Embedded newform 225.4.u.a.4.3

• Label: 225.4.u.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11854 - 5.26231i) q^{2} +(4.92317 + 1.66204i) q^{3} +(-19.1324 + 8.51831i) q^{4} +(11.0988 - 1.34760i) q^{5} +(3.23940 - 27.7663i) q^{6} +(-14.9230 + 8.61579i) q^{7} +(40.9287 + 56.3335i) q^{8} +(21.4753 + 16.3650i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11854	−	5.26231i	−0.395463	−	1.86051i	−0.500056	−	0.865993i	\(-0.666687\pi\)
							0.104592	−	0.994515i	\(-0.466646\pi\)
\(3\)	4.92317	+	1.66204i	0.947465	+	0.319859i
							\(5\)	11.0988	−	1.34760i	0.992709	−	0.120533i
\(7\)	−14.9230	+	8.61579i	−0.805765	+	0.465209i								−0.845483	−	0.534002i	\(-0.820687\pi\)
							0.0397178	+	0.999211i	\(0.487354\pi\)
\(11\)	47.9842	−	10.1994i	1.31525	−	0.279566i	0.503718	−	0.863868i	\(-0.331965\pi\)
							0.811536	+	0.584302i	\(0.198632\pi\)
\(13\)	−15.2715	+	71.8469i	−0.325813	+	1.53283i	0.444862	+	0.895599i	\(0.353253\pi\)
							−0.770675	+	0.637229i	\(0.780081\pi\)
\(17\)	28.1772	+	38.7826i	0.401999	+	0.553304i	0.961244	−	0.275698i	\(-0.0889090\pi\)
							−0.559245	+	0.829002i	\(0.688909\pi\)
\(19\)	−23.2828	+	16.9160i	−0.281129	+	0.204252i	−0.719410	−	0.694586i	\(-0.755588\pi\)
							0.438281	+	0.898838i	\(0.355588\pi\)
\(23\)	81.0452	−	72.9734i	0.734743	−	0.661566i	−0.214280	−	0.976772i	\(-0.568740\pi\)
							0.949023	+	0.315207i	\(0.102074\pi\)
\(29\)	13.1699	−	125.304i	0.0843309	−	0.802354i	−0.867851	−	0.496824i	\(-0.834500\pi\)
							0.952182	−	0.305531i	\(-0.0988338\pi\)
\(31\)	6.82261	+	64.9128i	0.0395283	+	0.376086i	0.996347	+	0.0854003i	\(0.0272169\pi\)
							−0.956818	+	0.290686i	\(0.906116\pi\)
\(37\)	335.852	−	109.125i	1.49226	−	0.484865i	0.554512	−	0.832176i	\(-0.312905\pi\)
							0.937750	+	0.347311i	\(0.112905\pi\)
\(41\)	−359.091	−	76.3272i	−1.36782	−	0.290739i	−0.535266	−	0.844683i	\(-0.679789\pi\)
							−0.832554	+	0.553944i	\(0.813122\pi\)
\(43\)	−280.307	+	161.835i	−0.994102	+	0.573945i	−0.906498	−	0.422211i	\(-0.861254\pi\)
							−0.0876038	+	0.996155i	\(0.527921\pi\)
\(47\)	14.1567	+	1.48793i	0.0439355	+	0.00461781i	0.126471	−	0.991970i	\(-0.459635\pi\)
							−0.0825352	+	0.996588i	\(0.526302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.3	✓	704
9.7	even	3	inner	225.4.u.a.79.3	yes	704
25.19	even	10	inner	225.4.u.a.94.3	yes	704
225.169	even	30	inner	225.4.u.a.169.3	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.3	✓	704	1.1	even	1	trivial
225.4.u.a.79.3	yes	704	9.7	even	3	inner
225.4.u.a.94.3	yes	704	25.19	even	10	inner
225.4.u.a.169.3	yes	704	225.169	even	30	inner