Embedded newform 225.4.u.a.169.1

• Label: 225.4.u.a.169.1
• 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			169.1
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13719 + 5.35004i) q^{2} +(-4.50950 - 2.58155i) q^{3} +(-20.0214 - 8.91409i) q^{4} +(2.25903 + 10.9497i) q^{5} +(18.9395 - 21.1903i) q^{6} +(24.9371 + 14.3974i) q^{7} +(44.7393 - 61.5784i) q^{8} +(13.6712 + 23.2830i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13719	+	5.35004i	−0.402056	+	1.89152i	0.0485847	+	0.998819i	\(0.484529\pi\)
							−0.450641	+	0.892705i	\(0.648804\pi\)
\(3\)	−4.50950	−	2.58155i	−0.867854	−	0.496819i
							\(5\)	2.25903	+	10.9497i	0.202053	+	0.979374i
\(7\)	24.9371	+	14.3974i	1.34647	+	0.777388i								0.987748	−	0.156055i	\(-0.0498777\pi\)
							0.358727	+	0.933443i	\(0.383211\pi\)
\(11\)	43.6921	+	9.28704i	1.19761	+	0.254559i	0.763202	−	0.646160i	\(-0.223626\pi\)
							0.434403	+	0.900719i	\(0.356959\pi\)
\(13\)	2.98869	+	14.0607i	0.0637625	+	0.299979i	0.998461	−	0.0554632i	\(-0.0176636\pi\)
							−0.934698	+	0.355442i	\(0.884330\pi\)
\(17\)	34.9316	−	48.0793i	0.498363	−	0.685937i	−0.483540	−	0.875322i	\(-0.660649\pi\)
							0.981903	+	0.189385i	\(0.0606494\pi\)
\(19\)	98.4561	+	71.5326i	1.18881	+	0.863721i	0.993138	−	0.116948i	\(-0.0373112\pi\)
							0.195672	+	0.980669i	\(0.437311\pi\)
\(23\)	−34.7244	−	31.2660i	−0.314806	−	0.283453i	0.496546	−	0.868010i	\(-0.334601\pi\)
							−0.811352	+	0.584558i	\(0.801268\pi\)
\(29\)	−12.6901	−	120.739i	−0.0812587	−	0.773124i	−0.956950	−	0.290253i	\(-0.906261\pi\)
							0.875691	−	0.482871i	\(-0.160406\pi\)
\(31\)	−14.3789	+	136.806i	−0.0833072	+	0.792615i	0.870494	+	0.492178i	\(0.163799\pi\)
							−0.953802	+	0.300437i	\(0.902867\pi\)
\(37\)	−13.1630	−	4.27692i	−0.0584860	−	0.0190033i	0.279628	−	0.960108i	\(-0.409789\pi\)
							−0.338114	+	0.941105i	\(0.609789\pi\)
\(41\)	307.965	−	65.4600i	1.17308	−	0.249345i	0.420152	−	0.907454i	\(-0.361977\pi\)
							0.752923	+	0.658109i	\(0.228643\pi\)
\(43\)	226.301	+	130.655i	0.802572	+	0.463365i	0.844370	−	0.535761i	\(-0.179975\pi\)
							−0.0417976	+	0.999126i	\(0.513308\pi\)
\(47\)	−285.686	+	30.0268i	−0.886630	+	0.0931886i	−0.536884	−	0.843656i	\(-0.680399\pi\)
							−0.349746	+	0.936845i	\(0.613732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.169.1	yes	704
9.4	even	3	inner	225.4.u.a.94.1	yes	704
25.4	even	10	inner	225.4.u.a.79.1	yes	704
225.4	even	30	inner	225.4.u.a.4.1	✓	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.1	✓	704	225.4	even	30	inner
225.4.u.a.79.1	yes	704	25.4	even	10	inner
225.4.u.a.94.1	yes	704	9.4	even	3	inner
225.4.u.a.169.1	yes	704	1.1	even	1	trivial