Embedded newform 225.4.u.a.4.6

• Label: 225.4.u.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06743 - 5.02188i) q^{2} +(-0.0785966 + 5.19556i) q^{3} +(-16.7715 + 7.46716i) q^{4} +(-8.45497 + 7.31529i) q^{5} +(26.1754 - 5.15121i) q^{6} +(-9.64326 + 5.56754i) q^{7} +(31.2598 + 43.0254i) q^{8} +(-26.9876 - 0.816706i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06743	−	5.02188i	−0.377395	−	1.77550i	−0.599392	−	0.800456i	\(-0.704591\pi\)
							0.221997	−	0.975047i	\(-0.428743\pi\)
\(3\)	−0.0785966	+	5.19556i	−0.0151259	+	0.999886i
							\(5\)	−8.45497	+	7.31529i	−0.756236	+	0.654299i
\(7\)	−9.64326	+	5.56754i	−0.520687	+	0.300619i								−0.737216	−	0.675657i	\(-0.763860\pi\)
							0.216529	+	0.976276i	\(0.430527\pi\)
\(11\)	39.7998	−	8.45970i	1.09092	−	0.231881i	0.372875	−	0.927882i	\(-0.378372\pi\)
							0.718041	+	0.696000i	\(0.245039\pi\)
\(13\)	7.12033	−	33.4985i	0.151909	−	0.714678i	−0.834584	−	0.550880i	\(-0.814292\pi\)
							0.986494	−	0.163798i	\(-0.0523745\pi\)
\(17\)	−3.12864	−	4.30620i	−0.0446357	−	0.0614357i	0.786116	−	0.618079i	\(-0.212089\pi\)
							−0.830751	+	0.556644i	\(0.812089\pi\)
\(19\)	−19.4526	+	14.1332i	−0.234881	+	0.170651i	−0.699000	−	0.715122i	\(-0.746371\pi\)
							0.464119	+	0.885773i	\(0.346371\pi\)
\(23\)	−53.5088	+	48.1795i	−0.485103	+	0.436788i	−0.875038	−	0.484054i	\(-0.839164\pi\)
							0.389936	+	0.920842i	\(0.372497\pi\)
\(29\)	11.1770	−	106.342i	0.0715696	−	0.680939i	−0.898642	−	0.438682i	\(-0.855446\pi\)
							0.970212	−	0.242257i	\(-0.0778878\pi\)
\(31\)	−29.2365	−	278.167i	−0.169388	−	1.61162i	−0.667569	−	0.744548i	\(-0.732665\pi\)
							0.498181	−	0.867073i	\(-0.334002\pi\)
\(37\)	272.570	−	88.5635i	1.21109	−	0.393507i	0.367260	−	0.930118i	\(-0.380296\pi\)
							0.843829	+	0.536612i	\(0.180296\pi\)
\(41\)	−316.929	−	67.3654i	−1.20722	−	0.256603i	−0.440007	−	0.897994i	\(-0.645024\pi\)
							−0.767214	+	0.641391i	\(0.778357\pi\)
\(43\)	280.615	−	162.013i	0.995196	−	0.574577i	0.0883728	−	0.996087i	\(-0.471833\pi\)
							0.906823	+	0.421511i	\(0.138500\pi\)
\(47\)	42.4571	+	4.46242i	0.131766	+	0.0138492i	0.170182	−	0.985413i	\(-0.445564\pi\)
							−0.0384159	+	0.999262i	\(0.512231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.6	✓	704
9.7	even	3	inner	225.4.u.a.79.6	yes	704
25.19	even	10	inner	225.4.u.a.94.6	yes	704
225.169	even	30	inner	225.4.u.a.169.6	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.6	✓	704	1.1	even	1	trivial
225.4.u.a.79.6	yes	704	9.7	even	3	inner
225.4.u.a.94.6	yes	704	25.19	even	10	inner
225.4.u.a.169.6	yes	704	225.169	even	30	inner