Embedded newform 43.9.h.a.3.2

• Label: 43.9.h.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.1501 + 25.2300i) q^{2} +(111.026 + 8.32026i) q^{3} +(-329.314 - 412.947i) q^{4} +(-217.435 + 704.908i) q^{5} +(-1558.90 + 2700.10i) q^{6} +(939.408 - 542.367i) q^{7} +(7430.79 - 1696.03i) q^{8} +(5769.85 + 869.665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.1501	+	25.2300i	−0.759383	+	1.57688i	0.0563254	+	0.998412i	\(0.482062\pi\)
							−0.815709	+	0.578463i	\(0.803653\pi\)
\(3\)	111.026	+	8.32026i	1.37069	+	0.102719i	0.739631	−	0.673013i	\(-0.235000\pi\)
							0.631062	+	0.775732i	\(0.282619\pi\)
\(5\)	−217.435	+	704.908i	−0.347897	+	1.12785i	0.597483	+	0.801882i	\(0.296168\pi\)
							−0.945380	+	0.325972i	\(0.894309\pi\)
\(7\)	939.408	−	542.367i	0.391257	−	0.225892i	−0.291448	−	0.956587i	\(-0.594137\pi\)
							0.682705	+	0.730695i	\(0.260804\pi\)
\(11\)	−7087.43	+	8887.36i	−0.484081	+	0.607019i	−0.962556	−	0.271083i	\(-0.912618\pi\)
							0.478475	+	0.878101i	\(0.341190\pi\)
\(13\)	−20811.5	+	19310.3i	−0.728668	+	0.676106i	−0.954551	−	0.298049i	\(-0.903664\pi\)
							0.225882	+	0.974155i	\(0.427474\pi\)
\(17\)	−138487.	+	42717.5i	−1.65811	+	0.511458i	−0.975625	−	0.219444i	\(-0.929576\pi\)
							−0.682482	+	0.730903i	\(0.739099\pi\)
\(19\)	−32821.3	−	217755.i	−0.251850	−	1.67091i	−0.654884	−	0.755730i	\(-0.727282\pi\)
							0.403034	−	0.915185i	\(-0.367956\pi\)
\(23\)	94732.3	+	241374.i	0.338522	+	0.862540i	0.994205	+	0.107503i	\(0.0342856\pi\)
							−0.655683	+	0.755036i	\(0.727619\pi\)
\(29\)	587879.	−	44055.4i	0.831181	−	0.0622884i	0.347660	−	0.937621i	\(-0.386976\pi\)
							0.483522	+	0.875332i	\(0.339357\pi\)
\(31\)	128785.	−	87804.4i	0.139450	−	0.0950756i	−0.491585	−	0.870829i	\(-0.663582\pi\)
							0.631036	+	0.775754i	\(0.282630\pi\)
\(37\)	−1.25700e6	−	725732.i	−0.670702	−	0.387230i	0.125640	−	0.992076i	\(-0.459901\pi\)
							−0.796343	+	0.604846i	\(0.793235\pi\)
\(41\)	939517.	+	452447.i	0.332483	+	0.160115i	0.592675	−	0.805441i	\(-0.298072\pi\)
							−0.260193	+	0.965557i	\(0.583786\pi\)
\(43\)	3.40148e6	+	343691.i	0.994934	+	0.100530i
							\(47\)	−3.16046e6	−	3.96310e6i	−0.647678	−	0.812163i	0.344261	−	0.938874i	\(-0.388129\pi\)
−0.991940	+	0.126711i	\(0.959558\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.2	✓	336
43.29	odd	42	inner	43.9.h.a.29.2	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.2	✓	336	1.1	even	1	trivial
43.9.h.a.29.2	yes	336	43.29	odd	42	inner