Embedded newform 672.2.ch.a.11.18

• Label: 672.2.ch.a.11.18
• Level: $672$
• Weight: $2$
• Character: 672.11
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(124\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			11.18
Character	\(\chi\)	\(=\)	672.11
Dual form			672.2.ch.a.611.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30144 - 0.553394i) q^{2} +(1.57447 + 0.721834i) q^{3} +(1.38751 + 1.44042i) q^{4} +(0.510185 + 3.87524i) q^{5} +(-1.64963 - 1.81073i) q^{6} +(-2.48556 + 0.906631i) q^{7} +(-1.00865 - 2.64247i) q^{8} +(1.95791 + 2.27301i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 4 q^{3} - 8 q^{4} - 16 q^{6} - 16 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30144	−	0.553394i	−0.920260	−	0.391309i
							\(3\)	1.57447	+	0.721834i	0.909021	+	0.416751i
\(5\)	0.510185	+	3.87524i	0.228161	+	1.73306i								0.595838	+	0.803105i	\(0.296820\pi\)
							−0.367677	+	0.929954i	\(0.619847\pi\)
\(7\)	−2.48556	+	0.906631i	−0.939454	+	0.342674i
							\(11\)	−3.07093	+	2.35641i	−0.925920	+	0.710483i	−0.957322	−	0.289024i	\(-0.906669\pi\)
0.0314022	+	0.999507i	\(0.490003\pi\)
\(13\)	−1.60469	−	3.87405i	−0.445060	−	1.07447i	−0.974150	−	0.225904i	\(-0.927467\pi\)
							0.529090	−	0.848566i	\(-0.322533\pi\)
\(17\)	1.14277	−	1.97933i	0.277162	−	0.480059i	−0.693516	−	0.720441i	\(-0.743939\pi\)
							0.970678	+	0.240382i	\(0.0772728\pi\)
\(19\)	5.03373	+	3.86251i	1.15482	+	0.886122i	0.994886	−	0.101007i	\(-0.0322063\pi\)
							0.159930	+	0.987128i	\(0.448873\pi\)
\(23\)	0.730240	+	0.195667i	0.152266	+	0.0407994i	0.334147	−	0.942521i	\(-0.391552\pi\)
							−0.181881	+	0.983321i	\(0.558219\pi\)
\(29\)	−3.19690	−	7.71800i	−0.593650	−	1.43320i	−0.879954	−	0.475059i	\(-0.842427\pi\)
							0.286304	−	0.958139i	\(-0.407573\pi\)
\(31\)	1.16309	+	0.671512i	0.208898	+	0.120607i	0.600799	−	0.799400i	\(-0.294849\pi\)
							−0.391901	+	0.920007i	\(0.628183\pi\)
\(37\)	−7.76869	+	1.02277i	−1.27717	+	0.168142i	−0.738428	−	0.674332i	\(-0.764432\pi\)
							−0.538737	+	0.842474i	\(0.681098\pi\)
\(41\)	4.67370	+	4.67370i	0.729909	+	0.729909i	0.970601	−	0.240693i	\(-0.0773746\pi\)
							−0.240693	+	0.970601i	\(0.577375\pi\)
\(43\)	−1.86966	+	4.51375i	−0.285120	+	0.688340i	−0.999940	−	0.0109683i	\(-0.996509\pi\)
							0.714820	+	0.699309i	\(0.246509\pi\)
\(47\)	2.31009	−	1.33373i	0.336962	−	0.194545i	−0.321966	−	0.946751i	\(-0.604344\pi\)
							0.658928	+	0.752206i	\(0.271010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.ch.a.11.18	✓	992
3.2	odd	2	inner	672.2.ch.a.11.107	yes	992
7.2	even	3	inner	672.2.ch.a.107.66	yes	992
21.2	odd	6	inner	672.2.ch.a.107.59	yes	992
32.3	odd	8	inner	672.2.ch.a.515.59	yes	992
96.35	even	8	inner	672.2.ch.a.515.66	yes	992
224.163	odd	24	inner	672.2.ch.a.611.107	yes	992
672.611	even	24	inner	672.2.ch.a.611.18	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.ch.a.11.18	✓	992	1.1	even	1	trivial
672.2.ch.a.11.107	yes	992	3.2	odd	2	inner
672.2.ch.a.107.59	yes	992	21.2	odd	6	inner
672.2.ch.a.107.66	yes	992	7.2	even	3	inner
672.2.ch.a.515.59	yes	992	32.3	odd	8	inner
672.2.ch.a.515.66	yes	992	96.35	even	8	inner
672.2.ch.a.611.18	yes	992	672.611	even	24	inner
672.2.ch.a.611.107	yes	992	224.163	odd	24	inner