Embedded newform 224.2.x.b.27.3

• Label: 224.2.x.b.27.3
• Level: $224$
• Weight: $2$
• Character: 224.27
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.x (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			27.3
Character	\(\chi\)	\(=\)	224.27
Dual form			224.2.x.b.83.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36868 + 0.355955i) q^{2} +(-1.01638 - 2.45376i) q^{3} +(1.74659 - 0.974380i) q^{4} +(1.11188 - 2.68430i) q^{5} +(2.26454 + 2.99664i) q^{6} +(0.0435134 - 2.64539i) q^{7} +(-2.04370 + 1.95533i) q^{8} +(-2.86660 + 2.86660i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36868	+	0.355955i	−0.967806	+	0.251698i
							\(3\)	−1.01638	−	2.45376i	−0.586809	−	1.41668i	−0.886538	−	0.462657i	\(-0.846896\pi\)
0.299729	−	0.954024i	\(-0.403104\pi\)
\(5\)	1.11188	−	2.68430i	0.497246	−	1.20046i	−0.453715	−	0.891147i	\(-0.649902\pi\)
							0.950961	−	0.309311i	\(-0.100098\pi\)
\(7\)	0.0435134	−	2.64539i	0.0164465	−	0.999865i
							\(11\)	1.44195	+	0.597277i	0.434765	+	0.180086i	0.589322	−	0.807898i	\(-0.299395\pi\)
−0.154557	+	0.987984i	\(0.549395\pi\)
\(13\)	1.41757	+	3.42231i	0.393163	+	0.949179i	0.989247	+	0.146257i	\(0.0467225\pi\)
							−0.596084	+	0.802922i	\(0.703277\pi\)
\(17\)	−2.54645			−0.617604			−0.308802	−	0.951126i	\(-0.599928\pi\)
							−0.308802	+	0.951126i	\(0.599928\pi\)
\(19\)	1.94460	−	0.805480i	0.446122	−	0.184790i	−0.148301	−	0.988942i	\(-0.547380\pi\)
							0.594423	+	0.804152i	\(0.297380\pi\)
\(23\)	−6.51022	−	6.51022i	−1.35748	−	1.35748i	−0.877018	−	0.480458i	\(-0.840471\pi\)
							−0.480458	−	0.877018i	\(-0.659529\pi\)
\(29\)	2.87982	+	6.95251i	0.534770	+	1.29105i	0.928333	+	0.371751i	\(0.121242\pi\)
							−0.393563	+	0.919298i	\(0.628758\pi\)
\(31\)	3.98396			0.715540			0.357770	−	0.933810i	\(-0.383537\pi\)
							0.357770	+	0.933810i	\(0.383537\pi\)
\(37\)	6.26182	+	2.59373i	1.02944	+	0.426407i	0.832509	−	0.554011i	\(-0.186903\pi\)
							0.196928	+	0.980418i	\(0.436903\pi\)
\(41\)	−1.24468	−	1.24468i	−0.194386	−	0.194386i	0.603202	−	0.797588i	\(-0.293891\pi\)
							−0.797588	+	0.603202i	\(0.793891\pi\)
\(43\)	0.455327	+	0.188603i	0.0694367	+	0.0287616i	0.417131	−	0.908846i	\(-0.363035\pi\)
							−0.347695	+	0.937608i	\(0.613035\pi\)
\(47\)			11.5542i			1.68536i	0.538418	+	0.842678i	\(0.319022\pi\)
							−0.538418	+	0.842678i	\(0.680978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.x.b.27.3	✓	112
4.3	odd	2		896.2.x.b.783.25		112
7.6	odd	2	inner	224.2.x.b.27.4	yes	112
28.27	even	2		896.2.x.b.783.4		112
32.13	even	8		896.2.x.b.111.4		112
32.19	odd	8	inner	224.2.x.b.83.4	yes	112
224.13	odd	8		896.2.x.b.111.25		112
224.83	even	8	inner	224.2.x.b.83.3	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.x.b.27.3	✓	112	1.1	even	1	trivial
224.2.x.b.27.4	yes	112	7.6	odd	2	inner
224.2.x.b.83.3	yes	112	224.83	even	8	inner
224.2.x.b.83.4	yes	112	32.19	odd	8	inner
896.2.x.b.111.4		112	32.13	even	8
896.2.x.b.111.25		112	224.13	odd	8
896.2.x.b.783.4		112	28.27	even	2
896.2.x.b.783.25		112	4.3	odd	2