Embedded newform 43.10.g.a.24.7

• Label: 43.10.g.a.24.7
• Level: $43$
• Weight: $10$
• Character: 43.24
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(32\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			24.7
Character	\(\chi\)	\(=\)	43.24
Dual form			43.10.g.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.0000 + 22.5713i) q^{2} +(152.742 - 23.0222i) q^{3} +(-71.5330 - 313.406i) q^{4} +(827.323 - 564.059i) q^{5} +(-2229.72 + 3861.99i) q^{6} +(-595.853 - 1032.05i) q^{7} +(-4955.94 - 2386.65i) q^{8} +(3991.59 - 1231.24i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.0000	+	22.5713i	−0.795497	+	0.997521i	0.204330	+	0.978902i	\(0.434498\pi\)
							−0.999827	+	0.0186192i	\(0.994073\pi\)
\(3\)	152.742	−	23.0222i	1.08871	−	0.164097i	0.419931	−	0.907556i	\(-0.362054\pi\)
							0.668781	+	0.743459i	\(0.266816\pi\)
\(5\)	827.323	−	564.059i	0.591984	−	0.403608i	−0.229940	−	0.973205i	\(-0.573853\pi\)
							0.821924	+	0.569597i	\(0.192901\pi\)
\(7\)	−595.853	−	1032.05i	−0.0937989	−	0.162464i	0.815308	−	0.579028i	\(-0.196568\pi\)
							−0.909107	+	0.416563i	\(0.863234\pi\)
\(11\)	−6846.66	+	29997.2i	−0.140998	+	0.617751i	0.854207	+	0.519934i	\(0.174043\pi\)
							−0.995204	+	0.0978175i	\(0.968814\pi\)
\(13\)	3666.00	+	48919.4i	0.0355998	+	0.475046i	0.986303	+	0.164944i	\(0.0527443\pi\)
							−0.950703	+	0.310102i	\(0.899637\pi\)
\(17\)	385503.	+	262831.i	1.11946	+	0.763233i	0.974093	−	0.226146i	\(-0.0726126\pi\)
							0.145363	+	0.989378i	\(0.453565\pi\)
\(19\)	604783.	+	186551.i	1.06465	+	0.328402i	0.777110	−	0.629365i	\(-0.216685\pi\)
							0.287544	+	0.957767i	\(0.407161\pi\)
\(23\)	741649.	+	688150.i	0.552616	+	0.512752i	0.906313	−	0.422608i	\(-0.138885\pi\)
							−0.353697	+	0.935360i	\(0.615076\pi\)
\(29\)	2.02898e6	+	305819.i	0.532704	+	0.0802923i	0.409886	−	0.912137i	\(-0.365569\pi\)
							0.122819	+	0.992429i	\(0.460807\pi\)
\(31\)	1.01838e6	+	2.59478e6i	0.198052	+	0.504630i	0.995053	−	0.0993497i	\(-0.0316762\pi\)
							−0.797000	+	0.603979i	\(0.793581\pi\)
\(37\)	−7.06192e6	+	1.22316e7i	−0.619462	+	1.07294i	0.370121	+	0.928983i	\(0.379316\pi\)
							−0.989584	+	0.143957i	\(0.954017\pi\)
\(41\)	133545.	−	167461.i	0.00738077	−	0.00925520i	−0.778127	−	0.628106i	\(-0.783830\pi\)
							0.785508	+	0.618851i	\(0.212402\pi\)
\(43\)	2.17389e7	−	5.47851e6i	0.969681	−	0.244374i
							\(47\)	−2.12091e6	−	9.29232e6i	−0.0633990	−	0.277769i	0.933285	−	0.359136i	\(-0.116929\pi\)
−0.996684	+	0.0813670i	\(0.974071\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.24.7	yes	384
43.9	even	21	inner	43.10.g.a.9.7	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.7	✓	384	43.9	even	21	inner
43.10.g.a.24.7	yes	384	1.1	even	1	trivial