Embedded newform 224.3.w.a.99.11

• Label: 224.3.w.a.99.11
• 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			99.11
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55094 - 1.26277i) q^{2} +(-1.89928 + 4.58527i) q^{3} +(0.810816 + 3.91696i) q^{4} +(-2.09096 - 5.04803i) q^{5} +(8.73581 - 4.71311i) q^{6} +(-1.87083 - 1.87083i) q^{7} +(3.68870 - 7.09884i) q^{8} +(-11.0535 - 11.0535i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55094	−	1.26277i	−0.775469	−	0.631386i
							\(3\)	−1.89928	+	4.58527i	−0.633094	+	1.52842i	0.202620	+	0.979257i	\(0.435054\pi\)
−0.835714	+	0.549166i	\(0.814946\pi\)
\(5\)	−2.09096	−	5.04803i	−0.418193	−	1.00961i	−0.982871	−	0.184295i	\(-0.941000\pi\)
							0.564678	−	0.825311i	\(-0.309000\pi\)
\(7\)	−1.87083	−	1.87083i	−0.267261	−	0.267261i
							\(11\)	0.560618	+	1.35345i	0.0509652	+	0.123041i	0.947312	−	0.320314i	\(-0.103788\pi\)
−0.896346	+	0.443355i	\(0.853788\pi\)
\(13\)	0.285358	−	0.688915i	0.0219506	−	0.0529934i	−0.912524	−	0.409023i	\(-0.865870\pi\)
							0.934475	+	0.356029i	\(0.115870\pi\)
\(17\)		−	2.08433i		−	0.122608i	−0.998119	−	0.0613039i	\(-0.980474\pi\)
							0.998119	−	0.0613039i	\(-0.0195259\pi\)
\(19\)	26.8257	+	11.1116i	1.41188	+	0.584820i	0.952806	−	0.303579i	\(-0.0981816\pi\)
							0.459073	+	0.888398i	\(0.348182\pi\)
\(23\)	26.5738	−	26.5738i	1.15538	−	1.15538i	0.169928	−	0.985456i	\(-0.445646\pi\)
							0.985456	−	0.169928i	\(-0.0543536\pi\)
\(29\)	22.2065	+	9.19824i	0.765742	+	0.317181i	0.731146	−	0.682221i	\(-0.238986\pi\)
							0.0345958	+	0.999401i	\(0.488986\pi\)
\(31\)			22.4567i			0.724410i	0.932098	+	0.362205i	\(0.117976\pi\)
							−0.932098	+	0.362205i	\(0.882024\pi\)
\(37\)	1.44778	+	3.49525i	0.0391292	+	0.0944663i	0.942236	−	0.334950i	\(-0.108719\pi\)
							−0.903107	+	0.429416i	\(0.858719\pi\)
\(41\)	−45.8249	−	45.8249i	−1.11768	−	1.11768i	−0.992081	−	0.125599i	\(-0.959915\pi\)
							−0.125599	−	0.992081i	\(-0.540085\pi\)
\(43\)	−13.3506	−	32.2312i	−0.310479	−	0.749563i	−0.999687	−	0.0249996i	\(-0.992042\pi\)
							0.689208	−	0.724563i	\(-0.257958\pi\)
\(47\)	75.2413			1.60088			0.800439	−	0.599414i	\(-0.204600\pi\)
							0.800439	+	0.599414i	\(0.204600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.99.11	yes	192
32.11	odd	8	inner	224.3.w.a.43.11	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.11	✓	192	32.11	odd	8	inner
224.3.w.a.99.11	yes	192	1.1	even	1	trivial