Embedded newform 224.2.u.b.85.4

• Label: 224.2.u.b.85.4
• Level: $224$
• Weight: $2$
• Character: 224.85
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			85.4
Character	\(\chi\)	\(=\)	224.85
Dual form			224.2.u.b.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.420598 + 1.35022i) q^{2} +(2.28471 - 0.946358i) q^{3} +(-1.64619 - 1.13580i) q^{4} +(0.446012 - 1.07677i) q^{5} +(0.316848 + 3.48290i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(2.22597 - 1.74501i) q^{8} +(2.20299 - 2.20299i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{2} + 4 q^{3} + 8 q^{5} + 12 q^{6} - 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.420598	+	1.35022i	−0.297408	+	0.954751i
							\(3\)	2.28471	−	0.946358i	1.31908	−	0.546380i	0.391558	−	0.920153i	\(-0.371936\pi\)
0.927520	+	0.373773i	\(0.121936\pi\)
\(5\)	0.446012	−	1.07677i	0.199462	−	0.481545i	−0.792223	−	0.610232i	\(-0.791076\pi\)
							0.991685	+	0.128687i	\(0.0410762\pi\)
\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
							\(11\)	2.89205	+	1.19792i	0.871985	+	0.361188i	0.773383	−	0.633939i	\(-0.218563\pi\)
0.0986018	+	0.995127i	\(0.468563\pi\)
\(13\)	1.10101	+	2.65808i	0.305366	+	0.737218i	0.999843	+	0.0177038i	\(0.00563558\pi\)
							−0.694478	+	0.719514i	\(0.744364\pi\)
\(17\)		−	3.12612i		−	0.758195i	−0.925357	−	0.379097i	\(-0.876235\pi\)
							0.925357	−	0.379097i	\(-0.123765\pi\)
\(19\)	−1.63944	−	3.95797i	−0.376114	−	0.908021i	−0.992686	−	0.120722i	\(-0.961479\pi\)
							0.616572	−	0.787299i	\(-0.288521\pi\)
\(23\)	−1.37407	+	1.37407i	−0.286514	+	0.286514i	−0.835700	−	0.549186i	\(-0.814938\pi\)
							0.549186	+	0.835700i	\(0.314938\pi\)
\(29\)	−8.15384	+	3.37743i	−1.51413	+	0.627173i	−0.976405	−	0.215948i	\(-0.930716\pi\)
							−0.537724	+	0.843121i	\(0.680716\pi\)
\(31\)	−2.07787			−0.373196			−0.186598	−	0.982436i	\(-0.559746\pi\)
							−0.186598	+	0.982436i	\(0.559746\pi\)
\(37\)	−2.47273	+	5.96970i	−0.406515	+	0.981413i	0.579533	+	0.814949i	\(0.303235\pi\)
							−0.986047	+	0.166464i	\(0.946765\pi\)
\(41\)	−5.68545	+	5.68545i	−0.887918	+	0.887918i	−0.994323	−	0.106405i	\(-0.966066\pi\)
							0.106405	+	0.994323i	\(0.466066\pi\)
\(43\)	5.63921	+	2.33584i	0.859972	+	0.356212i	0.768697	−	0.639614i	\(-0.220906\pi\)
							0.0912753	+	0.995826i	\(0.470906\pi\)
\(47\)		−	8.41845i		−	1.22796i	−0.789322	−	0.613979i	\(-0.789568\pi\)
							0.789322	−	0.613979i	\(-0.210432\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.u.b.85.4	yes	40
4.3	odd	2		896.2.u.b.561.2		40
32.3	odd	8		896.2.u.b.337.2		40
32.29	even	8	inner	224.2.u.b.29.4	✓	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.u.b.29.4	✓	40	32.29	even	8	inner
224.2.u.b.85.4	yes	40	1.1	even	1	trivial
896.2.u.b.337.2		40	32.3	odd	8
896.2.u.b.561.2		40	4.3	odd	2