Embedded newform 672.2.cf.a.187.29

• Label: 672.2.cf.a.187.29
• 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			187.29
Character	\(\chi\)	\(=\)	672.187
Dual form			672.2.cf.a.115.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0456468 + 1.41348i) q^{2} +(-0.608761 + 0.793353i) q^{3} +(-1.99583 - 0.129041i) q^{4} +(2.11354 + 2.75442i) q^{5} +(-1.09360 - 0.896684i) q^{6} +(2.45568 - 0.984703i) q^{7} +(0.273500 - 2.81517i) q^{8} +(-0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0456468	+	1.41348i	−0.0322772	+	0.999479i
							\(3\)	−0.608761	+	0.793353i	−0.351469	+	0.458043i
\(5\)	2.11354	+	2.75442i	0.945204	+	1.23181i								0.972670	+	0.232192i	\(0.0745897\pi\)
							−0.0274656	+	0.999623i	\(0.508744\pi\)
\(7\)	2.45568	−	0.984703i	0.928159	−	0.372183i
							\(11\)	0.570276	+	4.33168i	0.171945	+	1.30605i	0.831976	+	0.554812i	\(0.187210\pi\)
−0.660031	+	0.751238i	\(0.729457\pi\)
\(13\)	0.414385	+	1.00041i	0.114930	+	0.277465i	0.970869	−	0.239612i	\(-0.0770202\pi\)
							−0.855939	+	0.517077i	\(0.827020\pi\)
\(17\)	1.39153	+	2.41020i	0.337495	+	0.584559i	0.983961	−	0.178384i	\(-0.0570869\pi\)
							−0.646466	+	0.762943i	\(0.723754\pi\)
\(19\)	0.148048	−	1.12454i	0.0339646	−	0.257987i	−0.966035	−	0.258410i	\(-0.916801\pi\)
							1.00000		0.000423431i	\(0.000134782\pi\)
\(23\)	2.58691	−	0.693161i	0.539408	−	0.144534i	0.0211791	−	0.999776i	\(-0.493258\pi\)
							0.518229	+	0.855242i	\(0.326591\pi\)
\(29\)	1.73434	+	4.18707i	0.322059	+	0.777519i	0.999134	+	0.0416049i	\(0.0132471\pi\)
							−0.677075	+	0.735914i	\(0.736753\pi\)
\(31\)	−4.27165	−	7.39872i	−0.767212	−	1.32885i	−0.939069	−	0.343728i	\(-0.888310\pi\)
							0.171858	−	0.985122i	\(-0.445023\pi\)
\(37\)	0.587264	−	0.450624i	0.0965456	−	0.0740821i	−0.559361	−	0.828924i	\(-0.688953\pi\)
							0.655906	+	0.754842i	\(0.272287\pi\)
\(41\)	−1.90214	−	1.90214i	−0.297065	−	0.297065i	0.542798	−	0.839863i	\(-0.317365\pi\)
							−0.839863	+	0.542798i	\(0.817365\pi\)
\(43\)	−8.53377	−	3.53480i	−1.30139	−	0.539052i	−0.379028	−	0.925385i	\(-0.623742\pi\)
							−0.922359	+	0.386333i	\(0.873742\pi\)
\(47\)	−10.4578	−	6.03781i	−1.52543	−	0.880705i	−0.999546	−	0.0301457i	\(-0.990403\pi\)
							−0.525880	−	0.850559i	\(-0.676264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cf.a.187.29	yes	512
7.3	odd	6	inner	672.2.cf.a.283.12	yes	512
32.19	odd	8	inner	672.2.cf.a.19.12	✓	512
224.115	even	24	inner	672.2.cf.a.115.29	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cf.a.19.12	✓	512	32.19	odd	8	inner
672.2.cf.a.115.29	yes	512	224.115	even	24	inner
672.2.cf.a.187.29	yes	512	1.1	even	1	trivial
672.2.cf.a.283.12	yes	512	7.3	odd	6	inner