Embedded newform 43.8.g.a.24.18

• Label: 43.8.g.a.24.18
• Level: $43$
• Weight: $8$
• Character: 43.24
• 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			24.18
Character	\(\chi\)	\(=\)	43.24
Dual form			43.8.g.a.9.18

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.18	✓	300	43.9	even	21	inner
43.8.g.a.24.18	yes	300	1.1	even	1	trivial