Embedded newform 225.4.u.a.4.13

• Label: 225.4.u.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.912762 - 4.29421i) q^{2} +(-3.93731 - 3.39081i) q^{3} +(-10.2987 + 4.58529i) q^{4} +(5.02582 - 9.98705i) q^{5} +(-10.9670 + 20.0026i) q^{6} +(-4.48355 + 2.58858i) q^{7} +(8.44680 + 11.6260i) q^{8} +(4.00486 + 26.7013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.912762	−	4.29421i	−0.322710	−	1.51823i	−0.778214	−	0.627999i	\(-0.783874\pi\)
							0.455504	−	0.890234i	\(-0.349459\pi\)
\(3\)	−3.93731	−	3.39081i	−0.757736	−	0.652561i
							\(5\)	5.02582	−	9.98705i	0.449523	−	0.893269i
\(7\)	−4.48355	+	2.58858i	−0.242089	+	0.139770i								−0.616137	−	0.787639i	\(-0.711303\pi\)
							0.374047	+	0.927410i	\(0.377970\pi\)
\(11\)	−19.6306	+	4.17262i	−0.538078	+	0.114372i	−0.468931	−	0.883235i	\(-0.655361\pi\)
							−0.0691463	+	0.997607i	\(0.522028\pi\)
\(13\)	−8.37005	+	39.3780i	−0.178572	+	0.840114i	0.794074	+	0.607821i	\(0.207956\pi\)
							−0.972646	+	0.232293i	\(0.925377\pi\)
\(17\)	34.0702	+	46.8936i	0.486073	+	0.669022i	0.979658	−	0.200675i	\(-0.0643137\pi\)
							−0.493585	+	0.869698i	\(0.664314\pi\)
\(19\)	−63.1063	+	45.8494i	−0.761978	+	0.553609i	−0.899516	−	0.436887i	\(-0.856081\pi\)
							0.137538	+	0.990496i	\(0.456081\pi\)
\(23\)	57.8119	−	52.0541i	0.524114	−	0.471914i	−0.364090	−	0.931364i	\(-0.618620\pi\)
							0.888204	+	0.459449i	\(0.151953\pi\)
\(29\)	−5.52543	+	52.5709i	−0.0353809	+	0.336627i	0.962485	+	0.271334i	\(0.0874648\pi\)
							−0.997866	+	0.0652928i	\(0.979202\pi\)
\(31\)	−16.1419	−	153.580i	−0.0935216	−	0.889798i	−0.936221	−	0.351413i	\(-0.885701\pi\)
							0.842699	−	0.538385i	\(-0.180965\pi\)
\(37\)	−251.420	+	81.6912i	−1.11711	+	0.362971i	−0.808664	−	0.588270i	\(-0.799809\pi\)
							−0.308447	+	0.951242i	\(0.599809\pi\)
\(41\)	−453.148	−	96.3195i	−1.72609	−	0.366892i	−0.765195	−	0.643798i	\(-0.777358\pi\)
							−0.960897	+	0.276906i	\(0.910691\pi\)
\(43\)	361.931	−	208.961i	1.28358	−	0.741076i	0.306080	−	0.952006i	\(-0.400982\pi\)
							0.977501	+	0.210930i	\(0.0676491\pi\)
\(47\)	191.675	+	20.1459i	0.594866	+	0.0625229i	0.397177	−	0.917742i	\(-0.369990\pi\)
							0.197689	+	0.980265i	\(0.436656\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.13	✓	704
9.7	even	3	inner	225.4.u.a.79.13	yes	704
25.19	even	10	inner	225.4.u.a.94.13	yes	704
225.169	even	30	inner	225.4.u.a.169.13	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.13	✓	704	1.1	even	1	trivial
225.4.u.a.79.13	yes	704	9.7	even	3	inner
225.4.u.a.94.13	yes	704	25.19	even	10	inner
225.4.u.a.169.13	yes	704	225.169	even	30	inner