Embedded newform 43.5.h.a.3.1

• Label: 43.5.h.a.3.1
• Level: $43$
• Weight: $5$
• Character: 43.3
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.21040 + 6.66647i) q^{2} +(13.4095 + 1.00491i) q^{3} +(-24.1593 - 30.2948i) q^{4} +(-6.13495 + 19.8890i) q^{5} +(-49.7491 + 86.1681i) q^{6} +(-69.5021 + 40.1271i) q^{7} +(164.101 - 37.4550i) q^{8} +(98.7103 + 14.8782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.21040	+	6.66647i	−0.802601	+	1.66662i	−0.0587487	+	0.998273i	\(0.518711\pi\)
							−0.743852	+	0.668345i	\(0.767003\pi\)
\(3\)	13.4095	+	1.00491i	1.48995	+	0.111656i	0.794651	−	0.607067i	\(-0.207654\pi\)
							0.695297	+	0.718723i	\(0.255273\pi\)
\(5\)	−6.13495	+	19.8890i	−0.245398	+	0.795561i	0.746145	+	0.665783i	\(0.231902\pi\)
							−0.991543	+	0.129777i	\(0.958574\pi\)
\(7\)	−69.5021	+	40.1271i	−1.41841	+	0.818920i	−0.996159	−	0.0875591i	\(-0.972093\pi\)
							−0.422251	+	0.906479i	\(0.638760\pi\)
\(11\)	−66.3060	+	83.1451i	−0.547983	+	0.687149i	−0.976286	−	0.216486i	\(-0.930540\pi\)
							0.428302	+	0.903636i	\(0.359112\pi\)
\(13\)	54.4014	−	50.4771i	0.321902	−	0.298681i	−0.502625	−	0.864505i	\(-0.667632\pi\)
							0.824526	+	0.565824i	\(0.191442\pi\)
\(17\)	499.214	−	153.987i	1.72738	−	0.532828i	0.737580	−	0.675260i	\(-0.235968\pi\)
							0.989805	+	0.142432i	\(0.0454922\pi\)
\(19\)	54.5780	+	362.101i	0.151185	+	1.00305i	0.927801	+	0.373075i	\(0.121697\pi\)
							−0.776616	+	0.629975i	\(0.783065\pi\)
\(23\)	51.1748	+	130.391i	0.0967387	+	0.246486i	0.971049	−	0.238882i	\(-0.0767809\pi\)
							−0.874310	+	0.485368i	\(0.838686\pi\)
\(29\)	−514.892	+	38.5858i	−0.612238	+	0.0458809i	−0.377241	−	0.926115i	\(-0.623127\pi\)
							−0.234997	+	0.971996i	\(0.575508\pi\)
\(31\)	495.679	−	337.948i	0.515795	−	0.351663i	−0.277278	−	0.960790i	\(-0.589432\pi\)
							0.793073	+	0.609127i	\(0.208480\pi\)
\(37\)	−350.877	−	202.579i	−0.256302	−	0.147976i	0.366345	−	0.930479i	\(-0.380609\pi\)
							−0.622646	+	0.782503i	\(0.713942\pi\)
\(41\)	2387.87	+	1149.94i	1.42050	+	0.684079i	0.977205	−	0.212296i	\(-0.0680940\pi\)
							0.443298	+	0.896374i	\(0.353808\pi\)
\(43\)	762.311	+	1684.54i	0.412283	+	0.911056i
							\(47\)	−998.888	−	1252.57i	−0.452190	−	0.567029i	0.502520	−	0.864566i	\(-0.332406\pi\)
−0.954710	+	0.297537i	\(0.903835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.1	✓	168
43.29	odd	42	inner	43.5.h.a.29.1	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.1	✓	168	1.1	even	1	trivial
43.5.h.a.29.1	yes	168	43.29	odd	42	inner