Embedded newform 224.3.v.b.13.20

• Label: 224.3.v.b.13.20
• 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			13.20
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28270 - 1.53450i) q^{2} +(0.416824 + 0.172654i) q^{3} +(-0.709381 + 3.93659i) q^{4} +(-1.21861 - 2.94198i) q^{5} +(-0.269721 - 0.861080i) q^{6} +(6.07185 + 3.48320i) q^{7} +(6.95063 - 3.96091i) q^{8} +(-6.22003 - 6.22003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28270	−	1.53450i	−0.641348	−	0.767250i
							\(3\)	0.416824	+	0.172654i	0.138941	+	0.0575514i	0.451071	−	0.892488i	\(-0.351042\pi\)
−0.312129	+	0.950040i	\(0.601042\pi\)
\(5\)	−1.21861	−	2.94198i	−0.243722	−	0.588397i	0.753925	−	0.656961i	\(-0.228158\pi\)
							−0.997647	+	0.0685640i	\(0.978158\pi\)
\(7\)	6.07185	+	3.48320i	0.867407	+	0.497599i
							\(11\)	6.49632	+	15.6835i	0.590574	+	1.42577i	0.882949	+	0.469468i	\(0.155554\pi\)
−0.292375	+	0.956304i	\(0.594446\pi\)
\(13\)	3.18050	−	7.67840i	0.244654	−	0.590646i	−0.753080	−	0.657929i	\(-0.771433\pi\)
							0.997734	+	0.0672823i	\(0.0214328\pi\)
\(17\)	19.7748			1.16322			0.581612	−	0.813466i	\(-0.302422\pi\)
							0.581612	+	0.813466i	\(0.302422\pi\)
\(19\)	12.0416	−	29.0709i	0.633766	−	1.53005i	−0.201086	−	0.979574i	\(-0.564447\pi\)
							0.834853	−	0.550474i	\(-0.185553\pi\)
\(23\)	7.09691	+	7.09691i	0.308561	+	0.308561i	0.844351	−	0.535790i	\(-0.179986\pi\)
							−0.535790	+	0.844351i	\(0.679986\pi\)
\(29\)	−6.56941	+	15.8600i	−0.226531	+	0.546895i	−0.995751	−	0.0920896i	\(-0.970645\pi\)
							0.769219	+	0.638985i	\(0.220645\pi\)
\(31\)		−	57.4959i		−	1.85471i	−0.374188	−	0.927353i	\(-0.622079\pi\)
							0.374188	−	0.927353i	\(-0.377921\pi\)
\(37\)	1.99927	−	0.828124i	0.0540342	−	0.0223817i	−0.355503	−	0.934675i	\(-0.615690\pi\)
							0.409537	+	0.912294i	\(0.365690\pi\)
\(41\)	−10.4173	+	10.4173i	−0.254080	+	0.254080i	−0.822641	−	0.568561i	\(-0.807500\pi\)
							0.568561	+	0.822641i	\(0.307500\pi\)
\(43\)	−6.95080	−	16.7807i	−0.161647	−	0.390249i	0.822216	−	0.569176i	\(-0.192738\pi\)
							−0.983862	+	0.178926i	\(0.942738\pi\)
\(47\)	58.0268			1.23461			0.617307	−	0.786723i	\(-0.288224\pi\)
							0.617307	+	0.786723i	\(0.288224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.20	yes	240
7.6	odd	2	inner	224.3.v.b.13.19	✓	240
32.5	even	8	inner	224.3.v.b.69.19	yes	240
224.69	odd	8	inner	224.3.v.b.69.20	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.19	✓	240	7.6	odd	2	inner
224.3.v.b.13.20	yes	240	1.1	even	1	trivial
224.3.v.b.69.19	yes	240	32.5	even	8	inner
224.3.v.b.69.20	yes	240	224.69	odd	8	inner