Embedded newform 225.2.q.a.196.16

• Label: 225.2.q.a.196.16
• Level: $225$
• Weight: $2$
• Character: 225.196
• 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			196.16
Character	\(\chi\)	\(=\)	225.196
Dual form			225.2.q.a.31.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.349998 + 0.388712i) q^{2} +(-0.835354 + 1.51730i) q^{3} +(0.180458 - 1.71695i) q^{4} +(-0.130882 + 2.23223i) q^{5} +(-0.882163 + 0.206338i) q^{6} +(-1.68864 + 2.92481i) q^{7} +(1.57689 - 1.14568i) q^{8} +(-1.60437 - 2.53496i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 3 q^{2} - 8 q^{3} + 23 q^{4} - 8 q^{5} - 10 q^{6} - 8 q^{7} - 20 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.349998	+	0.388712i	0.247486	+	0.274861i	0.854070	−	0.520159i	\(-0.174127\pi\)
							−0.606584	+	0.795020i	\(0.707461\pi\)
\(3\)	−0.835354	+	1.51730i	−0.482292	+	0.876011i
							\(5\)	−0.130882	+	2.23223i	−0.0585323	+	0.998286i
\(7\)	−1.68864	+	2.92481i	−0.638246	+	1.10547i								0.347571	+	0.937654i	\(0.387006\pi\)
							−0.985817	+	0.167821i	\(0.946327\pi\)
\(11\)	2.76572	+	3.07165i	0.833897	+	0.926137i	0.998182	−	0.0602761i	\(-0.0191981\pi\)
							−0.164284	+	0.986413i	\(0.552531\pi\)
\(13\)	−3.39970	+	3.77575i	−0.942906	+	1.04720i	0.0559039	+	0.998436i	\(0.482196\pi\)
							−0.998810	+	0.0487673i	\(0.984471\pi\)
\(17\)	0.0810977	−	0.0589209i	0.0196691	−	0.0142904i	−0.577907	−	0.816102i	\(-0.696130\pi\)
							0.597576	+	0.801812i	\(0.296130\pi\)
\(19\)	2.55513	−	1.85641i	0.586188	−	0.425890i	−0.254762	−	0.967004i	\(-0.581997\pi\)
							0.840950	+	0.541113i	\(0.181997\pi\)
\(23\)	1.92791	−	0.409791i	0.401998	−	0.0854473i	−0.00247582	−	0.999997i	\(-0.500788\pi\)
							0.404474	+	0.914550i	\(0.367455\pi\)
\(29\)	6.19799	−	2.75952i	1.15094	−	0.512431i	0.259576	−	0.965723i	\(-0.416417\pi\)
							0.891362	+	0.453292i	\(0.149750\pi\)
\(31\)	2.22457	+	0.990444i	0.399545	+	0.177889i	0.596665	−	0.802490i	\(-0.296492\pi\)
							−0.197120	+	0.980379i	\(0.563159\pi\)
\(37\)	0.930291	+	2.86314i	0.152939	+	0.470698i	0.997946	−	0.0640571i	\(-0.0204040\pi\)
							−0.845007	+	0.534755i	\(0.820404\pi\)
\(41\)	−3.88818	+	4.31827i	−0.607232	+	0.674400i	−0.965855	−	0.259083i	\(-0.916580\pi\)
							0.358623	+	0.933483i	\(0.383246\pi\)
\(43\)	5.61908	−	9.73253i	0.856902	−	1.48420i	−0.0179676	−	0.999839i	\(-0.505720\pi\)
							0.874869	−	0.484359i	\(-0.160947\pi\)
\(47\)	8.50718	−	3.78764i	1.24090	−	0.552484i	0.321912	−	0.946770i	\(-0.395675\pi\)
							0.918988	+	0.394286i	\(0.129008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.q.a.196.16	yes	224
3.2	odd	2		675.2.r.a.46.13		224
9.4	even	3	inner	225.2.q.a.121.13	yes	224
9.5	odd	6		675.2.r.a.496.16		224
25.6	even	5	inner	225.2.q.a.106.13	yes	224
75.56	odd	10		675.2.r.a.181.16		224
225.31	even	15	inner	225.2.q.a.31.16	✓	224
225.131	odd	30		675.2.r.a.631.13		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.q.a.31.16	✓	224	225.31	even	15	inner
225.2.q.a.106.13	yes	224	25.6	even	5	inner
225.2.q.a.121.13	yes	224	9.4	even	3	inner
225.2.q.a.196.16	yes	224	1.1	even	1	trivial
675.2.r.a.46.13		224	3.2	odd	2
675.2.r.a.181.16		224	75.56	odd	10
675.2.r.a.496.16		224	9.5	odd	6
675.2.r.a.631.13		224	225.131	odd	30