Embedded newform 43.8.g.a.9.18

• Label: 43.8.g.a.9.18
• 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			9.18
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.51857 + 8.17403i) q^{2} +(-88.4724 - 13.3351i) q^{3} +(4.15968 - 18.2247i) q^{4} +(-356.783 - 243.251i) q^{5} +(-467.712 - 810.101i) q^{6} +(-739.305 + 1280.51i) q^{7} +(1381.79 - 665.437i) q^{8} +(5559.70 + 1714.94i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.51857	+	8.17403i	0.576166	+	0.722489i	0.981454	−	0.191700i	\(-0.0614001\pi\)
							−0.405288	+	0.914189i	\(0.632829\pi\)
\(3\)	−88.4724	−	13.3351i	−1.89183	−	0.285148i	−0.902487	−	0.430717i	\(-0.858261\pi\)
							−0.989348	+	0.145569i	\(0.953499\pi\)
\(5\)	−356.783	−	243.251i	−1.27647	−	0.870281i	−0.280487	−	0.959858i	\(-0.590496\pi\)
							−0.995980	+	0.0895771i	\(0.971448\pi\)
\(7\)	−739.305	+	1280.51i	−0.814667	+	1.41104i	0.0948994	+	0.995487i	\(0.469747\pi\)
							−0.909567	+	0.415558i	\(0.863586\pi\)
\(11\)	516.396	+	2262.48i	0.116979	+	0.512519i	0.999136	+	0.0415630i	\(0.0132337\pi\)
							−0.882157	+	0.470956i	\(0.843909\pi\)
\(13\)	603.639	−	8055.01i	0.0762036	−	1.01687i	−0.819751	−	0.572721i	\(-0.805888\pi\)
							0.895954	−	0.444146i	\(-0.146493\pi\)
\(17\)	7530.29	−	5134.07i	0.371741	−	0.253449i	−0.363011	−	0.931785i	\(-0.618251\pi\)
							0.734752	+	0.678336i	\(0.237299\pi\)
\(19\)	−12841.5	+	3961.09i	−0.429516	+	0.132488i	−0.501973	−	0.864883i	\(-0.667392\pi\)
							0.0724566	+	0.997372i	\(0.476916\pi\)
\(23\)	−26366.5	+	24464.5i	−0.451860	+	0.419265i	−0.872992	−	0.487734i	\(-0.837823\pi\)
							0.421132	+	0.906999i	\(0.361633\pi\)
\(29\)	180502.	−	27206.3i	1.37432	−	0.207146i	0.579987	−	0.814626i	\(-0.303058\pi\)
							0.794335	+	0.607480i	\(0.207820\pi\)
\(31\)	4998.03	−	12734.8i	0.0301323	−	0.0767759i	−0.915016	−	0.403418i	\(-0.867822\pi\)
							0.945148	+	0.326642i	\(0.105917\pi\)
\(37\)	42672.6	+	73911.1i	0.138498	+	0.239885i	0.926928	−	0.375239i	\(-0.122439\pi\)
							−0.788430	+	0.615124i	\(0.789106\pi\)
\(41\)	421307.	+	528302.i	0.954674	+	1.19712i	0.980313	+	0.197452i	\(0.0632667\pi\)
							−0.0256385	+	0.999671i	\(0.508162\pi\)
\(43\)	409503.	+	322685.i	0.785448	+	0.618927i
							\(47\)	−74580.8	+	326760.i	−0.104781	+	0.459078i	0.895130	+	0.445804i	\(0.147082\pi\)
−0.999912	+	0.0132732i	\(0.995775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.18	✓	300
43.24	even	21	inner	43.8.g.a.24.18	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.18	✓	300	1.1	even	1	trivial
43.8.g.a.24.18	yes	300	43.24	even	21	inner