Embedded newform 224.3.w.a.99.2

• Label: 224.3.w.a.99.2
• 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			99.2
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99845 + 0.0786486i) q^{2} +(1.73419 - 4.18671i) q^{3} +(3.98763 - 0.314351i) q^{4} +(-2.33059 - 5.62654i) q^{5} +(-3.13642 + 8.50332i) q^{6} +(-1.87083 - 1.87083i) q^{7} +(-7.94437 + 0.941838i) q^{8} +(-8.15712 - 8.15712i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99845	+	0.0786486i	−0.999226	+	0.0393243i
							\(3\)	1.73419	−	4.18671i	0.578063	−	1.39557i	−0.316485	−	0.948597i	\(-0.602503\pi\)
0.894549	−	0.446971i	\(-0.147497\pi\)
\(5\)	−2.33059	−	5.62654i	−0.466118	−	1.12531i	−0.965844	−	0.259125i	\(-0.916566\pi\)
							0.499726	−	0.866184i	\(-0.333434\pi\)
\(7\)	−1.87083	−	1.87083i	−0.267261	−	0.267261i
							\(11\)	−3.50516	−	8.46221i	−0.318651	−	0.769292i	−0.999326	−	0.0367061i	\(-0.988313\pi\)
0.680675	−	0.732585i	\(-0.261687\pi\)
\(13\)	0.514028	−	1.24097i	0.0395406	−	0.0954595i	−0.902875	−	0.429902i	\(-0.858548\pi\)
							0.942416	+	0.334443i	\(0.108548\pi\)
\(17\)			23.8792i			1.40466i	0.711853	+	0.702329i	\(0.247856\pi\)
							−0.711853	+	0.702329i	\(0.752144\pi\)
\(19\)	17.2181	+	7.13196i	0.906215	+	0.375366i	0.786606	−	0.617455i	\(-0.211836\pi\)
							0.119608	+	0.992821i	\(0.461836\pi\)
\(23\)	2.59794	−	2.59794i	0.112954	−	0.112954i	−0.648371	−	0.761325i	\(-0.724549\pi\)
							0.761325	+	0.648371i	\(0.224549\pi\)
\(29\)	−20.4589	−	8.47437i	−0.705481	−	0.292220i	0.000952287	−	1.00000i	\(-0.499697\pi\)
							−0.706433	+	0.707780i	\(0.749697\pi\)
\(31\)			4.26728i			0.137654i	0.997629	+	0.0688271i	\(0.0219257\pi\)
							−0.997629	+	0.0688271i	\(0.978074\pi\)
\(37\)	−10.6032	−	25.5984i	−0.286573	−	0.691848i	0.713388	−	0.700770i	\(-0.247160\pi\)
							−0.999960	+	0.00892210i	\(0.997160\pi\)
\(41\)	−52.9347	−	52.9347i	−1.29109	−	1.29109i	−0.934113	−	0.356978i	\(-0.883807\pi\)
							−0.356978	−	0.934113i	\(-0.616193\pi\)
\(43\)	18.1278	+	43.7645i	0.421578	+	1.01778i	0.981882	+	0.189492i	\(0.0606841\pi\)
							−0.560305	+	0.828287i	\(0.689316\pi\)
\(47\)	−51.5941			−1.09775			−0.548874	−	0.835905i	\(-0.684943\pi\)
							−0.548874	+	0.835905i	\(0.684943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.99.2	yes	192
32.11	odd	8	inner	224.3.w.a.43.2	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.2	✓	192	32.11	odd	8	inner
224.3.w.a.99.2	yes	192	1.1	even	1	trivial