Embedded newform 43.9.h.a.3.9

• Label: 43.9.h.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.93493 + 14.4005i) q^{2} +(26.0262 + 1.95039i) q^{3} +(0.331155 + 0.415255i) q^{4} +(249.037 - 807.358i) q^{5} +(-208.577 + 361.265i) q^{6} +(1041.77 - 601.469i) q^{7} +(-3997.44 + 912.389i) q^{8} +(-5814.16 - 876.344i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.93493	+	14.4005i	−0.433433	+	0.900034i	0.563815	+	0.825901i	\(0.309333\pi\)
							−0.997248	+	0.0741326i	\(0.976381\pi\)
\(3\)	26.0262	+	1.95039i	0.321311	+	0.0240789i	0.234409	−	0.972138i	\(-0.424684\pi\)
							0.0869015	+	0.996217i	\(0.472303\pi\)
\(5\)	249.037	−	807.358i	0.398459	−	1.29177i	−0.505008	−	0.863115i	\(-0.668510\pi\)
							0.903467	−	0.428658i	\(-0.141013\pi\)
\(7\)	1041.77	−	601.469i	0.433892	−	0.250508i	−0.267111	−	0.963666i	\(-0.586069\pi\)
							0.701003	+	0.713158i	\(0.252736\pi\)
\(11\)	−1452.25	+	1821.07i	−0.0991909	+	0.124381i	−0.828950	−	0.559323i	\(-0.811061\pi\)
							0.729759	+	0.683705i	\(0.239632\pi\)
\(13\)	−38000.5	+	35259.3i	−1.33050	+	1.23453i	−0.379556	+	0.925169i	\(0.623923\pi\)
							−0.950946	+	0.309356i	\(0.899886\pi\)
\(17\)	−114955.	+	35458.8i	−1.37636	+	0.424550i	−0.892677	−	0.450696i	\(-0.851176\pi\)
							−0.483679	+	0.875246i	\(0.660700\pi\)
\(19\)	12827.6	+	85105.6i	0.0984309	+	0.653046i	0.982580	+	0.185841i	\(0.0595009\pi\)
							−0.884149	+	0.467205i	\(0.845261\pi\)
\(23\)	−40998.4	−	104462.i	−0.146506	−	0.373291i	0.838717	−	0.544567i	\(-0.183306\pi\)
							−0.985223	+	0.171276i	\(0.945211\pi\)
\(29\)	−1.01796e6	+	76285.4i	−1.43926	+	0.107857i	−0.771361	−	0.636398i	\(-0.780424\pi\)
							−0.667895	+	0.744256i	\(0.732804\pi\)
\(31\)	1.14195e6	−	778568.i	1.23652	−	0.843043i	0.244562	−	0.969634i	\(-0.421356\pi\)
							0.991955	+	0.126591i	\(0.0404034\pi\)
\(37\)	−1.30669e6	−	754418.i	−0.697213	−	0.402536i	0.109096	−	0.994031i	\(-0.465205\pi\)
							−0.806309	+	0.591495i	\(0.798538\pi\)
\(41\)	415476.	+	200083.i	0.147032	+	0.0708067i	0.505953	−	0.862561i	\(-0.331141\pi\)
							−0.358921	+	0.933368i	\(0.616855\pi\)
\(43\)	2.83156e6	+	1.91585e6i	0.828231	+	0.560387i
							\(47\)	2.62338e6	+	3.28961e6i	0.537612	+	0.674144i	0.974244	−	0.225496i	\(-0.0724002\pi\)
−0.436632	+	0.899640i	\(0.643829\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.9	✓	336
43.29	odd	42	inner	43.9.h.a.29.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.9	✓	336	1.1	even	1	trivial
43.9.h.a.29.9	yes	336	43.29	odd	42	inner