Embedded newform 224.3.w.a.99.9

• Label: 224.3.w.a.99.9
• Level: $224$
• Weight: $3$
• Character: 224.99
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			99.9
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72313 - 1.01530i) q^{2} +(0.819473 - 1.97838i) q^{3} +(1.93834 + 3.49898i) q^{4} +(1.60646 + 3.87835i) q^{5} +(-3.42071 + 2.57700i) q^{6} +(-1.87083 - 1.87083i) q^{7} +(0.212503 - 7.99718i) q^{8} +(3.12150 + 3.12150i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72313	−	1.01530i	−0.861564	−	0.507649i
							\(3\)	0.819473	−	1.97838i	0.273158	−	0.659461i	−0.726457	−	0.687212i	\(-0.758834\pi\)
0.999615	+	0.0277508i	\(0.00883447\pi\)
\(5\)	1.60646	+	3.87835i	0.321293	+	0.775670i	0.999179	+	0.0405030i	\(0.0128960\pi\)
							−0.677887	+	0.735167i	\(0.737104\pi\)
\(7\)	−1.87083	−	1.87083i	−0.267261	−	0.267261i
							\(11\)	2.19493	+	5.29904i	0.199539	+	0.481731i	0.991699	−	0.128584i	\(-0.0410431\pi\)
−0.792159	+	0.610315i	\(0.791043\pi\)
\(13\)	−7.92625	+	19.1357i	−0.609712	+	1.47197i	0.253603	+	0.967308i	\(0.418384\pi\)
							−0.863315	+	0.504666i	\(0.831616\pi\)
\(17\)			8.66263i			0.509566i	0.966998	+	0.254783i	\(0.0820041\pi\)
							−0.966998	+	0.254783i	\(0.917996\pi\)
\(19\)	9.92137	+	4.10957i	0.522177	+	0.216293i	0.628173	−	0.778074i	\(-0.283803\pi\)
							−0.105996	+	0.994367i	\(0.533803\pi\)
\(23\)	−1.21606	+	1.21606i	−0.0528721	+	0.0528721i	−0.733048	−	0.680176i	\(-0.761903\pi\)
							0.680176	+	0.733048i	\(0.261903\pi\)
\(29\)	26.8363	+	11.1159i	0.925388	+	0.383308i	0.793927	−	0.608013i	\(-0.208033\pi\)
							0.131461	+	0.991321i	\(0.458033\pi\)
\(31\)		−	9.35016i		−	0.301618i	−0.988563	−	0.150809i	\(-0.951812\pi\)
							0.988563	−	0.150809i	\(-0.0481878\pi\)
\(37\)	−10.6588	−	25.7325i	−0.288075	−	0.695473i	0.711902	−	0.702279i	\(-0.247834\pi\)
							−0.999977	+	0.00680503i	\(0.997834\pi\)
\(41\)	35.1006	+	35.1006i	0.856111	+	0.856111i	0.990877	−	0.134766i	\(-0.0430283\pi\)
							−0.134766	+	0.990877i	\(0.543028\pi\)
\(43\)	2.82656	+	6.82393i	0.0657340	+	0.158696i	0.953333	−	0.301921i	\(-0.0976280\pi\)
							−0.887599	+	0.460617i	\(0.847628\pi\)
\(47\)	−11.2490			−0.239341			−0.119670	−	0.992814i	\(-0.538184\pi\)
							−0.119670	+	0.992814i	\(0.538184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.99.9	yes	192
32.11	odd	8	inner	224.3.w.a.43.9	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.9	✓	192	32.11	odd	8	inner
224.3.w.a.99.9	yes	192	1.1	even	1	trivial