Embedded newform 225.4.u.a.169.5

• Label: 225.4.u.a.169.5
• 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			169.5
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07574 + 5.06097i) q^{2} +(0.736845 + 5.14364i) q^{3} +(-17.1478 - 7.63470i) q^{4} +(-5.20342 + 9.89568i) q^{5} +(-26.8245 - 1.80408i) q^{6} +(-29.1252 - 16.8154i) q^{7} +(32.7558 - 45.0845i) q^{8} +(-25.9141 + 7.58014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07574	+	5.06097i	−0.380332	+	1.78932i	0.205250	+	0.978710i	\(0.434199\pi\)
							−0.585582	+	0.810613i	\(0.699134\pi\)
\(3\)	0.736845	+	5.14364i	0.141806	+	0.989894i
							\(5\)	−5.20342	+	9.89568i	−0.465408	+	0.885096i
\(7\)	−29.1252	−	16.8154i	−1.57261	−	0.907948i								−0.995847	−	0.0910377i	\(-0.970982\pi\)
							−0.576765	−	0.816910i	\(-0.695685\pi\)
\(11\)	44.3419	+	9.42516i	1.21542	+	0.258345i	0.770625	−	0.637289i	\(-0.219944\pi\)
							0.444792	+	0.895634i	\(0.353278\pi\)
\(13\)	13.2249	+	62.2183i	0.282149	+	1.32740i	0.859595	+	0.510976i	\(0.170716\pi\)
							−0.577446	+	0.816429i	\(0.695951\pi\)
\(17\)	−48.0610	+	66.1503i	−0.685677	+	0.943753i	−0.999984	−	0.00556808i	\(-0.998228\pi\)
							0.314308	+	0.949321i	\(0.398228\pi\)
\(19\)	42.4448	+	30.8379i	0.512500	+	0.372353i	0.813771	−	0.581186i	\(-0.197411\pi\)
							−0.301271	+	0.953538i	\(0.597411\pi\)
\(23\)	28.7962	+	25.9282i	0.261062	+	0.235061i	0.789259	−	0.614061i	\(-0.210465\pi\)
							−0.528197	+	0.849122i	\(0.677132\pi\)
\(29\)	−6.27981	−	59.7484i	−0.0402114	−	0.382586i	−0.996057	−	0.0887166i	\(-0.971723\pi\)
							0.955845	−	0.293870i	\(-0.0949432\pi\)
\(31\)	24.1652	−	229.917i	0.140006	−	1.33207i	−0.668556	−	0.743662i	\(-0.733087\pi\)
							0.808562	−	0.588410i	\(-0.200246\pi\)
\(37\)	−160.969	−	52.3021i	−0.715222	−	0.232390i	−0.0712714	−	0.997457i	\(-0.522706\pi\)
							−0.643950	+	0.765067i	\(0.722706\pi\)
\(41\)	−49.5936	+	10.5414i	−0.188908	+	0.0401536i	−0.301394	−	0.953500i	\(-0.597452\pi\)
							0.112486	+	0.993653i	\(0.464119\pi\)
\(43\)	80.6135	+	46.5422i	0.285894	+	0.165061i	0.636089	−	0.771616i	\(-0.280551\pi\)
							−0.350195	+	0.936677i	\(0.613885\pi\)
\(47\)	−212.599	+	22.3450i	−0.659802	+	0.0693480i	−0.428511	−	0.903537i	\(-0.640962\pi\)
							−0.231291	+	0.972885i	\(0.574295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.169.5	yes	704
9.4	even	3	inner	225.4.u.a.94.5	yes	704
25.4	even	10	inner	225.4.u.a.79.5	yes	704
225.4	even	30	inner	225.4.u.a.4.5	✓	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.5	✓	704	225.4	even	30	inner
225.4.u.a.79.5	yes	704	25.4	even	10	inner
225.4.u.a.94.5	yes	704	9.4	even	3	inner
225.4.u.a.169.5	yes	704	1.1	even	1	trivial