Embedded newform 225.4.u.a.169.9

• Label: 225.4.u.a.169.9
• 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			169.9
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01603 + 4.78006i) q^{2} +(3.24528 - 4.05810i) q^{3} +(-14.5083 - 6.45951i) q^{4} +(-5.06856 + 9.96543i) q^{5} +(16.1007 + 19.6358i) q^{6} +(11.0356 + 6.37141i) q^{7} +(22.6385 - 31.1592i) q^{8} +(-5.93634 - 26.3393i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01603	+	4.78006i	−0.359222	+	1.69001i	0.313077	+	0.949728i	\(0.398640\pi\)
							−0.672299	+	0.740280i	\(0.734693\pi\)
\(3\)	3.24528	−	4.05810i	0.624554	−	0.780982i
							\(5\)	−5.06856	+	9.96543i	−0.453345	+	0.891335i
\(7\)	11.0356	+	6.37141i	0.595867	+	0.344024i								0.767414	−	0.641152i	\(-0.221543\pi\)
							−0.171547	+	0.985176i	\(0.554877\pi\)
\(11\)	−32.9215	−	6.99768i	−0.902382	−	0.191807i	−0.266719	−	0.963774i	\(-0.585940\pi\)
							−0.635662	+	0.771967i	\(0.719273\pi\)
\(13\)	−0.362239	−	1.70420i	−0.00772824	−	0.0363585i	0.974115	−	0.226055i	\(-0.0725829\pi\)
							−0.981843	+	0.189697i	\(0.939250\pi\)
\(17\)	−75.1732	+	103.467i	−1.07248	+	1.47614i	−0.204942	+	0.978774i	\(0.565701\pi\)
							−0.867539	+	0.497369i	\(0.834299\pi\)
\(19\)	−93.2835	−	67.7744i	−1.12635	−	0.818343i	−0.141193	−	0.989982i	\(-0.545094\pi\)
							−0.985160	+	0.171639i	\(0.945094\pi\)
\(23\)	42.4892	+	38.2575i	0.385201	+	0.346836i	0.838868	−	0.544335i	\(-0.183218\pi\)
							−0.453667	+	0.891171i	\(0.649885\pi\)
\(29\)	−19.2512	−	183.163i	−0.123271	−	1.17284i	−0.864869	−	0.501998i	\(-0.832598\pi\)
							0.741598	−	0.670844i	\(-0.234068\pi\)
\(31\)	6.00277	−	57.1125i	0.0347783	−	0.330894i	−0.963274	−	0.268519i	\(-0.913466\pi\)
							0.998053	−	0.0623750i	\(-0.0198675\pi\)
\(37\)	2.72181	+	0.884371i	0.0120936	+	0.00392945i	0.315058	−	0.949073i	\(-0.397976\pi\)
							−0.302964	+	0.953002i	\(0.597976\pi\)
\(41\)	5.01337	−	1.06563i	0.0190965	−	0.00405909i	−0.198354	−	0.980131i	\(-0.563559\pi\)
							0.217450	+	0.976071i	\(0.430226\pi\)
\(43\)	−177.427	−	102.437i	−0.629240	−	0.363292i	0.151218	−	0.988500i	\(-0.451681\pi\)
							−0.780458	+	0.625209i	\(0.785014\pi\)
\(47\)	212.010	−	22.2832i	0.657976	−	0.0691560i	0.230344	−	0.973109i	\(-0.426015\pi\)
							0.427631	+	0.903953i	\(0.359348\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.169.9	yes	704
9.4	even	3	inner	225.4.u.a.94.9	yes	704
25.4	even	10	inner	225.4.u.a.79.9	yes	704
225.4	even	30	inner	225.4.u.a.4.9	✓	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.9	✓	704	225.4	even	30	inner
225.4.u.a.79.9	yes	704	25.4	even	10	inner
225.4.u.a.94.9	yes	704	9.4	even	3	inner
225.4.u.a.169.9	yes	704	1.1	even	1	trivial