Embedded newform 224.3.w.a.43.11

• Label: 224.3.w.a.43.11
• 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			43.11
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.11

$q$-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 \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55094	+	1.26277i	−0.775469	+	0.631386i
							\(3\)	−1.89928	−	4.58527i	−0.633094	−	1.52842i	−0.835714	−	0.549166i	\(-0.814946\pi\)
0.202620	−	0.979257i	\(-0.435054\pi\)
\(5\)	−2.09096	+	5.04803i	−0.418193	+	1.00961i	0.564678	+	0.825311i	\(0.309000\pi\)
							−0.982871	+	0.184295i	\(0.941000\pi\)
\(7\)	−1.87083	+	1.87083i	−0.267261	+	0.267261i
							\(11\)	0.560618	−	1.35345i	0.0509652	−	0.123041i	−0.896346	−	0.443355i	\(-0.853788\pi\)
0.947312	+	0.320314i	\(0.103788\pi\)
\(13\)	0.285358	+	0.688915i	0.0219506	+	0.0529934i	0.934475	−	0.356029i	\(-0.115870\pi\)
							−0.912524	+	0.409023i	\(0.865870\pi\)
\(17\)			2.08433i			0.122608i	0.998119	+	0.0613039i	\(0.0195259\pi\)
							−0.998119	+	0.0613039i	\(0.980474\pi\)
\(19\)	26.8257	−	11.1116i	1.41188	−	0.584820i	0.459073	−	0.888398i	\(-0.348182\pi\)
							0.952806	+	0.303579i	\(0.0981816\pi\)
\(23\)	26.5738	+	26.5738i	1.15538	+	1.15538i	0.985456	+	0.169928i	\(0.0543536\pi\)
							0.169928	+	0.985456i	\(0.445646\pi\)
\(29\)	22.2065	−	9.19824i	0.765742	−	0.317181i	0.0345958	−	0.999401i	\(-0.488986\pi\)
							0.731146	+	0.682221i	\(0.238986\pi\)
\(31\)		−	22.4567i		−	0.724410i	−0.932098	−	0.362205i	\(-0.882024\pi\)
							0.932098	−	0.362205i	\(-0.117976\pi\)
\(37\)	1.44778	−	3.49525i	0.0391292	−	0.0944663i	−0.903107	−	0.429416i	\(-0.858719\pi\)
							0.942236	+	0.334950i	\(0.108719\pi\)
\(41\)	−45.8249	+	45.8249i	−1.11768	+	1.11768i	−0.125599	+	0.992081i	\(0.540085\pi\)
							−0.992081	+	0.125599i	\(0.959915\pi\)
\(43\)	−13.3506	+	32.2312i	−0.310479	+	0.749563i	0.689208	+	0.724563i	\(0.257958\pi\)
							−0.999687	+	0.0249996i	\(0.992042\pi\)
\(47\)	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.43.11	✓	192
32.3	odd	8	inner	224.3.w.a.99.11	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.11	✓	192	1.1	even	1	trivial
224.3.w.a.99.11	yes	192	32.3	odd	8	inner