Embedded newform 224.3.v.b.13.1

• Label: 224.3.v.b.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96919 + 0.349710i) q^{2} +(-0.313666 - 0.129925i) q^{3} +(3.75541 - 1.37729i) q^{4} +(-3.56967 - 8.61795i) q^{5} +(0.663104 + 0.146154i) q^{6} +(-6.10386 + 3.42679i) q^{7} +(-6.91345 + 4.02544i) q^{8} +(-6.28246 - 6.28246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96919	+	0.349710i	−0.984594	+	0.174855i
							\(3\)	−0.313666	−	0.129925i	−0.104555	−	0.0433082i	0.329793	−	0.944053i	\(-0.393021\pi\)
−0.434348	+	0.900745i	\(0.643021\pi\)
\(5\)	−3.56967	−	8.61795i	−0.713934	−	1.72359i	−0.689928	−	0.723878i	\(-0.742358\pi\)
							−0.0240066	−	0.999712i	\(-0.507642\pi\)
\(7\)	−6.10386	+	3.42679i	−0.871980	+	0.489541i
							\(11\)	4.82010	+	11.6368i	0.438191	+	1.05789i	0.976573	+	0.215186i	\(0.0690357\pi\)
−0.538382	+	0.842701i	\(0.680964\pi\)
\(13\)	−2.75909	+	6.66103i	−0.212238	+	0.512387i	−0.993766	−	0.111482i	\(-0.964440\pi\)
							0.781529	+	0.623869i	\(0.214440\pi\)
\(17\)	28.4801			1.67530			0.837649	−	0.546209i	\(-0.183930\pi\)
							0.837649	+	0.546209i	\(0.183930\pi\)
\(19\)	−9.23350	+	22.2916i	−0.485974	+	1.17324i	0.470756	+	0.882264i	\(0.343981\pi\)
							−0.956729	+	0.290980i	\(0.906019\pi\)
\(23\)	7.97013	+	7.97013i	0.346527	+	0.346527i	0.858814	−	0.512287i	\(-0.171202\pi\)
							−0.512287	+	0.858814i	\(0.671202\pi\)
\(29\)	−1.69500	+	4.09210i	−0.0584484	+	0.141107i	−0.950406	−	0.311013i	\(-0.899332\pi\)
							0.891957	+	0.452119i	\(0.149332\pi\)
\(31\)		−	20.5650i		−	0.663388i	−0.943387	−	0.331694i	\(-0.892380\pi\)
							0.943387	−	0.331694i	\(-0.107620\pi\)
\(37\)	−48.6158	+	20.1373i	−1.31394	+	0.544252i	−0.926032	−	0.377446i	\(-0.876803\pi\)
							−0.387909	+	0.921698i	\(0.626803\pi\)
\(41\)	−3.17782	+	3.17782i	−0.0775079	+	0.0775079i	−0.744798	−	0.667290i	\(-0.767454\pi\)
							0.667290	+	0.744798i	\(0.267454\pi\)
\(43\)	9.26217	+	22.3609i	0.215399	+	0.520020i	0.994237	−	0.107206i	\(-0.0341904\pi\)
							−0.778838	+	0.627226i	\(0.784190\pi\)
\(47\)	−62.0152			−1.31947			−0.659736	−	0.751497i	\(-0.729332\pi\)
							−0.659736	+	0.751497i	\(0.729332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.1	✓	240
7.6	odd	2	inner	224.3.v.b.13.2	yes	240
32.5	even	8	inner	224.3.v.b.69.2	yes	240
224.69	odd	8	inner	224.3.v.b.69.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.1	✓	240	1.1	even	1	trivial
224.3.v.b.13.2	yes	240	7.6	odd	2	inner
224.3.v.b.69.1	yes	240	224.69	odd	8	inner
224.3.v.b.69.2	yes	240	32.5	even	8	inner