Embedded newform 225.4.u.a.4.19

• Label: 225.4.u.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.770399 - 3.62444i) q^{2} +(-4.10935 + 3.18013i) q^{3} +(-5.23470 + 2.33064i) q^{4} +(9.67594 - 5.60144i) q^{5} +(14.6920 + 12.4441i) q^{6} +(-20.0051 + 11.5499i) q^{7} +(-4.94384 - 6.80461i) q^{8} +(6.77352 - 26.1366i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.770399	−	3.62444i	−0.272377	−	1.28143i	−0.875279	−	0.483618i	\(-0.839322\pi\)
							0.602902	−	0.797815i	\(-0.294011\pi\)
\(3\)	−4.10935	+	3.18013i	−0.790845	+	0.612017i
							\(5\)	9.67594	−	5.60144i	0.865443	−	0.501008i
\(7\)	−20.0051	+	11.5499i	−1.08017	+	0.623638i								−0.930943	−	0.365165i	\(-0.881013\pi\)
							−0.149229	+	0.988803i	\(0.547679\pi\)
\(11\)	38.0240	−	8.08225i	1.04224	−	0.221535i	0.345173	−	0.938539i	\(-0.387820\pi\)
							0.697069	+	0.717004i	\(0.254487\pi\)
\(13\)	2.41823	−	11.3769i	0.0515920	−	0.242721i	−0.944794	−	0.327666i	\(-0.893738\pi\)
							0.996386	+	0.0849445i	\(0.0270713\pi\)
\(17\)	−51.7479	−	71.2249i	−0.738277	−	1.01615i	−0.998716	−	0.0506606i	\(-0.983867\pi\)
							0.260439	−	0.965490i	\(-0.416133\pi\)
\(19\)	−117.016	+	85.0171i	−1.41291	+	1.02654i	−0.420019	+	0.907515i	\(0.637977\pi\)
							−0.992891	+	0.119025i	\(0.962023\pi\)
\(23\)	−111.491	+	100.387i	−1.01076	+	0.910095i	−0.995960	−	0.0897950i	\(-0.971379\pi\)
							−0.0148031	+	0.999890i	\(0.504712\pi\)
\(29\)	−12.7498	+	121.306i	−0.0816405	+	0.776757i	0.874732	+	0.484607i	\(0.161037\pi\)
							−0.956372	+	0.292150i	\(0.905629\pi\)
\(31\)	7.81281	+	74.3339i	0.0452652	+	0.430670i	0.993562	+	0.113286i	\(0.0361377\pi\)
							−0.948297	+	0.317384i	\(0.897196\pi\)
\(37\)	128.096	−	41.6209i	0.569158	−	0.184931i	−0.0102799	−	0.999947i	\(-0.503272\pi\)
							0.579438	+	0.815017i	\(0.303272\pi\)
\(41\)	151.115	+	32.1204i	0.575613	+	0.122350i	0.486515	−	0.873672i	\(-0.338268\pi\)
							0.0890985	+	0.996023i	\(0.471601\pi\)
\(43\)	−227.195	+	131.171i	−0.805742	+	0.465195i	−0.845475	−	0.534015i	\(-0.820682\pi\)
							0.0397333	+	0.999210i	\(0.487349\pi\)
\(47\)	−181.489	−	19.0752i	−0.563252	−	0.0592002i	−0.181376	−	0.983414i	\(-0.558055\pi\)
							−0.381876	+	0.924214i	\(0.624722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.19	✓	704
9.7	even	3	inner	225.4.u.a.79.19	yes	704
25.19	even	10	inner	225.4.u.a.94.19	yes	704
225.169	even	30	inner	225.4.u.a.169.19	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.19	✓	704	1.1	even	1	trivial
225.4.u.a.79.19	yes	704	9.7	even	3	inner
225.4.u.a.94.19	yes	704	25.19	even	10	inner
225.4.u.a.169.19	yes	704	225.169	even	30	inner