Embedded newform 672.2.cf.a.187.26

• Label: 672.2.cf.a.187.26
• Level: $672$
• Weight: $2$
• Character: 672.187
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.cf (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(64\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			187.26
Character	\(\chi\)	\(=\)	672.187
Dual form			672.2.cf.a.115.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.347308 - 1.37090i) q^{2} +(0.608761 - 0.793353i) q^{3} +(-1.75875 + 0.952252i) q^{4} +(-0.430993 - 0.561681i) q^{5} +(-1.29904 - 0.559015i) q^{6} +(2.64427 + 0.0884090i) q^{7} +(1.91627 + 2.08036i) q^{8} +(-0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.347308	−	1.37090i	−0.245584	−	0.969375i
							\(3\)	0.608761	−	0.793353i	0.351469	−	0.458043i
\(5\)	−0.430993	−	0.561681i	−0.192746	−	0.251191i								0.686976	−	0.726680i	\(-0.258938\pi\)
							−0.879722	+	0.475489i	\(0.842271\pi\)
\(7\)	2.64427	+	0.0884090i	0.999442	+	0.0334155i
							\(11\)	0.333866	+	2.53596i	0.100664	+	0.764622i	0.965101	+	0.261880i	\(0.0843423\pi\)
−0.864436	+	0.502743i	\(0.832324\pi\)
\(13\)	−2.05523	−	4.96176i	−0.570018	−	1.37615i	−0.901539	−	0.432698i	\(-0.857562\pi\)
							0.331521	−	0.943448i	\(-0.392438\pi\)
\(17\)	−2.22735	−	3.85789i	−0.540213	−	0.935676i	−0.998891	−	0.0470737i	\(-0.985010\pi\)
							0.458679	−	0.888602i	\(-0.348323\pi\)
\(19\)	0.512786	−	3.89499i	0.117641	−	0.893573i	−0.826247	−	0.563308i	\(-0.809529\pi\)
							0.943888	−	0.330265i	\(-0.107138\pi\)
\(23\)	5.08546	−	1.36264i	1.06039	−	0.284131i	0.313852	−	0.949472i	\(-0.398381\pi\)
							0.746539	+	0.665341i	\(0.231714\pi\)
\(29\)	2.91965	+	7.04865i	0.542165	+	1.30890i	0.923193	+	0.384337i	\(0.125570\pi\)
							−0.381028	+	0.924563i	\(0.624430\pi\)
\(31\)	−3.58813	−	6.21482i	−0.644446	−	1.11621i	−0.984429	−	0.175782i	\(-0.943755\pi\)
							0.339983	−	0.940432i	\(-0.389579\pi\)
\(37\)	−3.42306	+	2.62661i	−0.562748	+	0.431812i	−0.850562	−	0.525874i	\(-0.823738\pi\)
							0.287814	+	0.957686i	\(0.407072\pi\)
\(41\)	5.80281	+	5.80281i	0.906247	+	0.906247i	0.995967	−	0.0897199i	\(-0.0285972\pi\)
							−0.0897199	+	0.995967i	\(0.528597\pi\)
\(43\)	−5.36835	−	2.22364i	−0.818665	−	0.339102i	−0.0662596	−	0.997802i	\(-0.521107\pi\)
							−0.752405	+	0.658700i	\(0.771107\pi\)
\(47\)	−6.54145	−	3.77671i	−0.954168	−	0.550889i	−0.0597951	−	0.998211i	\(-0.519045\pi\)
							−0.894373	+	0.447321i	\(0.852378\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cf.a.187.26	yes	512
7.3	odd	6	inner	672.2.cf.a.283.59	yes	512
32.19	odd	8	inner	672.2.cf.a.19.59	✓	512
224.115	even	24	inner	672.2.cf.a.115.26	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cf.a.19.59	✓	512	32.19	odd	8	inner
672.2.cf.a.115.26	yes	512	224.115	even	24	inner
672.2.cf.a.187.26	yes	512	1.1	even	1	trivial
672.2.cf.a.283.59	yes	512	7.3	odd	6	inner