Embedded newform 43.9.h.a.12.22

• Label: 43.9.h.a.12.22
• Level: $43$
• Weight: $9$
• Character: 43.12
• 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			12.22
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.51590 + 19.7600i) q^{2} +(39.4634 + 57.8822i) q^{3} +(-140.290 + 175.919i) q^{4} +(-579.761 + 624.834i) q^{5} +(-768.221 + 1330.60i) q^{6} +(-1391.53 + 803.398i) q^{7} +(662.661 + 151.248i) q^{8} +(604.013 - 1539.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.51590	+	19.7600i	0.594744	+	1.23500i	0.953451	+	0.301548i	\(0.0975033\pi\)
							−0.358708	+	0.933450i	\(0.616782\pi\)
\(3\)	39.4634	+	57.8822i	0.487203	+	0.714595i	0.988591	−	0.150627i	\(-0.0481294\pi\)
							−0.501388	+	0.865223i	\(0.667177\pi\)
\(5\)	−579.761	+	624.834i	−0.927618	+	0.999734i	0.0723797	+	0.997377i	\(0.476941\pi\)
							−0.999997	+	0.00235682i	\(0.999250\pi\)
\(7\)	−1391.53	+	803.398i	−0.579561	+	0.334610i	−0.760959	−	0.648800i	\(-0.775271\pi\)
							0.181398	+	0.983410i	\(0.441938\pi\)
\(11\)	−2789.02	−	3497.33i	−0.190494	−	0.238872i	0.677408	−	0.735608i	\(-0.263103\pi\)
							−0.867902	+	0.496736i	\(0.834532\pi\)
\(13\)	3416.45	−	1053.83i	0.119619	−	0.0368977i	−0.234368	−	0.972148i	\(-0.575302\pi\)
							0.353987	+	0.935250i	\(0.384826\pi\)
\(17\)	−22791.2	+	21147.2i	−0.272880	+	0.253196i	−0.804727	−	0.593645i	\(-0.797689\pi\)
							0.531847	+	0.846840i	\(0.321498\pi\)
\(19\)	8583.57	−	3368.80i	0.0658648	−	0.0258500i	−0.332180	−	0.943216i	\(-0.607784\pi\)
							0.398045	+	0.917366i	\(0.369689\pi\)
\(23\)	−148657.	+	22406.4i	−0.531218	+	0.0800683i	−0.409174	−	0.912456i	\(-0.634183\pi\)
							−0.122044	+	0.992525i	\(0.538945\pi\)
\(29\)	−310386.	+	455253.i	−0.438844	+	0.643666i	−0.980404	−	0.196996i	\(-0.936881\pi\)
							0.541560	+	0.840662i	\(0.317834\pi\)
\(31\)	62355.7	−	832079.i	0.0675195	−	0.900986i	−0.855887	−	0.517163i	\(-0.826988\pi\)
							0.923407	−	0.383823i	\(-0.125393\pi\)
\(37\)	1.37111e6	+	791612.i	0.731587	+	0.422382i	0.819002	−	0.573790i	\(-0.194527\pi\)
							−0.0874154	+	0.996172i	\(0.527861\pi\)
\(41\)	−2.02688e6	+	976094.i	−0.717286	+	0.345427i	−0.756670	−	0.653797i	\(-0.773175\pi\)
							0.0393833	+	0.999224i	\(0.487461\pi\)
\(43\)	−1.77107e6	+	2.92430e6i	−0.518038	+	0.855358i
							\(47\)	−881778.	+	1.10572e6i	−0.180704	+	0.226596i	−0.863931	−	0.503611i	\(-0.832005\pi\)
0.683227	+	0.730206i	\(0.260576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.22	✓	336
43.18	odd	42	inner	43.9.h.a.18.22	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.22	✓	336	1.1	even	1	trivial
43.9.h.a.18.22	yes	336	43.18	odd	42	inner