Embedded newform 672.2.cj.a.613.64

• Label: 672.2.cj.a.613.64
• Level: $672$
• Weight: $2$
• Character: 672.613
• 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.cj (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			613.64
Character	\(\chi\)	\(=\)	672.613
Dual form			672.2.cj.a.205.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41349 + 0.0452294i) q^{2} +(-0.991445 - 0.130526i) q^{3} +(1.99591 + 0.127863i) q^{4} +(0.117178 + 0.890058i) q^{5} +(-1.39549 - 0.229340i) q^{6} +(-0.419165 - 2.61234i) q^{7} +(2.81541 + 0.271006i) q^{8} +(0.965926 + 0.258819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41349	+	0.0452294i	0.999488	+	0.0319820i
							\(3\)	−0.991445	−	0.130526i	−0.572411	−	0.0753593i
\(5\)	0.117178	+	0.890058i	0.0524038	+	0.398046i								0.997465	+	0.0711538i	\(0.0226681\pi\)
							−0.945062	+	0.326892i	\(0.893999\pi\)
\(7\)	−0.419165	−	2.61234i	−0.158429	−	0.987370i
							\(11\)	−3.16414	−	4.12359i	−0.954026	−	1.24331i	−0.969917	−	0.243437i	\(-0.921725\pi\)
0.0158913	−	0.999874i	\(-0.494941\pi\)
\(13\)	4.68517	−	1.94066i	1.29943	−	0.538243i	0.377650	−	0.925949i	\(-0.376732\pi\)
							0.921783	+	0.387706i	\(0.126732\pi\)
\(17\)	2.75456	+	1.59035i	0.668079	+	0.385715i	0.795348	−	0.606153i	\(-0.207288\pi\)
							−0.127270	+	0.991868i	\(0.540621\pi\)
\(19\)	2.43670	+	1.86975i	0.559018	+	0.428950i	0.849238	−	0.528010i	\(-0.177062\pi\)
							−0.290220	+	0.956960i	\(0.593728\pi\)
\(23\)	0.642442	+	0.172142i	0.133958	+	0.0358941i	0.325175	−	0.945654i	\(-0.394577\pi\)
							−0.191217	+	0.981548i	\(0.561243\pi\)
\(29\)	−2.31484	−	5.58853i	−0.429856	−	1.03776i	−0.979333	−	0.202255i	\(-0.935173\pi\)
							0.549477	−	0.835509i	\(-0.314827\pi\)
\(31\)	−4.65400	+	8.06096i	−0.835882	+	1.44779i	0.0574279	+	0.998350i	\(0.481710\pi\)
							−0.893310	+	0.449441i	\(0.851623\pi\)
\(37\)	0.618961	+	4.70147i	0.101757	+	0.772918i	0.963902	+	0.266257i	\(0.0857870\pi\)
							−0.862146	+	0.506661i	\(0.830880\pi\)
\(41\)	−5.31923	+	5.31923i	−0.830723	+	0.830723i	−0.987616	−	0.156892i	\(-0.949852\pi\)
							0.156892	+	0.987616i	\(0.449852\pi\)
\(43\)	2.82065	−	6.80964i	0.430144	−	1.03846i	−0.549096	−	0.835759i	\(-0.685028\pi\)
							0.979241	−	0.202701i	\(-0.0649721\pi\)
\(47\)	1.87024	−	1.07978i	0.272802	−	0.157502i	−0.357358	−	0.933967i	\(-0.616323\pi\)
							0.630160	+	0.776465i	\(0.282989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cj.a.613.64	yes	512
7.2	even	3	inner	672.2.cj.a.37.23	✓	512
32.13	even	8	inner	672.2.cj.a.109.23	yes	512
224.205	even	24	inner	672.2.cj.a.205.64	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cj.a.37.23	✓	512	7.2	even	3	inner
672.2.cj.a.109.23	yes	512	32.13	even	8	inner
672.2.cj.a.205.64	yes	512	224.205	even	24	inner
672.2.cj.a.613.64	yes	512	1.1	even	1	trivial