Embedded newform 225.4.u.a.4.5

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

$q$-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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.5	✓	704	1.1	even	1	trivial
225.4.u.a.79.5	yes	704	9.7	even	3	inner
225.4.u.a.94.5	yes	704	25.19	even	10	inner
225.4.u.a.169.5	yes	704	225.169	even	30	inner