Embedded newform 43.9.h.a.18.12

• Label: 43.9.h.a.18.12
• 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			18.12
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.46248 + 7.18992i) q^{2} +(-10.4678 + 15.3534i) q^{3} +(119.907 + 150.359i) q^{4} +(61.3321 + 66.1003i) q^{5} +(-74.1454 - 128.424i) q^{6} +(-2369.34 - 1367.94i) q^{7} +(-3487.96 + 796.104i) q^{8} +(2270.85 + 5786.03i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.46248	+	7.18992i	−0.216405	+	0.449370i	−0.980706	−	0.195490i	\(-0.937370\pi\)
							0.764300	+	0.644860i	\(0.223084\pi\)
\(3\)	−10.4678	+	15.3534i	−0.129232	+	0.189548i	−0.885401	−	0.464828i	\(-0.846116\pi\)
							0.756169	+	0.654376i	\(0.227069\pi\)
\(5\)	61.3321	+	66.1003i	0.0981313	+	0.105760i	0.780208	−	0.625520i	\(-0.215113\pi\)
							−0.682076	+	0.731281i	\(0.738923\pi\)
\(7\)	−2369.34	−	1367.94i	−0.986813	−	0.569737i	−0.0824931	−	0.996592i	\(-0.526288\pi\)
							−0.904320	+	0.426855i	\(0.859622\pi\)
\(11\)	−2242.64	+	2812.18i	−0.153175	+	0.192076i	−0.852498	−	0.522730i	\(-0.824914\pi\)
							0.699323	+	0.714806i	\(0.253485\pi\)
\(13\)	16825.6	+	5190.02i	0.589112	+	0.181717i	0.574958	−	0.818183i	\(-0.305018\pi\)
							0.0141539	+	0.999900i	\(0.495495\pi\)
\(17\)	−46125.0	−	42797.7i	−0.552256	−	0.512419i	0.353945	−	0.935266i	\(-0.384840\pi\)
							−0.906201	+	0.422848i	\(0.861031\pi\)
\(19\)	−175762.	−	68981.3i	−1.34868	−	0.529319i	−0.422489	−	0.906368i	\(-0.638843\pi\)
							−0.926193	+	0.377049i	\(0.876939\pi\)
\(23\)	−93067.4	−	14027.7i	−0.332573	−	0.0501273i	−0.0193666	−	0.999812i	\(-0.506165\pi\)
							−0.313206	+	0.949685i	\(0.601403\pi\)
\(29\)	−403998.	−	592557.i	−0.571199	−	0.837796i	0.426577	−	0.904451i	\(-0.359719\pi\)
							−0.997776	+	0.0666558i	\(0.978767\pi\)
\(31\)	38370.3	+	512017.i	0.0415479	+	0.554418i	0.978673	+	0.205424i	\(0.0658574\pi\)
							−0.937125	+	0.348994i	\(0.886524\pi\)
\(37\)	816146.	−	471202.i	0.435473	−	0.251420i	−0.266202	−	0.963917i	\(-0.585769\pi\)
							0.701675	+	0.712497i	\(0.252436\pi\)
\(41\)	−1.80966e6	−	871488.i	−0.640417	−	0.308408i	0.0853483	−	0.996351i	\(-0.472800\pi\)
							−0.725765	+	0.687943i	\(0.758514\pi\)
\(43\)	−2.84113e6	+	1.90162e6i	−0.831031	+	0.556226i
							\(47\)	3.51888e6	+	4.41254e6i	0.721130	+	0.904269i	0.998401	−	0.0565231i	\(-0.0180015\pi\)
−0.277271	+	0.960792i	\(0.589430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.18.12	yes	336
43.12	odd	42	inner	43.9.h.a.12.12	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.12	✓	336	43.12	odd	42	inner
43.9.h.a.18.12	yes	336	1.1	even	1	trivial