Embedded newform 224.3.w.a.43.6

• Label: 224.3.w.a.43.6
• Level: $224$
• Weight: $3$
• Character: 224.43
• 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			43.6
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86483 - 0.722783i) q^{2} +(-1.45943 - 3.52338i) q^{3} +(2.95517 + 2.69573i) q^{4} +(-3.21097 + 7.75196i) q^{5} +(0.174951 + 7.62534i) q^{6} +(1.87083 - 1.87083i) q^{7} +(-3.56245 - 7.16302i) q^{8} +(-3.92028 + 3.92028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86483	−	0.722783i	−0.932414	−	0.361391i
							\(3\)	−1.45943	−	3.52338i	−0.486477	−	1.17446i	−0.956481	−	0.291794i	\(-0.905748\pi\)
0.470004	−	0.882664i	\(-0.344252\pi\)
\(5\)	−3.21097	+	7.75196i	−0.642194	+	1.55039i	0.181520	+	0.983387i	\(0.441898\pi\)
							−0.823714	+	0.567006i	\(0.808102\pi\)
\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
							\(11\)	3.78882	−	9.14701i	0.344438	−	0.831546i	−0.652818	−	0.757515i	\(-0.726413\pi\)
0.997256	−	0.0740316i	\(-0.0235866\pi\)
\(13\)	6.24954	+	15.0877i	0.480734	+	1.16059i	0.959261	+	0.282521i	\(0.0911707\pi\)
							−0.478527	+	0.878073i	\(0.658829\pi\)
\(17\)		−	19.0578i		−	1.12105i	−0.828138	−	0.560524i	\(-0.810600\pi\)
							0.828138	−	0.560524i	\(-0.189400\pi\)
\(19\)	24.1823	−	10.0166i	1.27275	−	0.527191i	0.358953	−	0.933356i	\(-0.383134\pi\)
							0.913800	+	0.406164i	\(0.133134\pi\)
\(23\)	−21.8802	−	21.8802i	−0.951315	−	0.951315i	0.0475535	−	0.998869i	\(-0.484858\pi\)
							−0.998869	+	0.0475535i	\(0.984858\pi\)
\(29\)	20.7285	−	8.58602i	0.714775	−	0.296069i	0.00449611	−	0.999990i	\(-0.498569\pi\)
							0.710279	+	0.703920i	\(0.248569\pi\)
\(31\)			13.8010i			0.445192i	0.974911	+	0.222596i	\(0.0714531\pi\)
							−0.974911	+	0.222596i	\(0.928547\pi\)
\(37\)	−1.32733	+	3.20446i	−0.0358738	+	0.0866069i	−0.940801	−	0.338959i	\(-0.889925\pi\)
							0.904927	+	0.425566i	\(0.139925\pi\)
\(41\)	31.4208	−	31.4208i	0.766360	−	0.766360i	−0.211104	−	0.977464i	\(-0.567706\pi\)
							0.977464	+	0.211104i	\(0.0677058\pi\)
\(43\)	22.9809	−	55.4808i	0.534440	−	1.29025i	−0.394117	−	0.919060i	\(-0.628950\pi\)
							0.928557	−	0.371191i	\(-0.121050\pi\)
\(47\)	−16.2865			−0.346522			−0.173261	−	0.984876i	\(-0.555430\pi\)
							−0.173261	+	0.984876i	\(0.555430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.6	✓	192
32.3	odd	8	inner	224.3.w.a.99.6	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.6	✓	192	1.1	even	1	trivial
224.3.w.a.99.6	yes	192	32.3	odd	8	inner