Embedded newform 224.3.v.b.13.18

• Label: 224.3.v.b.13.18
• Level: $224$
• Weight: $3$
• Character: 224.13
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.18
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33217 - 1.49175i) q^{2} +(2.62896 + 1.08895i) q^{3} +(-0.450649 + 3.97453i) q^{4} +(-2.38974 - 5.76934i) q^{5} +(-1.87778 - 5.37243i) q^{6} +(-5.98528 + 3.62994i) q^{7} +(6.52936 - 4.62250i) q^{8} +(-0.638332 - 0.638332i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33217	−	1.49175i	−0.666085	−	0.745876i
							\(3\)	2.62896	+	1.08895i	0.876321	+	0.362984i	0.775069	−	0.631877i	\(-0.217715\pi\)
0.101252	+	0.994861i	\(0.467715\pi\)
\(5\)	−2.38974	−	5.76934i	−0.477948	−	1.15387i	−0.960570	−	0.278039i	\(-0.910315\pi\)
							0.482622	−	0.875829i	\(-0.339685\pi\)
\(7\)	−5.98528	+	3.62994i	−0.855040	+	0.518562i
							\(11\)	−4.95364	−	11.9592i	−0.450331	−	1.08720i	−0.972196	−	0.234167i	\(-0.924764\pi\)
0.521865	−	0.853028i	\(-0.325236\pi\)
\(13\)	−6.06710	+	14.6473i	−0.466700	+	1.12671i	0.498895	+	0.866663i	\(0.333740\pi\)
							−0.965595	+	0.260051i	\(0.916260\pi\)
\(17\)	−8.22837			−0.484022			−0.242011	−	0.970274i	\(-0.577807\pi\)
							−0.242011	+	0.970274i	\(0.577807\pi\)
\(19\)	8.30680	−	20.0544i	0.437200	−	1.05549i	−0.539711	−	0.841850i	\(-0.681467\pi\)
							0.976912	−	0.213644i	\(-0.0685334\pi\)
\(23\)	−26.4453	−	26.4453i	−1.14979	−	1.14979i	−0.986593	−	0.163202i	\(-0.947818\pi\)
							−0.163202	−	0.986593i	\(-0.552182\pi\)
\(29\)	7.68667	−	18.5573i	0.265057	−	0.639905i	−0.734180	−	0.678955i	\(-0.762433\pi\)
							0.999237	+	0.0390498i	\(0.0124331\pi\)
\(31\)			59.4680i			1.91832i	0.282859	+	0.959162i	\(0.408717\pi\)
							−0.282859	+	0.959162i	\(0.591283\pi\)
\(37\)	−20.1216	+	8.33464i	−0.543827	+	0.225261i	−0.637647	−	0.770329i	\(-0.720092\pi\)
							0.0938199	+	0.995589i	\(0.470092\pi\)
\(41\)	−24.3694	+	24.3694i	−0.594376	+	0.594376i	−0.938810	−	0.344435i	\(-0.888071\pi\)
							0.344435	+	0.938810i	\(0.388071\pi\)
\(43\)	−14.7173	−	35.5306i	−0.342262	−	0.826294i	−0.997486	−	0.0708592i	\(-0.977426\pi\)
							0.655224	−	0.755434i	\(-0.272574\pi\)
\(47\)	5.31825			0.113154			0.0565771	−	0.998398i	\(-0.481981\pi\)
							0.0565771	+	0.998398i	\(0.481981\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.18	yes	240
7.6	odd	2	inner	224.3.v.b.13.17	✓	240
32.5	even	8	inner	224.3.v.b.69.17	yes	240
224.69	odd	8	inner	224.3.v.b.69.18	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.17	✓	240	7.6	odd	2	inner
224.3.v.b.13.18	yes	240	1.1	even	1	trivial
224.3.v.b.69.17	yes	240	32.5	even	8	inner
224.3.v.b.69.18	yes	240	224.69	odd	8	inner