Embedded newform 224.3.w.a.99.4

• Label: 224.3.w.a.99.4
• 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			99.4
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92658 - 0.536908i) q^{2} +(-0.340996 + 0.823236i) q^{3} +(3.42346 + 2.06880i) q^{4} +(-1.83063 - 4.41953i) q^{5} +(1.09896 - 1.40295i) q^{6} +(1.87083 + 1.87083i) q^{7} +(-5.48483 - 5.82380i) q^{8} +(5.80252 + 5.80252i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92658	−	0.536908i	−0.963292	−	0.268454i
							\(3\)	−0.340996	+	0.823236i	−0.113665	+	0.274412i	−0.970467	−	0.241234i	\(-0.922448\pi\)
0.856802	+	0.515646i	\(0.172448\pi\)
\(5\)	−1.83063	−	4.41953i	−0.366126	−	0.883907i	−0.994377	−	0.105896i	\(-0.966229\pi\)
							0.628251	−	0.778011i	\(-0.283771\pi\)
\(7\)	1.87083	+	1.87083i	0.267261	+	0.267261i
							\(11\)	−5.69950	−	13.7598i	−0.518136	−	1.25089i	−0.939047	−	0.343790i	\(-0.888289\pi\)
0.420910	−	0.907102i	\(-0.361711\pi\)
\(13\)	−2.66083	+	6.42381i	−0.204679	+	0.494139i	−0.992570	−	0.121676i	\(-0.961173\pi\)
							0.787891	+	0.615815i	\(0.211173\pi\)
\(17\)		−	11.3003i		−	0.664725i	−0.943152	−	0.332362i	\(-0.892154\pi\)
							0.943152	−	0.332362i	\(-0.107846\pi\)
\(19\)	−4.35334	−	1.80321i	−0.229123	−	0.0949058i	0.265168	−	0.964202i	\(-0.414572\pi\)
							−0.494291	+	0.869296i	\(0.664572\pi\)
\(23\)	29.2047	−	29.2047i	1.26977	−	1.26977i	0.323563	−	0.946207i	\(-0.395119\pi\)
							0.946207	−	0.323563i	\(-0.104881\pi\)
\(29\)	−8.46573	−	3.50662i	−0.291922	−	0.120918i	0.231917	−	0.972736i	\(-0.425500\pi\)
							−0.523838	+	0.851818i	\(0.675500\pi\)
\(31\)		−	59.7873i		−	1.92862i	−0.264770	−	0.964312i	\(-0.585296\pi\)
							0.264770	−	0.964312i	\(-0.414704\pi\)
\(37\)	−23.5684	−	56.8992i	−0.636985	−	1.53782i	−0.830677	−	0.556755i	\(-0.812046\pi\)
							0.193692	−	0.981062i	\(-0.437954\pi\)
\(41\)	39.1521	+	39.1521i	0.954930	+	0.954930i	0.999027	−	0.0440976i	\(-0.0140413\pi\)
							−0.0440976	+	0.999027i	\(0.514041\pi\)
\(43\)	−14.7274	−	35.5552i	−0.342499	−	0.826865i	−0.997462	−	0.0712040i	\(-0.977316\pi\)
							0.654963	−	0.755661i	\(-0.272684\pi\)
\(47\)	−59.3761			−1.26332			−0.631661	−	0.775245i	\(-0.717626\pi\)
							−0.631661	+	0.775245i	\(0.717626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.99.4	yes	192
32.11	odd	8	inner	224.3.w.a.43.4	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.4	✓	192	32.11	odd	8	inner
224.3.w.a.99.4	yes	192	1.1	even	1	trivial