Embedded newform 43.8.g.a.9.16

• Label: 43.8.g.a.9.16
• Level: $43$
• Weight: $8$
• Character: 43.9
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.16
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.34377 + 4.19295i) q^{2} +(-17.0374 - 2.56798i) q^{3} +(22.0826 - 96.7502i) q^{4} +(150.846 + 102.845i) q^{5} +(-46.2018 - 80.0239i) q^{6} +(-185.160 + 320.707i) q^{7} +(1097.99 - 528.764i) q^{8} +(-1806.16 - 557.126i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.34377	+	4.19295i	0.295550	+	0.370608i	0.907329	−	0.420420i	\(-0.138117\pi\)
							−0.611779	+	0.791029i	\(0.709546\pi\)
\(3\)	−17.0374	−	2.56798i	−0.364317	−	0.0549120i	−0.0356672	−	0.999364i	\(-0.511356\pi\)
							−0.328650	+	0.944452i	\(0.606594\pi\)
\(5\)	150.846	+	102.845i	0.539683	+	0.367950i	0.802272	−	0.596958i	\(-0.203624\pi\)
							−0.262590	+	0.964908i	\(0.584577\pi\)
\(7\)	−185.160	+	320.707i	−0.204035	+	0.353399i	−0.949825	−	0.312782i	\(-0.898739\pi\)
							0.745790	+	0.666181i	\(0.232072\pi\)
\(11\)	−308.923	−	1353.48i	−0.0699804	−	0.306604i	0.927809	−	0.373055i	\(-0.121690\pi\)
							−0.997790	+	0.0664507i	\(0.978832\pi\)
\(13\)	1096.63	−	14633.5i	0.138439	−	1.84734i	−0.306609	−	0.951835i	\(-0.599195\pi\)
							0.445049	−	0.895506i	\(-0.353186\pi\)
\(17\)	13951.1	−	9511.70i	0.688711	−	0.469555i	−0.167708	−	0.985837i	\(-0.553637\pi\)
							0.856419	+	0.516281i	\(0.172684\pi\)
\(19\)	44497.6	−	13725.7i	1.48833	−	0.459089i	0.559259	−	0.828993i	\(-0.311086\pi\)
							0.929070	+	0.369905i	\(0.120610\pi\)
\(23\)	−32582.1	+	30231.8i	−0.558383	+	0.518104i	−0.908101	−	0.418751i	\(-0.862468\pi\)
							0.349718	+	0.936855i	\(0.386277\pi\)
\(29\)	−121439.	+	18304.0i	−0.924622	+	0.139364i	−0.594061	−	0.804420i	\(-0.702476\pi\)
							−0.330562	+	0.943784i	\(0.607238\pi\)
\(31\)	64359.4	−	163985.i	0.388013	−	0.988641i	−0.594773	−	0.803894i	\(-0.702758\pi\)
							0.982786	−	0.184747i	\(-0.0591467\pi\)
\(37\)	−180369.	−	312409.i	−0.585405	−	1.01395i	−0.994825	−	0.101605i	\(-0.967602\pi\)
							0.409420	−	0.912346i	\(-0.365731\pi\)
\(41\)	244787.	+	306953.i	0.554682	+	0.695549i	0.977565	−	0.210635i	\(-0.0675532\pi\)
							−0.422883	+	0.906184i	\(0.638982\pi\)
\(43\)	−471339.	+	222842.i	−0.904052	+	0.427422i
							\(47\)	−33165.7	+	145308.i	−0.0465958	+	0.204150i	−0.992867	−	0.119223i	\(-0.961960\pi\)
0.946272	+	0.323373i	\(0.104817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.16	✓	300
43.24	even	21	inner	43.8.g.a.24.16	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.16	✓	300	1.1	even	1	trivial
43.8.g.a.24.16	yes	300	43.24	even	21	inner