Embedded newform 225.4.u.a.4.14

• Label: 225.4.u.a.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.849046 - 3.99445i) q^{2} +(0.343429 - 5.18479i) q^{3} +(-7.92637 + 3.52905i) q^{4} +(5.82974 + 9.54013i) q^{5} +(-21.0020 + 3.03032i) q^{6} +(-17.1647 + 9.91002i) q^{7} +(1.62379 + 2.23496i) q^{8} +(-26.7641 - 3.56121i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.849046	−	3.99445i	−0.300183	−	1.41225i	−0.826967	−	0.562251i	\(-0.809936\pi\)
							0.526784	−	0.849999i	\(-0.323398\pi\)
\(3\)	0.343429	−	5.18479i	0.0660929	−	0.997813i
							\(5\)	5.82974	+	9.54013i	0.521428	+	0.853296i
\(7\)	−17.1647	+	9.91002i	−0.926804	+	0.535091i								−0.885799	−	0.464068i	\(-0.846389\pi\)
							−0.0410049	+	0.999159i	\(0.513056\pi\)
\(11\)	−0.600903	+	0.127726i	−0.0164708	+	0.00350098i	−0.216140	−	0.976362i	\(-0.569347\pi\)
							0.199669	+	0.979863i	\(0.436013\pi\)
\(13\)	−8.91167	+	41.9261i	−0.190127	+	0.894478i	0.774850	+	0.632145i	\(0.217825\pi\)
							−0.964977	+	0.262333i	\(0.915508\pi\)
\(17\)	47.0031	+	64.6942i	0.670584	+	0.922980i	0.999773	−	0.0212843i	\(-0.00677550\pi\)
							−0.329190	+	0.944264i	\(0.606776\pi\)
\(19\)	32.5043	−	23.6157i	0.392473	−	0.285148i	−0.373995	−	0.927431i	\(-0.622012\pi\)
							0.766468	+	0.642282i	\(0.222012\pi\)
\(23\)	−87.9775	+	79.2153i	−0.797591	+	0.718154i	−0.963419	−	0.268001i	\(-0.913637\pi\)
							0.165828	+	0.986155i	\(0.446970\pi\)
\(29\)	−4.07461	+	38.7673i	−0.0260909	+	0.248238i	0.973701	+	0.227830i	\(0.0731632\pi\)
							−0.999792	+	0.0204079i	\(0.993504\pi\)
\(31\)	−33.0037	−	314.009i	−0.191214	−	1.81928i	−0.497637	−	0.867386i	\(-0.665799\pi\)
							0.306422	−	0.951896i	\(-0.400868\pi\)
\(37\)	219.193	−	71.2202i	0.973923	−	0.316447i	0.221524	−	0.975155i	\(-0.428897\pi\)
							0.752399	+	0.658708i	\(0.228897\pi\)
\(41\)	64.7742	+	13.7682i	0.246733	+	0.0524446i	0.329618	−	0.944114i	\(-0.393080\pi\)
							−0.0828852	+	0.996559i	\(0.526413\pi\)
\(43\)	−358.878	+	207.199i	−1.27275	+	0.734825i	−0.975506	−	0.219975i	\(-0.929402\pi\)
							−0.297249	+	0.954800i	\(0.596069\pi\)
\(47\)	−541.844	−	56.9501i	−1.68162	−	0.176745i	−0.784984	−	0.619516i	\(-0.787329\pi\)
							−0.896635	+	0.442771i	\(0.853995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.14	✓	704
9.7	even	3	inner	225.4.u.a.79.14	yes	704
25.19	even	10	inner	225.4.u.a.94.14	yes	704
225.169	even	30	inner	225.4.u.a.169.14	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.14	✓	704	1.1	even	1	trivial
225.4.u.a.79.14	yes	704	9.7	even	3	inner
225.4.u.a.94.14	yes	704	25.19	even	10	inner
225.4.u.a.169.14	yes	704	225.169	even	30	inner