Embedded newform 225.4.u.a.169.19

• Label: 225.4.u.a.169.19
• 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			169.19
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.19

$q$-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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.19	✓	704	225.4	even	30	inner
225.4.u.a.79.19	yes	704	25.4	even	10	inner
225.4.u.a.94.19	yes	704	9.4	even	3	inner
225.4.u.a.169.19	yes	704	1.1	even	1	trivial