Embedded newform 225.2.u.a.169.28

• Label: 225.2.u.a.169.28
• Level: $225$
• Weight: $2$
• Character: 225.169
• Analytic conductor: $1.797$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			169.28
Character	\(\chi\)	\(=\)	225.169
Dual form			225.2.u.a.4.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.578019 - 2.71937i) q^{2} +(1.68946 - 0.381726i) q^{3} +(-5.23375 - 2.33022i) q^{4} +(2.07883 + 0.823700i) q^{5} +(-0.0615114 - 4.81491i) q^{6} +(-1.90860 - 1.10193i) q^{7} +(-6.09369 + 8.38725i) q^{8} +(2.70857 - 1.28982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 5 q^{2} - 10 q^{3} - 29 q^{4} - 2 q^{6} - 20 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.578019	−	2.71937i	0.408721	−	1.92288i	0.0243493	−	0.999704i	\(-0.492249\pi\)
							0.384372	−	0.923178i	\(-0.374418\pi\)
\(3\)	1.68946	−	0.381726i	0.975412	−	0.220390i
							\(5\)	2.07883	+	0.823700i	0.929679	+	0.368370i
\(7\)	−1.90860	−	1.10193i	−0.721384	−	0.416491i								0.0938779	−	0.995584i	\(-0.470074\pi\)
							−0.815262	+	0.579093i	\(0.803407\pi\)
\(11\)	−1.05211	−	0.223634i	−0.317224	−	0.0674281i	0.0465479	−	0.998916i	\(-0.485178\pi\)
							−0.363772	+	0.931488i	\(0.618511\pi\)
\(13\)	0.368258	+	1.73252i	0.102136	+	0.480514i	0.999250	+	0.0387148i	\(0.0123264\pi\)
							−0.897114	+	0.441799i	\(0.854340\pi\)
\(17\)	1.03062	−	1.41852i	0.249961	−	0.344042i	−0.665536	−	0.746365i	\(-0.731797\pi\)
							0.915498	+	0.402323i	\(0.131797\pi\)
\(19\)	1.98216	+	1.44012i	0.454738	+	0.330387i	0.791464	−	0.611216i	\(-0.209319\pi\)
							−0.336725	+	0.941603i	\(0.609319\pi\)
\(23\)	−1.76519	−	1.58938i	−0.368067	−	0.331409i	0.464255	−	0.885702i	\(-0.346322\pi\)
							−0.832322	+	0.554293i	\(0.812989\pi\)
\(29\)	0.934356	+	8.88980i	0.173506	+	1.65080i	0.641543	+	0.767087i	\(0.278294\pi\)
							−0.468038	+	0.883708i	\(0.655039\pi\)
\(31\)	0.129168	−	1.22896i	0.0231993	−	0.220727i	−0.976780	−	0.214246i	\(-0.931270\pi\)
							0.999979	−	0.00648063i	\(-0.00206286\pi\)
\(37\)	6.36228	+	2.06723i	1.04595	+	0.339851i	0.781079	−	0.624432i	\(-0.214670\pi\)
							0.264874	+	0.964283i	\(0.414670\pi\)
\(41\)	−4.99693	+	1.06213i	−0.780389	+	0.165877i	−0.580847	−	0.814013i	\(-0.697279\pi\)
							−0.199541	+	0.979889i	\(0.563945\pi\)
\(43\)	−2.57023	−	1.48392i	−0.391957	−	0.226296i	0.291051	−	0.956708i	\(-0.405995\pi\)
							−0.683008	+	0.730411i	\(0.739328\pi\)
\(47\)	−10.6067	+	1.11481i	−1.54714	+	0.162611i	−0.839264	−	0.543723i	\(-0.817014\pi\)
							−0.707878	+	0.706335i	\(0.750347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.u.a.169.28	yes	224
3.2	odd	2		675.2.y.a.19.1		224
9.4	even	3	inner	225.2.u.a.94.28	yes	224
9.5	odd	6		675.2.y.a.469.1		224
25.4	even	10	inner	225.2.u.a.79.28	yes	224
75.29	odd	10		675.2.y.a.154.1		224
225.4	even	30	inner	225.2.u.a.4.28	✓	224
225.104	odd	30		675.2.y.a.604.1		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.u.a.4.28	✓	224	225.4	even	30	inner
225.2.u.a.79.28	yes	224	25.4	even	10	inner
225.2.u.a.94.28	yes	224	9.4	even	3	inner
225.2.u.a.169.28	yes	224	1.1	even	1	trivial
675.2.y.a.19.1		224	3.2	odd	2
675.2.y.a.154.1		224	75.29	odd	10
675.2.y.a.469.1		224	9.5	odd	6
675.2.y.a.604.1		224	225.104	odd	30