Embedded newform 224.3.v.b.69.7

• Label: 224.3.v.b.69.7
• Level: $224$
• Weight: $3$
• Character: 224.69
• 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			69.7
Character	\(\chi\)	\(=\)	224.69
Dual form			224.3.v.b.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89176 - 0.649047i) q^{2} +(-5.08830 + 2.10764i) q^{3} +(3.15748 + 2.45567i) q^{4} +(-0.976625 + 2.35778i) q^{5} +(10.9938 - 0.684602i) q^{6} +(4.79964 - 5.09544i) q^{7} +(-4.37933 - 6.69489i) q^{8} +(15.0847 - 15.0847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89176	−	0.649047i	−0.945878	−	0.324523i
							\(3\)	−5.08830	+	2.10764i	−1.69610	+	0.702548i	−0.999883	−	0.0152817i	\(-0.995135\pi\)
−0.696218	+	0.717830i	\(0.745135\pi\)
\(5\)	−0.976625	+	2.35778i	−0.195325	+	0.471556i	−0.990950	−	0.134234i	\(-0.957143\pi\)
							0.795625	+	0.605790i	\(0.207143\pi\)
\(7\)	4.79964	−	5.09544i	0.685663	−	0.727919i
							\(11\)	2.51817	−	6.07940i	0.228925	−	0.552673i	−0.767122	−	0.641501i	\(-0.778312\pi\)
0.996047	+	0.0888278i	\(0.0283121\pi\)
\(13\)	9.18830	+	22.1825i	0.706792	+	1.70635i	0.707865	+	0.706348i	\(0.249658\pi\)
							−0.00107294	+	0.999999i	\(0.500342\pi\)
\(17\)	−12.9448			−0.761457			−0.380729	−	0.924687i	\(-0.624327\pi\)
							−0.380729	+	0.924687i	\(0.624327\pi\)
\(19\)	−5.85517	−	14.1356i	−0.308167	−	0.743981i	−0.999765	−	0.0216978i	\(-0.993093\pi\)
							0.691598	−	0.722283i	\(-0.256907\pi\)
\(23\)	3.80500	−	3.80500i	0.165435	−	0.165435i	−0.619535	−	0.784969i	\(-0.712679\pi\)
							0.784969	+	0.619535i	\(0.212679\pi\)
\(29\)	10.4805	+	25.3022i	0.361398	+	0.872491i	0.995096	+	0.0989110i	\(0.0315359\pi\)
							−0.633699	+	0.773580i	\(0.718464\pi\)
\(31\)			26.1430i			0.843323i	0.906753	+	0.421661i	\(0.138553\pi\)
							−0.906753	+	0.421661i	\(0.861447\pi\)
\(37\)	−22.3665	−	9.26451i	−0.604500	−	0.250392i	0.0593748	−	0.998236i	\(-0.481089\pi\)
							−0.663875	+	0.747844i	\(0.731089\pi\)
\(41\)	1.69723	+	1.69723i	0.0413957	+	0.0413957i	0.727502	−	0.686106i	\(-0.240681\pi\)
							−0.686106	+	0.727502i	\(0.740681\pi\)
\(43\)	−23.2062	+	56.0247i	−0.539679	+	1.30290i	0.385269	+	0.922804i	\(0.374109\pi\)
							−0.924947	+	0.380095i	\(0.875891\pi\)
\(47\)	−63.8209			−1.35789			−0.678946	−	0.734188i	\(-0.737563\pi\)
							−0.678946	+	0.734188i	\(0.737563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.69.7	yes	240
7.6	odd	2	inner	224.3.v.b.69.8	yes	240
32.13	even	8	inner	224.3.v.b.13.8	yes	240
224.13	odd	8	inner	224.3.v.b.13.7	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.7	✓	240	224.13	odd	8	inner
224.3.v.b.13.8	yes	240	32.13	even	8	inner
224.3.v.b.69.7	yes	240	1.1	even	1	trivial
224.3.v.b.69.8	yes	240	7.6	odd	2	inner