Embedded newform 225.4.u.a.169.16

• Label: 225.4.u.a.169.16
• 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			169.16
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.814957 + 3.83407i) q^{2} +(-3.33680 - 3.98318i) q^{3} +(-6.72759 - 2.99532i) q^{4} +(-9.34277 + 6.14107i) q^{5} +(17.9912 - 9.54742i) q^{6} +(-7.03825 - 4.06354i) q^{7} +(-1.46471 + 2.01600i) q^{8} +(-4.73148 + 26.5822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.814957	+	3.83407i	−0.288131	+	1.35555i	0.561196	+	0.827683i	\(0.310341\pi\)
							−0.849327	+	0.527867i	\(0.822992\pi\)
\(3\)	−3.33680	−	3.98318i	−0.642168	−	0.766564i
							\(5\)	−9.34277	+	6.14107i	−0.835642	+	0.549274i
\(7\)	−7.03825	−	4.06354i	−0.380030	−	0.219410i								0.297801	−	0.954628i	\(-0.403747\pi\)
							−0.677831	+	0.735218i	\(0.737080\pi\)
\(11\)	−24.8076	−	5.27302i	−0.679979	−	0.144534i	−0.145042	−	0.989426i	\(-0.546332\pi\)
							−0.534938	+	0.844892i	\(0.679665\pi\)
\(13\)	1.31609	+	6.19173i	0.0280784	+	0.132098i	0.989954	−	0.141388i	\(-0.0451565\pi\)
							−0.961876	+	0.273486i	\(0.911823\pi\)
\(17\)	51.9374	−	71.4857i	0.740981	−	1.01987i	−0.257581	−	0.966257i	\(-0.582925\pi\)
							0.998562	−	0.0536161i	\(-0.0170747\pi\)
\(19\)	−20.4471	−	14.8557i	−0.246889	−	0.179375i	0.457458	−	0.889231i	\(-0.348760\pi\)
							−0.704347	+	0.709856i	\(0.748760\pi\)
\(23\)	148.887	+	134.059i	1.34979	+	1.21535i	0.955008	+	0.296579i	\(0.0958457\pi\)
							0.394780	+	0.918776i	\(0.370821\pi\)
\(29\)	10.0631	+	95.7438i	0.0644368	+	0.613075i	0.978320	+	0.207098i	\(0.0664019\pi\)
							−0.913883	+	0.405977i	\(0.866931\pi\)
\(31\)	28.9150	−	275.108i	0.167525	−	1.59390i	−0.511173	−	0.859478i	\(-0.670789\pi\)
							0.678698	−	0.734418i	\(-0.262545\pi\)
\(37\)	161.279	+	52.4026i	0.716596	+	0.232836i	0.644547	−	0.764565i	\(-0.277046\pi\)
							0.0720494	+	0.997401i	\(0.477046\pi\)
\(41\)	27.8301	−	5.91547i	0.106008	−	0.0225327i	−0.154602	−	0.987977i	\(-0.549410\pi\)
							0.260610	+	0.965444i	\(0.416076\pi\)
\(43\)	33.3218	+	19.2383i	0.118175	+	0.0682284i	0.557922	−	0.829893i	\(-0.311599\pi\)
							−0.439747	+	0.898122i	\(0.644932\pi\)
\(47\)	−304.528	+	32.0072i	−0.945105	+	0.0993346i	−0.564528	−	0.825414i	\(-0.690942\pi\)
							−0.380578	+	0.924749i	\(0.624275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.169.16	yes	704
9.4	even	3	inner	225.4.u.a.94.16	yes	704
25.4	even	10	inner	225.4.u.a.79.16	yes	704
225.4	even	30	inner	225.4.u.a.4.16	✓	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.16	✓	704	225.4	even	30	inner
225.4.u.a.79.16	yes	704	25.4	even	10	inner
225.4.u.a.94.16	yes	704	9.4	even	3	inner
225.4.u.a.169.16	yes	704	1.1	even	1	trivial