Embedded newform 225.4.u.a.169.20

• Label: 225.4.u.a.169.20
• Level: $225$
• Weight: $4$
• Character: 225.169
• 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			169.20
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.20

$q$-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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.20	✓	704	225.4	even	30	inner
225.4.u.a.79.20	yes	704	25.4	even	10	inner
225.4.u.a.94.20	yes	704	9.4	even	3	inner
225.4.u.a.169.20	yes	704	1.1	even	1	trivial