Embedded newform 43.7.h.a.12.8

• Label: 43.7.h.a.12.8
• Level: $43$
• Weight: $7$
• Character: 43.12
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			12.8
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.67452 - 5.55371i) q^{2} +(24.4389 + 35.8453i) q^{3} +(16.2128 - 20.3302i) q^{4} +(-20.9922 + 22.6242i) q^{5} +(133.712 - 231.596i) q^{6} +(375.666 - 216.891i) q^{7} +(-540.884 - 123.453i) q^{8} +(-421.291 + 1073.43i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.67452	−	5.55371i	−0.334316	−	0.694214i	0.664264	−	0.747498i	\(-0.268745\pi\)
							−0.998579	+	0.0532847i	\(0.983031\pi\)
\(3\)	24.4389	+	35.8453i	0.905145	+	1.32760i	0.944546	+	0.328380i	\(0.106503\pi\)
							−0.0394013	+	0.999223i	\(0.512545\pi\)
\(5\)	−20.9922	+	22.6242i	−0.167937	+	0.180993i	−0.811413	−	0.584473i	\(-0.801301\pi\)
							0.643476	+	0.765466i	\(0.277492\pi\)
\(7\)	375.666	−	216.891i	1.09524	−	0.632336i	0.160271	−	0.987073i	\(-0.448763\pi\)
							0.934966	+	0.354737i	\(0.115430\pi\)
\(11\)	1489.70	+	1868.02i	1.11923	+	1.40347i	0.904327	+	0.426840i	\(0.140373\pi\)
							0.214907	+	0.976635i	\(0.431055\pi\)
\(13\)	223.335	−	68.8897i	0.101654	−	0.0313562i	−0.243511	−	0.969898i	\(-0.578299\pi\)
							0.345165	+	0.938542i	\(0.387823\pi\)
\(17\)	6.51765	−	6.04750i	0.00132661	−	0.00123092i	−0.679509	−	0.733667i	\(-0.737807\pi\)
							0.680836	+	0.732436i	\(0.261617\pi\)
\(19\)	9248.40	−	3629.73i	1.34836	−	0.529192i	0.422262	−	0.906474i	\(-0.361236\pi\)
							0.926099	+	0.377282i	\(0.123141\pi\)
\(23\)	849.967	−	128.112i	0.0698584	−	0.0105295i	−0.114020	−	0.993478i	\(-0.536373\pi\)
							0.183878	+	0.982949i	\(0.441135\pi\)
\(29\)	−6239.34	+	9151.44i	−0.255826	+	0.375228i	−0.932569	−	0.360991i	\(-0.882438\pi\)
							0.676743	+	0.736219i	\(0.263391\pi\)
\(31\)	2192.93	−	29262.6i	0.0736106	−	0.982265i	−0.831068	−	0.556170i	\(-0.812270\pi\)
							0.904679	−	0.426094i	\(-0.140111\pi\)
\(37\)	−50576.6	−	29200.4i	−0.998492	−	0.576480i	−0.0906903	−	0.995879i	\(-0.528907\pi\)
							−0.907802	+	0.419399i	\(0.862241\pi\)
\(41\)	−32433.9	+	15619.4i	−0.470596	+	0.226627i	−0.654124	−	0.756387i	\(-0.726962\pi\)
							0.183528	+	0.983014i	\(0.441248\pi\)
\(43\)	−76240.7	+	22554.7i	−0.958919	+	0.283682i
							\(47\)	101675.	−	127497.i	0.979313	−	1.22802i	0.00565971	−	0.999984i	\(-0.498198\pi\)
0.973653	−	0.228035i	\(-0.0732301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.8	✓	252
43.18	odd	42	inner	43.7.h.a.18.8	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.8	✓	252	1.1	even	1	trivial
43.7.h.a.18.8	yes	252	43.18	odd	42	inner