Embedded newform 43.7.h.a.18.8

• Label: 43.7.h.a.18.8
• Level: $43$
• Weight: $7$
• Character: 43.18
• 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			18.8
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.8

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.8	✓	252	43.12	odd	42	inner
43.7.h.a.18.8	yes	252	1.1	even	1	trivial