Embedded newform 224.2.x.b.27.19

• Label: 224.2.x.b.27.19
• 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			27.19
Character	\(\chi\)	\(=\)	224.27
Dual form			224.2.x.b.83.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.589446 + 1.28552i) q^{2} +(-1.08865 - 2.62824i) q^{3} +(-1.30511 + 1.51548i) q^{4} +(-1.22393 + 2.95482i) q^{5} +(2.73695 - 2.94869i) q^{6} +(-1.81057 + 1.92921i) q^{7} +(-2.71747 - 0.784442i) q^{8} +(-3.60116 + 3.60116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.589446	+	1.28552i	0.416801	+	0.908998i
							\(3\)	−1.08865	−	2.62824i	−0.628534	−	1.51742i	−0.841444	−	0.540344i	\(-0.818294\pi\)
0.212910	−	0.977072i	\(-0.431706\pi\)
\(5\)	−1.22393	+	2.95482i	−0.547357	+	1.32144i	0.372080	+	0.928201i	\(0.378645\pi\)
							−0.919437	+	0.393237i	\(0.871355\pi\)
\(7\)	−1.81057	+	1.92921i	−0.684330	+	0.729172i
							\(11\)	3.55240	+	1.47145i	1.07109	+	0.443659i	0.847374	−	0.530996i	\(-0.178182\pi\)
0.223714	+	0.974655i	\(0.428182\pi\)
\(13\)	1.45755	+	3.51884i	0.404252	+	0.975950i	0.986622	+	0.163026i	\(0.0521255\pi\)
							−0.582370	+	0.812924i	\(0.697875\pi\)
\(17\)	−6.06598			−1.47122			−0.735608	−	0.677408i	\(-0.763103\pi\)
							−0.735608	+	0.677408i	\(0.763103\pi\)
\(19\)	0.707276	−	0.292963i	0.162260	−	0.0672103i	−0.300075	−	0.953916i	\(-0.597012\pi\)
							0.462335	+	0.886705i	\(0.347012\pi\)
\(23\)	−2.56037	−	2.56037i	−0.533875	−	0.533875i	0.387848	−	0.921723i	\(-0.373218\pi\)
							−0.921723	+	0.387848i	\(0.873218\pi\)
\(29\)	−0.434147	−	1.04812i	−0.0806191	−	0.194632i	0.878430	−	0.477870i	\(-0.158591\pi\)
							−0.959049	+	0.283239i	\(0.908591\pi\)
\(31\)	1.65461			0.297176			0.148588	−	0.988899i	\(-0.452527\pi\)
							0.148588	+	0.988899i	\(0.452527\pi\)
\(37\)	9.03555	+	3.74265i	1.48544	+	0.615288i	0.970318	−	0.241832i	\(-0.0777483\pi\)
							0.515117	+	0.857120i	\(0.327748\pi\)
\(41\)	1.64682	+	1.64682i	0.257189	+	0.257189i	0.823910	−	0.566721i	\(-0.191788\pi\)
							−0.566721	+	0.823910i	\(0.691788\pi\)
\(43\)	−2.84392	−	1.17799i	−0.433694	−	0.179642i	0.155146	−	0.987892i	\(-0.450415\pi\)
							−0.588840	+	0.808250i	\(0.700415\pi\)
\(47\)			4.63757i			0.676458i	0.941064	+	0.338229i	\(0.109828\pi\)
							−0.941064	+	0.338229i	\(0.890172\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.x.b.27.19	✓	112
4.3	odd	2		896.2.x.b.783.26		112
7.6	odd	2	inner	224.2.x.b.27.20	yes	112
28.27	even	2		896.2.x.b.783.3		112
32.13	even	8		896.2.x.b.111.3		112
32.19	odd	8	inner	224.2.x.b.83.20	yes	112
224.13	odd	8		896.2.x.b.111.26		112
224.83	even	8	inner	224.2.x.b.83.19	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.x.b.27.19	✓	112	1.1	even	1	trivial
224.2.x.b.27.20	yes	112	7.6	odd	2	inner
224.2.x.b.83.19	yes	112	224.83	even	8	inner
224.2.x.b.83.20	yes	112	32.19	odd	8	inner
896.2.x.b.111.3		112	32.13	even	8
896.2.x.b.111.26		112	224.13	odd	8
896.2.x.b.783.3		112	28.27	even	2
896.2.x.b.783.26		112	4.3	odd	2