Embedded newform 43.10.e.a.4.2

• Label: 43.10.e.a.4.2
• Level: $43$
• Weight: $10$
• Character: 43.4
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.2638 + 32.9338i) q^{2} +(-77.4797 - 97.1565i) q^{3} +(-280.916 - 1230.77i) q^{4} +(-1723.13 - 829.815i) q^{5} +5234.65 q^{6} +4745.47 q^{7} +(28480.4 + 13715.4i) q^{8} +(943.600 - 4134.18i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.2638	+	32.9338i	−1.16071	+	1.45548i	−0.294604	+	0.955619i	\(0.595188\pi\)
							−0.866104	+	0.499863i	\(0.833384\pi\)
\(3\)	−77.4797	−	97.1565i	−0.552259	−	0.692510i	0.424847	−	0.905265i	\(-0.360328\pi\)
							−0.977106	+	0.212755i	\(0.931756\pi\)
\(5\)	−1723.13	−	829.815i	−1.23297	−	0.593767i	−0.300075	−	0.953916i	\(-0.597012\pi\)
							−0.932895	+	0.360148i	\(0.882726\pi\)
\(7\)	4745.47			0.747029			0.373515	−	0.927624i	\(-0.378153\pi\)
							0.373515	+	0.927624i	\(0.378153\pi\)
\(11\)	7153.08	−	31339.7i	0.147308	−	0.645398i	−0.846319	−	0.532677i	\(-0.821186\pi\)
							0.993627	−	0.112721i	\(-0.0359567\pi\)
\(13\)	−20487.2	−	9866.13i	−0.198947	−	0.0958080i	0.331757	−	0.943365i	\(-0.392358\pi\)
							−0.530705	+	0.847557i	\(0.678073\pi\)
\(17\)	322793.	−	155449.i	0.937355	−	0.451407i	0.0981199	−	0.995175i	\(-0.468717\pi\)
							0.839236	+	0.543768i	\(0.183003\pi\)
\(19\)	−189163.	−	828775.i	−0.333000	−	1.45897i	−0.813290	−	0.581858i	\(-0.802326\pi\)
							0.480291	−	0.877109i	\(-0.340531\pi\)
\(23\)	−24028.3	+	105275.i	−0.0179039	+	0.0784422i	−0.983091	−	0.183120i	\(-0.941380\pi\)
							0.965187	+	0.261562i	\(0.0842376\pi\)
\(29\)	2.61110e6	−	3.27421e6i	0.685539	−	0.859639i	−0.310312	−	0.950635i	\(-0.600434\pi\)
							0.995851	+	0.0909960i	\(0.0290050\pi\)
\(31\)	5.54866e6	−	6.95779e6i	1.07910	−	1.35314i	0.147743	−	0.989026i	\(-0.452799\pi\)
							0.931353	−	0.364118i	\(-0.118629\pi\)
\(37\)	−1.52925e7			−1.34144			−0.670721	−	0.741709i	\(-0.734015\pi\)
							−0.670721	+	0.741709i	\(0.734015\pi\)
\(41\)	5.56568e6	−	6.97915e6i	0.307603	−	0.385722i	−0.603869	−	0.797084i	\(-0.706375\pi\)
							0.911472	+	0.411361i	\(0.134946\pi\)
\(43\)	1.86966e7	−	1.23705e7i	0.833978	−	0.551797i
							\(47\)	1.15478e7	+	5.05940e7i	0.345189	+	1.51237i	0.787954	+	0.615735i	\(0.211141\pi\)
−0.442764	+	0.896638i	\(0.646002\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.2	✓	192
43.11	even	7	inner	43.10.e.a.11.2	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.2	✓	192	1.1	even	1	trivial
43.10.e.a.11.2	yes	192	43.11	even	7	inner