Embedded newform 225.4.q.a.16.8

• Label: 225.4.q.a.16.8
• Level: $225$
• Weight: $4$
• Character: 225.16
• 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.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(704\)
Relative dimension:	\(88\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.8
Character	\(\chi\)	\(=\)	225.16
Dual form			225.4.q.a.211.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.41308 + 1.96483i) q^{2} +(3.87578 - 3.46097i) q^{3} +(10.2617 - 11.3968i) q^{4} +(-6.88435 - 8.80941i) q^{5} +(-10.3039 + 22.8888i) q^{6} +(-13.7602 + 23.8334i) q^{7} +(-10.9508 + 33.7031i) q^{8} +(3.04337 - 26.8279i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 3 q^{2} - 2 q^{3} + 341 q^{4} - 20 q^{5} - 22 q^{6} - 8 q^{7} - 44 q^{8} - 38 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.41308	+	1.96483i	−1.56026	+	0.694673i	−0.991772	−	0.128018i	\(-0.959138\pi\)
							−0.568489	+	0.822691i	\(0.692472\pi\)
\(3\)	3.87578	−	3.46097i	0.745895	−	0.666064i
							\(5\)	−6.88435	−	8.80941i	−0.615755	−	0.787938i
\(7\)	−13.7602	+	23.8334i	−0.742981	+	1.28688i								0.208152	+	0.978097i	\(0.433255\pi\)
							−0.951132	+	0.308784i	\(0.900078\pi\)
\(11\)	53.2553	−	23.7108i	1.45973	−	0.649916i	0.485251	−	0.874375i	\(-0.338728\pi\)
							0.974483	+	0.224459i	\(0.0720616\pi\)
\(13\)	−22.4361	−	9.98920i	−0.478666	−	0.213116i	0.153198	−	0.988196i	\(-0.451043\pi\)
							−0.631863	+	0.775080i	\(0.717710\pi\)
\(17\)	19.4696	−	59.9213i	0.277769	−	0.854886i	−0.710704	−	0.703491i	\(-0.751624\pi\)
							0.988473	−	0.151395i	\(-0.0483765\pi\)
\(19\)	13.3289	−	41.0222i	0.160940	−	0.495323i	−0.837774	−	0.546017i	\(-0.816143\pi\)
							0.998714	+	0.0506942i	\(0.0161434\pi\)
\(23\)	−5.84139	+	55.5771i	−0.0529572	+	0.503854i	0.935606	+	0.353046i	\(0.114854\pi\)
							−0.988563	+	0.150808i	\(0.951813\pi\)
\(29\)	−98.8700	−	21.0155i	−0.633093	−	0.134568i	−0.119823	−	0.992795i	\(-0.538233\pi\)
							−0.513270	+	0.858227i	\(0.671566\pi\)
\(31\)	−265.824	+	56.5026i	−1.54011	+	0.327360i	−0.898258	−	0.439468i	\(-0.855167\pi\)
							−0.641852	+	0.766829i	\(0.721834\pi\)
\(37\)	−153.120	+	111.248i	−0.680345	+	0.494299i	−0.873472	−	0.486874i	\(-0.838137\pi\)
							0.193127	+	0.981174i	\(0.438137\pi\)
\(41\)	−200.629	−	89.3257i	−0.764219	−	0.340252i	−0.0126614	−	0.999920i	\(-0.504030\pi\)
							−0.751557	+	0.659668i	\(0.770697\pi\)
\(43\)	17.1294	−	29.6691i	0.0607492	−	0.105221i	−0.834051	−	0.551687i	\(-0.813984\pi\)
							0.894800	+	0.446466i	\(0.147318\pi\)
\(47\)	−554.229	−	117.805i	−1.72006	−	0.365609i	−0.760984	−	0.648771i	\(-0.775283\pi\)
							−0.959072	+	0.283162i	\(0.908617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.q.a.16.8	✓	704
9.4	even	3	inner	225.4.q.a.166.81	yes	704
25.11	even	5	inner	225.4.q.a.61.81	yes	704
225.211	even	15	inner	225.4.q.a.211.8	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.q.a.16.8	✓	704	1.1	even	1	trivial
225.4.q.a.61.81	yes	704	25.11	even	5	inner
225.4.q.a.166.81	yes	704	9.4	even	3	inner
225.4.q.a.211.8	yes	704	225.211	even	15	inner