Embedded newform 225.4.u.a.169.12

• Label: 225.4.u.a.169.12
• 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			169.12
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.922831 + 4.34158i) q^{2} +(4.70291 + 2.20967i) q^{3} +(-10.6893 - 4.75919i) q^{4} +(1.62060 - 11.0623i) q^{5} +(-13.9335 + 18.3789i) q^{6} +(8.42574 + 4.86461i) q^{7} +(9.65541 - 13.2895i) q^{8} +(17.2347 + 20.7838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.922831	+	4.34158i	−0.326270	+	1.53498i	0.443274	+	0.896386i	\(0.353817\pi\)
							−0.769544	+	0.638594i	\(0.779517\pi\)
\(3\)	4.70291	+	2.20967i	0.905075	+	0.425252i
							\(5\)	1.62060	−	11.0623i	0.144951	−	0.989439i
\(7\)	8.42574	+	4.86461i	0.454947	+	0.262664i								0.709917	−	0.704285i	\(-0.248732\pi\)
							−0.254970	+	0.966949i	\(0.582066\pi\)
\(11\)	36.6712	+	7.79470i	1.00516	+	0.213654i	0.680949	−	0.732331i	\(-0.261567\pi\)
							0.324212	+	0.945984i	\(0.394901\pi\)
\(13\)	4.83473	+	22.7456i	0.103147	+	0.485269i	0.999150	+	0.0412211i	\(0.0131248\pi\)
							−0.896003	+	0.444048i	\(0.853542\pi\)
\(17\)	−74.2628	+	102.214i	−1.05949	+	1.45827i	−0.179215	+	0.983810i	\(0.557356\pi\)
							−0.880278	+	0.474457i	\(0.842644\pi\)
\(19\)	6.89709	+	5.01103i	0.0832790	+	0.0605058i	0.628646	−	0.777692i	\(-0.283610\pi\)
							−0.545367	+	0.838198i	\(0.683610\pi\)
\(23\)	129.216	+	116.346i	1.17145	+	1.05478i	0.997546	+	0.0700132i	\(0.0223041\pi\)
							0.173902	+	0.984763i	\(0.444363\pi\)
\(29\)	−20.1627	−	191.835i	−0.129107	−	1.22838i	−0.846760	−	0.531974i	\(-0.821450\pi\)
							0.717653	−	0.696401i	\(-0.245216\pi\)
\(31\)	−23.2683	+	221.383i	−0.134810	+	1.28263i	0.692719	+	0.721207i	\(0.256412\pi\)
							−0.827529	+	0.561423i	\(0.810254\pi\)
\(37\)	−165.146	−	53.6593i	−0.733781	−	0.238420i	−0.0817932	−	0.996649i	\(-0.526065\pi\)
							−0.651988	+	0.758229i	\(0.726065\pi\)
\(41\)	315.398	−	67.0400i	1.20139	−	0.255363i	0.436607	−	0.899652i	\(-0.356180\pi\)
							0.764782	+	0.644289i	\(0.222847\pi\)
\(43\)	260.770	+	150.555i	0.924814	+	0.533942i	0.885168	−	0.465272i	\(-0.154044\pi\)
							0.0396463	+	0.999214i	\(0.487377\pi\)
\(47\)	169.359	−	17.8003i	0.525606	−	0.0552434i	0.161990	−	0.986792i	\(-0.448209\pi\)
							0.363616	+	0.931549i	\(0.381542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.169.12	yes	704
9.4	even	3	inner	225.4.u.a.94.12	yes	704
25.4	even	10	inner	225.4.u.a.79.12	yes	704
225.4	even	30	inner	225.4.u.a.4.12	✓	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.12	✓	704	225.4	even	30	inner
225.4.u.a.79.12	yes	704	25.4	even	10	inner
225.4.u.a.94.12	yes	704	9.4	even	3	inner
225.4.u.a.169.12	yes	704	1.1	even	1	trivial