Embedded newform 224.2.bd.a.37.11

• Label: 224.2.bd.a.37.11
• 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			37.11
Character	\(\chi\)	\(=\)	224.37
Dual form			224.2.bd.a.109.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.480321 - 1.33015i) q^{2} +(-0.758838 + 0.988936i) q^{3} +(-1.53858 + 1.27780i) q^{4} +(0.171111 - 0.131298i) q^{5} +(1.67992 + 0.534359i) q^{6} +(1.01488 - 2.44336i) q^{7} +(2.43867 + 1.43279i) q^{8} +(0.374296 + 1.39689i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.480321	−	1.33015i	−0.339638	−	0.940556i
							\(3\)	−0.758838	+	0.988936i	−0.438115	+	0.570963i	−0.959267	−	0.282501i	\(-0.908836\pi\)
0.521152	+	0.853464i	\(0.325503\pi\)
\(5\)	0.171111	−	0.131298i	0.0765231	−	0.0587182i	−0.569792	−	0.821789i	\(-0.692976\pi\)
							0.646315	+	0.763071i	\(0.276309\pi\)
\(7\)	1.01488	−	2.44336i	0.383590	−	0.923503i
							\(11\)	2.10346	−	0.276926i	0.634218	−	0.0834963i	0.193439	−	0.981112i	\(-0.438036\pi\)
0.440778	+	0.897616i	\(0.354702\pi\)
\(13\)	3.53590	−	1.46462i	0.980681	−	0.406211i	0.166003	−	0.986125i	\(-0.446914\pi\)
							0.814678	+	0.579914i	\(0.196914\pi\)
\(17\)	6.24315	−	3.60448i	1.51419	−	0.874216i	0.514324	−	0.857596i	\(-0.328043\pi\)
							0.999862	−	0.0166201i	\(-0.00529058\pi\)
\(19\)	−0.462645	+	3.51414i	−0.106138	+	0.806198i	0.852731	+	0.522350i	\(0.174945\pi\)
							−0.958869	+	0.283848i	\(0.908389\pi\)
\(23\)	−1.34605	−	5.02353i	−0.280671	−	1.04748i	−0.951945	−	0.306269i	\(-0.900919\pi\)
							0.671274	−	0.741210i	\(-0.265747\pi\)
\(29\)	2.56271	+	6.18694i	0.475884	+	1.14889i	0.961523	+	0.274726i	\(0.0885871\pi\)
							−0.485639	+	0.874160i	\(0.661413\pi\)
\(31\)	−2.46695	−	4.27288i	−0.443077	−	0.767432i	0.554839	−	0.831958i	\(-0.312780\pi\)
							−0.997916	+	0.0645260i	\(0.979446\pi\)
\(37\)	−2.06576	+	1.58511i	−0.339609	+	0.260591i	−0.764459	−	0.644673i	\(-0.776994\pi\)
							0.424850	+	0.905264i	\(0.360327\pi\)
\(41\)	4.31161	−	4.31161i	0.673360	−	0.673360i	−0.285129	−	0.958489i	\(-0.592036\pi\)
							0.958489	+	0.285129i	\(0.0920365\pi\)
\(43\)	−4.15699	+	10.0359i	−0.633935	+	1.53046i	0.200700	+	0.979653i	\(0.435678\pi\)
							−0.834635	+	0.550803i	\(0.814322\pi\)
\(47\)	4.71097	+	2.71988i	0.687165	+	0.396735i	0.802549	−	0.596586i	\(-0.203476\pi\)
							−0.115384	+	0.993321i	\(0.536810\pi\)