Embedded newform 43.4.g.a.9.6

• Label: 43.4.g.a.9.6
• Level: $43$
• Weight: $4$
• Character: 43.9
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.6
Character	\(\chi\)	\(=\)	43.9
Dual form			43.4.g.a.24.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.277412 + 0.347863i) q^{2} +(-3.67378 - 0.553733i) q^{3} +(1.73612 - 7.60642i) q^{4} +(14.5525 + 9.92173i) q^{5} +(-0.826526 - 1.43158i) q^{6} +(18.2723 - 31.6486i) q^{7} +(6.33459 - 3.05058i) q^{8} +(-12.6104 - 3.88980i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.277412	+	0.347863i	0.0980798	+	0.122988i	0.828449	−	0.560064i	\(-0.189224\pi\)
							−0.730369	+	0.683052i	\(0.760652\pi\)
\(3\)	−3.67378	−	0.553733i	−0.707019	−	0.106566i	−0.214317	−	0.976764i	\(-0.568753\pi\)
							−0.492702	+	0.870198i	\(0.663991\pi\)
\(5\)	14.5525	+	9.92173i	1.30162	+	0.887427i	0.997864	−	0.0653242i	\(-0.0208082\pi\)
							0.303752	+	0.952751i	\(0.401761\pi\)
\(7\)	18.2723	−	31.6486i	0.986612	−	1.70886i	0.352072	−	0.935973i	\(-0.385477\pi\)
							0.634541	−	0.772890i	\(-0.281189\pi\)
\(11\)	9.82948	+	43.0658i	0.269427	+	1.18044i	0.910681	+	0.413109i	\(0.135557\pi\)
							−0.641254	+	0.767329i	\(0.721586\pi\)
\(13\)	−0.115920	+	1.54685i	−0.00247311	+	0.0330014i	−0.998304	−	0.0582192i	\(-0.981458\pi\)
							0.995831	+	0.0912206i	\(0.0290768\pi\)
\(17\)	−63.2410	+	43.1170i	−0.902246	+	0.615141i	−0.923066	−	0.384642i	\(-0.874325\pi\)
							0.0208194	+	0.999783i	\(0.493372\pi\)
\(19\)	55.5137	−	17.1237i	0.670300	−	0.206760i	0.0591191	−	0.998251i	\(-0.481171\pi\)
							0.611181	+	0.791491i	\(0.290695\pi\)
\(23\)	−25.4025	+	23.5700i	−0.230295	+	0.213682i	−0.786876	−	0.617111i	\(-0.788303\pi\)
							0.556582	+	0.830793i	\(0.312113\pi\)
\(29\)	−117.436	+	17.7007i	−0.751979	+	0.113343i	−0.513837	−	0.857888i	\(-0.671776\pi\)
							−0.238142	+	0.971230i	\(0.576538\pi\)
\(31\)	−14.0891	+	35.8985i	−0.0816284	+	0.207986i	−0.965861	−	0.259061i	\(-0.916587\pi\)
							0.884233	+	0.467047i	\(0.154682\pi\)
\(37\)	149.900	+	259.635i	0.666039	+	1.15361i	0.979002	+	0.203848i	\(0.0653449\pi\)
							−0.312963	+	0.949765i	\(0.601322\pi\)
\(41\)	−83.7405	−	105.007i	−0.318977	−	0.399985i	0.596331	−	0.802739i	\(-0.296625\pi\)
							−0.915308	+	0.402754i	\(0.868053\pi\)
\(43\)	−123.412	+	253.528i	−0.437680	+	0.899131i
							\(47\)	−14.7119	+	64.4571i	−0.0456586	+	0.200043i	−0.992613	−	0.121325i	\(-0.961286\pi\)
0.946954	+	0.321368i	\(0.104143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.9.6	✓	120
43.14	even	21		1849.4.a.l.1.32		60
43.24	even	21	inner	43.4.g.a.24.6	yes	120
43.29	odd	42		1849.4.a.k.1.29		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.9.6	✓	120	1.1	even	1	trivial
43.4.g.a.24.6	yes	120	43.24	even	21	inner
1849.4.a.k.1.29		60	43.29	odd	42
1849.4.a.l.1.32		60	43.14	even	21