Embedded newform 43.9.h.a.3.16

• Label: 43.9.h.a.3.16
• Level: $43$
• Weight: $9$
• Character: 43.3
• 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			3.16
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.44533 - 3.00125i) q^{2} +(-54.6425 - 4.09489i) q^{3} +(152.695 + 191.473i) q^{4} +(-330.560 + 1071.65i) q^{5} +(-91.2660 + 158.077i) q^{6} +(-764.563 + 441.421i) q^{7} +(1626.74 - 371.294i) q^{8} +(-3518.69 - 530.357i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.44533	−	3.00125i	0.0903329	−	0.187578i	−0.850907	−	0.525316i	\(-0.823947\pi\)
							0.941240	+	0.337737i	\(0.109662\pi\)
\(3\)	−54.6425	−	4.09489i	−0.674598	−	0.0505542i	−0.266974	−	0.963704i	\(-0.586024\pi\)
							−0.407625	+	0.913150i	\(0.633643\pi\)
\(5\)	−330.560	+	1071.65i	−0.528896	+	1.71464i	0.157596	+	0.987504i	\(0.449626\pi\)
							−0.686493	+	0.727137i	\(0.740851\pi\)
\(7\)	−764.563	+	441.421i	−0.318435	+	0.183849i	−0.650695	−	0.759339i	\(-0.725522\pi\)
							0.332260	+	0.943188i	\(0.392189\pi\)
\(11\)	10630.7	−	13330.5i	0.726094	−	0.910494i	−0.272571	−	0.962136i	\(-0.587874\pi\)
							0.998666	+	0.0516420i	\(0.0164455\pi\)
\(13\)	4143.98	−	3845.05i	0.145092	−	0.134626i	−0.604288	−	0.796766i	\(-0.706542\pi\)
							0.749380	+	0.662140i	\(0.230352\pi\)
\(17\)	−93881.8	+	28958.7i	−1.12405	+	0.346724i	−0.800408	−	0.599456i	\(-0.795384\pi\)
							−0.323643	+	0.946179i	\(0.604908\pi\)
\(19\)	−11203.8	−	74332.5i	−0.0859710	−	0.570380i	−0.989798	−	0.142481i	\(-0.954492\pi\)
							0.903827	−	0.427899i	\(-0.140746\pi\)
\(23\)	−39144.9	−	99739.5i	−0.139883	−	0.356415i	0.843715	−	0.536791i	\(-0.180364\pi\)
							−0.983598	+	0.180376i	\(0.942268\pi\)
\(29\)	90696.5	−	6796.76i	0.128233	−	0.00960970i	−0.0104592	−	0.999945i	\(-0.503329\pi\)
							0.138692	+	0.990336i	\(0.455710\pi\)
\(31\)	−112252.	+	76532.0i	−0.121548	+	0.0828698i	−0.622568	−	0.782566i	\(-0.713911\pi\)
							0.501020	+	0.865436i	\(0.332958\pi\)
\(37\)	−604377.	−	348937.i	−0.322479	−	0.186183i	0.330018	−	0.943975i	\(-0.392945\pi\)
							−0.652497	+	0.757791i	\(0.726278\pi\)
\(41\)	−4.75395e6	−	2.28938e6i	−1.68236	−	0.810183i	−0.996603	−	0.0823547i	\(-0.973756\pi\)
							−0.685759	−	0.727828i	\(-0.740530\pi\)
\(43\)	2.11827e6	+	2.68349e6i	0.619594	+	0.784923i
							\(47\)	−3.94983e6	−	4.95293e6i	−0.809444	−	1.01501i	−0.999448	−	0.0332278i	\(-0.989421\pi\)
0.190003	−	0.981783i	\(-0.439150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.16	✓	336
43.29	odd	42	inner	43.9.h.a.29.16	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.16	✓	336	1.1	even	1	trivial
43.9.h.a.29.16	yes	336	43.29	odd	42	inner