Embedded newform 225.2.q.a.121.7

• Label: 225.2.q.a.121.7
• Level: $225$
• Weight: $2$
• Character: 225.121
• 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.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			121.7
Character	\(\chi\)	\(=\)	225.121
Dual form			225.2.q.a.106.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39878 + 0.297321i) q^{2} +(-1.19801 + 1.25091i) q^{3} +(0.0411076 - 0.0183023i) q^{4} +(1.76390 + 1.37429i) q^{5} +(1.30384 - 2.10594i) q^{6} +(1.92019 + 3.32586i) q^{7} +(2.26178 - 1.64328i) q^{8} +(-0.129533 - 2.99720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 3 q^{2} - 8 q^{3} + 23 q^{4} - 8 q^{5} - 10 q^{6} - 8 q^{7} - 20 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39878	+	0.297321i	−0.989090	+	0.210238i	−0.673925	−	0.738799i	\(-0.735393\pi\)
							−0.315165	+	0.949037i	\(0.602060\pi\)
\(3\)	−1.19801	+	1.25091i	−0.691673	+	0.722211i
							\(5\)	1.76390	+	1.37429i	0.788838	+	0.614601i
\(7\)	1.92019	+	3.32586i	0.725762	+	1.25706i								0.958660	+	0.284556i	\(0.0918460\pi\)
							−0.232897	+	0.972501i	\(0.574821\pi\)
\(11\)	−5.11325	+	1.08686i	−1.54170	+	0.327699i	−0.898839	−	0.438280i	\(-0.855588\pi\)
							−0.642865	+	0.765979i	\(0.722255\pi\)
\(13\)	3.28926	+	0.699153i	0.912276	+	0.193910i	0.640059	−	0.768326i	\(-0.278910\pi\)
							0.272217	+	0.962236i	\(0.412243\pi\)
\(17\)	−3.90998	+	2.84077i	−0.948309	+	0.688987i	−0.950406	−	0.311011i	\(-0.899332\pi\)
							0.00209712	+	0.999998i	\(0.499332\pi\)
\(19\)	−1.90903	+	1.38699i	−0.437961	+	0.318197i	−0.784824	−	0.619719i	\(-0.787247\pi\)
							0.346863	+	0.937916i	\(0.387247\pi\)
\(23\)	−3.12041	−	3.46556i	−0.650650	−	0.722620i	0.324074	−	0.946032i	\(-0.394947\pi\)
							−0.974724	+	0.223412i	\(0.928281\pi\)
\(29\)	0.768054	−	7.30754i	0.142624	−	1.35698i	−0.655824	−	0.754913i	\(-0.727679\pi\)
							0.798448	−	0.602063i	\(-0.205654\pi\)
\(31\)	−0.0679615	−	0.646611i	−0.0122062	−	0.116135i	0.986722	−	0.162417i	\(-0.0519291\pi\)
							−0.998928	+	0.0462828i	\(0.985262\pi\)
\(37\)	0.315262	+	0.970276i	0.0518287	+	0.159512i	0.973621	−	0.228172i	\(-0.0732749\pi\)
							−0.921792	+	0.387685i	\(0.873275\pi\)
\(41\)	3.66775	+	0.779604i	0.572806	+	0.121754i	0.485203	−	0.874402i	\(-0.338746\pi\)
							0.0876036	+	0.996155i	\(0.472079\pi\)
\(43\)	−2.34851	−	4.06773i	−0.358144	−	0.620324i	0.629507	−	0.776995i	\(-0.283257\pi\)
							−0.987651	+	0.156671i	\(0.949924\pi\)
\(47\)	−1.24854	+	11.8791i	−0.182118	+	1.73274i	0.397281	+	0.917697i	\(0.369954\pi\)
							−0.579400	+	0.815044i	\(0.696713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.q.a.121.7	yes	224
3.2	odd	2		675.2.r.a.496.22		224
9.2	odd	6		675.2.r.a.46.7		224
9.7	even	3	inner	225.2.q.a.196.22	yes	224
25.6	even	5	inner	225.2.q.a.31.22	✓	224
75.56	odd	10		675.2.r.a.631.7		224
225.56	odd	30		675.2.r.a.181.22		224
225.106	even	15	inner	225.2.q.a.106.7	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.q.a.31.22	✓	224	25.6	even	5	inner
225.2.q.a.106.7	yes	224	225.106	even	15	inner
225.2.q.a.121.7	yes	224	1.1	even	1	trivial
225.2.q.a.196.22	yes	224	9.7	even	3	inner
675.2.r.a.46.7		224	9.2	odd	6
675.2.r.a.181.22		224	225.56	odd	30
675.2.r.a.496.22		224	3.2	odd	2
675.2.r.a.631.7		224	75.56	odd	10