143i	−0.464475	+	0.268165i	−0.713924	−	0.700223i	\(-0.753084\pi\)
							0.249449	+	0.968388i	\(0.419750\pi\)
\(47\)	324.502	+	34.1066i	1.00710	+	0.105850i	0.593688	−	0.804696i	\(-0.297672\pi\)
							0.413408	+	0.910546i	\(0.364338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.20	✓	704
9.7	even	3	inner	225.4.u.a.79.20	yes	704
25.19	even	10	inner	225.4.u.a.94.20	yes	704
225.169	even	30	inner	225.4.u.a.169.20	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.20	✓	704	1.1	even	1	trivial
225.4.u.a.79.20	yes	704	9.7	even	3	inner
225.4.u.a.94.20	yes	704	25.19	even	10	inner
225.4.u.a.169.20	yes	704	225.169	even	30	inner