Embedded newform 43.8.g.a.9.1

• Label: 43.8.g.a.9.1
• 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			9.1
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.3861 - 16.7857i) q^{2} +(9.91222 + 1.49403i) q^{3} +(-74.0876 + 324.599i) q^{4} +(247.262 + 168.580i) q^{5} +(-107.608 - 186.382i) q^{6} +(-560.464 + 970.753i) q^{7} +(3964.38 - 1909.15i) q^{8} +(-1993.82 - 615.011i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.3861	−	16.7857i	−1.18318	−	1.48366i	−0.838471	−	0.544946i	\(-0.816550\pi\)
							−0.344706	−	0.938711i	\(-0.612021\pi\)
\(3\)	9.91222	+	1.49403i	0.211956	+	0.0319473i	0.254162	−	0.967162i	\(-0.418200\pi\)
							−0.0422056	+	0.999109i	\(0.513438\pi\)
\(5\)	247.262	+	168.580i	0.884631	+	0.603131i	0.918118	−	0.396308i	\(-0.129709\pi\)
							−0.0334869	+	0.999439i	\(0.510661\pi\)
\(7\)	−560.464	+	970.753i	−0.617597	+	1.06971i	0.372326	+	0.928102i	\(0.378560\pi\)
							−0.989923	+	0.141607i	\(0.954773\pi\)
\(11\)	−736.137	−	3225.23i	−0.166757	−	0.730611i	−0.987279	−	0.158996i	\(-0.949174\pi\)
							0.820522	−	0.571615i	\(-0.193683\pi\)
\(13\)	725.353	−	9679.17i	0.0915689	−	1.22190i	−0.742349	−	0.670013i	\(-0.766288\pi\)
							0.833918	−	0.551888i	\(-0.186093\pi\)
\(17\)	2017.35	−	1375.41i	0.0995889	−	0.0678985i	−0.512502	−	0.858686i	\(-0.671281\pi\)
							0.612091	+	0.790788i	\(0.290329\pi\)
\(19\)	−28988.4	+	8941.74i	−0.969587	+	0.299078i	−0.738799	−	0.673926i	\(-0.764607\pi\)
							−0.230788	+	0.973004i	\(0.574130\pi\)
\(23\)	83611.8	−	77580.4i	1.43291	−	1.32955i	0.572549	−	0.819870i	\(-0.305955\pi\)
							0.860365	−	0.509679i	\(-0.170236\pi\)
\(29\)	−65789.8	+	9916.21i	−0.500917	+	0.0755010i	−0.394640	−	0.918836i	\(-0.629131\pi\)
							−0.106276	+	0.994337i	\(0.533893\pi\)
\(31\)	96263.4	−	245275.i	0.580357	−	1.47873i	−0.275391	−	0.961332i	\(-0.588807\pi\)
							0.855748	−	0.517393i	\(-0.173097\pi\)
\(37\)	84355.9	+	146109.i	0.273785	+	0.474209i	0.969828	−	0.243791i	\(-0.0783910\pi\)
							−0.696043	+	0.718000i	\(0.745058\pi\)
\(41\)	−68961.3	−	86474.8i	−0.156265	−	0.195950i	0.697536	−	0.716550i	\(-0.254280\pi\)
							−0.853801	+	0.520600i	\(0.825708\pi\)
\(43\)	−64625.6	−	517341.i	−0.123955	−	0.992288i
							\(47\)	−102094.	+	447305.i	−0.143436	+	0.628436i	0.851186	+	0.524865i	\(0.175884\pi\)
−0.994622	+	0.103571i	\(0.966973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.1	✓	300
43.24	even	21	inner	43.8.g.a.24.1	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.1	✓	300	1.1	even	1	trivial
43.8.g.a.24.1	yes	300	43.24	even	21	inner