Embedded newform 224.2.bd.a.37.19

• Label: 224.2.bd.a.37.19
• Level: $224$
• Weight: $2$
• Character: 224.37
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.bd (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			37.19
Character	\(\chi\)	\(=\)	224.37
Dual form			224.2.bd.a.109.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.674919 - 1.24277i) q^{2} +(-1.68352 + 2.19400i) q^{3} +(-1.08897 - 1.67754i) q^{4} +(-1.03656 + 0.795384i) q^{5} +(1.59041 + 3.57300i) q^{6} +(-2.17678 - 1.50387i) q^{7} +(-2.81977 + 0.221138i) q^{8} +(-1.20296 - 4.48951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.674919	−	1.24277i	0.477240	−	0.878773i
							\(3\)	−1.68352	+	2.19400i	−0.971979	+	1.26671i	−0.00821140	+	0.999966i	\(0.502614\pi\)
−0.963768	+	0.266742i	\(0.914053\pi\)
\(5\)	−1.03656	+	0.795384i	−0.463566	+	0.355706i	−0.813984	−	0.580887i	\(-0.802706\pi\)
							0.350419	+	0.936593i	\(0.386039\pi\)
\(7\)	−2.17678	−	1.50387i	−0.822746	−	0.568409i
							\(11\)	−4.26070	+	0.560932i	−1.28465	+	0.169127i	−0.741754	−	0.670672i	\(-0.766006\pi\)
−0.542897	+	0.839799i	\(0.682673\pi\)
\(13\)	−2.27797	+	0.943568i	−0.631796	+	0.261699i	−0.675516	−	0.737345i	\(-0.736079\pi\)
							0.0437199	+	0.999044i	\(0.486079\pi\)
\(17\)	3.88131	−	2.24088i	0.941356	−	0.543492i	0.0509711	−	0.998700i	\(-0.483768\pi\)
							0.890385	+	0.455208i	\(0.150435\pi\)
\(19\)	−0.511323	+	3.88389i	−0.117306	+	0.891024i	0.827057	+	0.562118i	\(0.190013\pi\)
							−0.944363	+	0.328906i	\(0.893320\pi\)
\(23\)	−0.0381893	−	0.142524i	−0.00796302	−	0.0297184i	0.961830	−	0.273648i	\(-0.0882302\pi\)
							−0.969793	+	0.243929i	\(0.921564\pi\)
\(29\)	−1.58979	−	3.83810i	−0.295217	−	0.712717i	−0.999995	−	0.00328877i	\(-0.998953\pi\)
							0.704777	−	0.709428i	\(-0.251047\pi\)
\(31\)	0.922443	+	1.59772i	0.165676	+	0.286959i	0.936895	−	0.349611i	\(-0.113686\pi\)
							−0.771219	+	0.636569i	\(0.780353\pi\)
\(37\)	8.33189	−	6.39328i	1.36975	−	1.05105i	0.377608	−	0.925966i	\(-0.376747\pi\)
							0.992146	−	0.125084i	\(-0.0399199\pi\)
\(41\)	−3.83132	+	3.83132i	−0.598352	+	0.598352i	−0.939874	−	0.341522i	\(-0.889058\pi\)
							0.341522	+	0.939874i	\(0.389058\pi\)
\(43\)	1.05197	−	2.53968i	0.160424	−	0.387297i	−0.823145	−	0.567831i	\(-0.807783\pi\)
							0.983569	+	0.180534i	\(0.0577826\pi\)
\(47\)	−7.48969	−	4.32417i	−1.09248	−	0.630746i	−0.158247	−	0.987400i	\(-0.550584\pi\)
							−0.934237	+	0.356654i	\(0.883918\pi\)