Embedded newform 43.9.h.a.18.16

• Label: 43.9.h.a.18.16
• Level: $43$
• Weight: $9$
• Character: 43.18
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			18.16
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.28196 - 6.81506i) q^{2} +(-72.1741 + 105.860i) q^{3} +(123.940 + 155.415i) q^{4} +(428.780 + 462.115i) q^{5} +(484.570 + 839.299i) q^{6} +(2434.71 + 1405.68i) q^{7} +(3353.80 - 765.484i) q^{8} +(-3600.22 - 9173.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.28196	−	6.81506i	0.205123	−	0.425941i	−0.772875	−	0.634559i	\(-0.781182\pi\)
							0.977997	+	0.208617i	\(0.0668962\pi\)
\(3\)	−72.1741	+	105.860i	−0.891038	+	1.30691i	0.0601147	+	0.998191i	\(0.480853\pi\)
							−0.951153	+	0.308721i	\(0.900099\pi\)
\(5\)	428.780	+	462.115i	0.686048	+	0.739384i	0.976027	−	0.217649i	\(-0.0698389\pi\)
							−0.289979	+	0.957033i	\(0.593648\pi\)
\(7\)	2434.71	+	1405.68i	1.01404	+	0.585455i	0.912372	−	0.409363i	\(-0.134249\pi\)
							0.101667	+	0.994818i	\(0.467582\pi\)
\(11\)	−4301.73	+	5394.20i	−0.293814	+	0.368431i	−0.906726	−	0.421720i	\(-0.861426\pi\)
							0.612912	+	0.790151i	\(0.289998\pi\)
\(13\)	38076.8	+	11745.1i	1.33317	+	0.411230i	0.877782	−	0.479061i	\(-0.159023\pi\)
							0.455393	+	0.890291i	\(0.349499\pi\)
\(17\)	19019.8	+	17647.8i	0.227724	+	0.211297i	0.785780	−	0.618506i	\(-0.212262\pi\)
							−0.558056	+	0.829804i	\(0.688452\pi\)
\(19\)	−150079.	−	58901.7i	−1.15161	−	0.451974i	−0.288745	−	0.957406i	\(-0.593238\pi\)
							−0.862866	+	0.505432i	\(0.831333\pi\)
\(23\)	381516.	+	57504.3i	1.36333	+	0.205489i	0.789634	−	0.613578i	\(-0.210271\pi\)
							0.573697	+	0.819067i	\(0.305509\pi\)
\(29\)	−511788.	−	750655.i	−0.723599	−	1.06132i	−0.995099	−	0.0988862i	\(-0.968472\pi\)
							0.271500	−	0.962438i	\(-0.412480\pi\)
\(31\)	14153.2	+	188862.i	0.0153253	+	0.204502i	0.999610	+	0.0279146i	\(0.00888665\pi\)
							−0.984285	+	0.176587i	\(0.943494\pi\)
\(37\)	−813581.	+	469721.i	−0.434104	+	0.250630i	−0.701093	−	0.713069i	\(-0.747304\pi\)
							0.266990	+	0.963699i	\(0.413971\pi\)
\(41\)	−3.33583e6	−	1.60645e6i	−1.18051	−	0.568501i	−0.262447	−	0.964946i	\(-0.584529\pi\)
							−0.918058	+	0.396445i	\(0.870244\pi\)
\(43\)	−1.18840e6	−	3.20561e6i	−0.347607	−	0.937640i
							\(47\)	−740795.	−	928928.i	−0.151812	−	0.190367i	0.700110	−	0.714035i	\(-0.253134\pi\)
−0.851922	+	0.523669i	\(0.824563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.18.16	yes	336
43.12	odd	42	inner	43.9.h.a.12.16	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.16	✓	336	43.12	odd	42	inner
43.9.h.a.18.16	yes	336	1.1	even	1	trivial