Embedded newform 224.3.v.b.13.6

• Label: 224.3.v.b.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91469 - 0.577879i) q^{2} +(2.09916 + 0.869500i) q^{3} +(3.33211 + 2.21292i) q^{4} +(0.289108 + 0.697967i) q^{5} +(-3.51678 - 2.87788i) q^{6} +(-5.35156 - 4.51230i) q^{7} +(-5.10118 - 6.16263i) q^{8} +(-2.71353 - 2.71353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91469	−	0.577879i	−0.957347	−	0.288939i
							\(3\)	2.09916	+	0.869500i	0.699719	+	0.289833i	0.704043	−	0.710157i	\(-0.251376\pi\)
−0.00432364	+	0.999991i	\(0.501376\pi\)
\(5\)	0.289108	+	0.697967i	0.0578215	+	0.139593i	0.950150	−	0.311793i	\(-0.100930\pi\)
							−0.892329	+	0.451387i	\(0.850930\pi\)
\(7\)	−5.35156	−	4.51230i	−0.764508	−	0.644614i
							\(11\)	−4.20281	−	10.1465i	−0.382073	−	0.922407i	−0.991565	−	0.129614i	\(-0.958626\pi\)
0.609491	−	0.792793i	\(-0.291374\pi\)
\(13\)	5.80299	−	14.0096i	0.446384	−	1.07767i	−0.527283	−	0.849690i	\(-0.676789\pi\)
							0.973667	−	0.227976i	\(-0.0732106\pi\)
\(17\)	6.67995			0.392938			0.196469	−	0.980510i	\(-0.437052\pi\)
							0.196469	+	0.980510i	\(0.437052\pi\)
\(19\)	4.52161	−	10.9161i	0.237980	−	0.574534i	−0.759094	−	0.650981i	\(-0.774358\pi\)
							0.997074	+	0.0764475i	\(0.0243577\pi\)
\(23\)	23.1208	+	23.1208i	1.00525	+	1.00525i	0.999986	+	0.00526649i	\(0.00167638\pi\)
							0.00526649	+	0.999986i	\(0.498324\pi\)
\(29\)	−12.5246	+	30.2370i	−0.431882	+	1.04265i	0.546799	+	0.837264i	\(0.315846\pi\)
							−0.978680	+	0.205390i	\(0.934154\pi\)
\(31\)		−	43.5456i		−	1.40470i	−0.711834	−	0.702348i	\(-0.752135\pi\)
							0.711834	−	0.702348i	\(-0.247865\pi\)
\(37\)	−37.6230	+	15.5839i	−1.01684	+	0.421187i	−0.827944	−	0.560810i	\(-0.810490\pi\)
							−0.188892	+	0.981998i	\(0.560490\pi\)
\(41\)	33.6938	−	33.6938i	0.821800	−	0.821800i	−0.164566	−	0.986366i	\(-0.552622\pi\)
							0.986366	+	0.164566i	\(0.0526223\pi\)
\(43\)	2.11801	+	5.11333i	0.0492560	+	0.118915i	0.946592	−	0.322433i	\(-0.104501\pi\)
							−0.897336	+	0.441347i	\(0.854501\pi\)
\(47\)	−80.2362			−1.70715			−0.853577	−	0.520967i	\(-0.825571\pi\)
							−0.853577	+	0.520967i	\(0.825571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.6	yes	240
7.6	odd	2	inner	224.3.v.b.13.5	✓	240
32.5	even	8	inner	224.3.v.b.69.5	yes	240
224.69	odd	8	inner	224.3.v.b.69.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.5	✓	240	7.6	odd	2	inner
224.3.v.b.13.6	yes	240	1.1	even	1	trivial
224.3.v.b.69.5	yes	240	32.5	even	8	inner
224.3.v.b.69.6	yes	240	224.69	odd	8	inner