Embedded newform 225.4.u.a.169.18

• Label: 225.4.u.a.169.18
• 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			169.18
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.792549 + 3.72865i) q^{2} +(2.16415 - 4.72403i) q^{3} +(-5.96634 - 2.65639i) q^{4} +(9.75102 + 5.46969i) q^{5} +(15.8991 + 11.8134i) q^{6} +(-4.93704 - 2.85040i) q^{7} +(-3.29151 + 4.53038i) q^{8} +(-17.6329 - 20.4470i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.792549	+	3.72865i	−0.280208	+	1.31828i	0.582605	+	0.812755i	\(0.302033\pi\)
							−0.862814	+	0.505522i	\(0.831300\pi\)
\(3\)	2.16415	−	4.72403i	0.416490	−	0.909140i
							\(5\)	9.75102	+	5.46969i	0.872158	+	0.489224i
\(7\)	−4.93704	−	2.85040i	−0.266575	−	0.153907i								0.360755	−	0.932661i	\(-0.382519\pi\)
							−0.627330	+	0.778753i	\(0.715852\pi\)
\(11\)	−3.91048	−	0.831199i	−0.107187	−	0.0227833i	0.154006	−	0.988070i	\(-0.450783\pi\)
							−0.261193	+	0.965287i	\(0.584116\pi\)
\(13\)	18.8877	+	88.8597i	0.402962	+	1.89579i	0.442872	+	0.896585i	\(0.353960\pi\)
							−0.0399100	+	0.999203i	\(0.512707\pi\)
\(17\)	27.5620	−	37.9359i	0.393222	−	0.541224i	−0.565805	−	0.824539i	\(-0.691434\pi\)
							0.959027	+	0.283316i	\(0.0914344\pi\)
\(19\)	123.602	+	89.8023i	1.49244	+	1.08432i	0.973275	+	0.229644i	\(0.0737561\pi\)
							0.519163	+	0.854675i	\(0.326244\pi\)
\(23\)	66.4280	+	59.8120i	0.602225	+	0.542246i	0.912855	−	0.408285i	\(-0.133873\pi\)
							−0.310629	+	0.950531i	\(0.600540\pi\)
\(29\)	3.36489	+	32.0148i	0.0215464	+	0.205000i	0.999999	−	0.00147173i	\(-0.000468468\pi\)
							−0.978453	+	0.206472i	\(0.933802\pi\)
\(31\)	−14.1845	+	134.956i	−0.0821808	+	0.781899i	0.873367	+	0.487063i	\(0.161932\pi\)
							−0.955548	+	0.294836i	\(0.904735\pi\)
\(37\)	−102.435	−	33.2831i	−0.455140	−	0.147884i	0.0724700	−	0.997371i	\(-0.476912\pi\)
							−0.527610	+	0.849487i	\(0.676912\pi\)
\(41\)	161.891	−	34.4110i	0.616662	−	0.131076i	0.111019	−	0.993818i	\(-0.464589\pi\)
							0.505643	+	0.862743i	\(0.331255\pi\)
\(43\)	196.031	+	113.179i	0.695221	+	0.401386i	0.805565	−	0.592507i	\(-0.201862\pi\)
							−0.110344	+	0.993893i	\(0.535195\pi\)
\(47\)	−162.330	+	17.0615i	−0.503791	+	0.0529506i	−0.353016	−	0.935617i	\(-0.614844\pi\)
							−0.150776	+	0.988568i	\(0.548177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.169.18	yes	704
9.4	even	3	inner	225.4.u.a.94.18	yes	704
25.4	even	10	inner	225.4.u.a.79.18	yes	704
225.4	even	30	inner	225.4.u.a.4.18	✓	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.18	✓	704	225.4	even	30	inner
225.4.u.a.79.18	yes	704	25.4	even	10	inner
225.4.u.a.94.18	yes	704	9.4	even	3	inner
225.4.u.a.169.18	yes	704	1.1	even	1	trivial