Embedded newform 43.5.h.a.3.13

• Label: 43.5.h.a.3.13
• Level: $43$
• Weight: $5$
• Character: 43.3
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.13
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.55695 - 5.30956i) q^{2} +(11.5931 + 0.868784i) q^{3} +(-11.6776 - 14.6433i) q^{4} +(0.622954 - 2.01957i) q^{5} +(34.2559 - 59.3329i) q^{6} +(-3.58135 + 2.06769i) q^{7} +(-15.6817 + 3.57925i) q^{8} +(53.5501 + 8.07138i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.55695	−	5.30956i	0.639238	−	1.32739i	−0.289687	−	0.957121i	\(-0.593551\pi\)
							0.928925	−	0.370269i	\(-0.120734\pi\)
\(3\)	11.5931	+	0.868784i	1.28812	+	0.0965315i	0.701114	−	0.713049i	\(-0.252686\pi\)
							0.587009	+	0.809580i	\(0.300305\pi\)
\(5\)	0.622954	−	2.01957i	0.0249182	−	0.0807827i	−0.942260	−	0.334883i	\(-0.891303\pi\)
							0.967178	+	0.254100i	\(0.0817794\pi\)
\(7\)	−3.58135	+	2.06769i	−0.0730887	+	0.0421978i	−0.536099	−	0.844155i	\(-0.680103\pi\)
							0.463010	+	0.886353i	\(0.346769\pi\)
\(11\)	−55.1945	+	69.2117i	−0.456153	+	0.571998i	−0.955720	−	0.294277i	\(-0.904921\pi\)
							0.499567	+	0.866275i	\(0.333492\pi\)
\(13\)	−115.507	+	107.175i	−0.683476	+	0.634173i	−0.943478	−	0.331435i	\(-0.892467\pi\)
							0.260002	+	0.965608i	\(0.416277\pi\)
\(17\)	−60.6163	+	18.6976i	−0.209745	+	0.0646977i	−0.397847	−	0.917452i	\(-0.630243\pi\)
							0.188102	+	0.982149i	\(0.439766\pi\)
\(19\)	−10.3219	−	68.4814i	−0.0285925	−	0.189699i	0.969975	−	0.243205i	\(-0.0781986\pi\)
							−0.998568	+	0.0535055i	\(0.982961\pi\)
\(23\)	249.120	+	634.748i	0.470927	+	1.19990i	0.947211	+	0.320610i	\(0.103888\pi\)
							−0.476284	+	0.879291i	\(0.658017\pi\)
\(29\)	339.526	−	25.4439i	0.403717	−	0.0302544i	0.128675	−	0.991687i	\(-0.458928\pi\)
							0.275041	+	0.961432i	\(0.411308\pi\)
\(31\)	953.715	−	650.232i	0.992420	−	0.676620i	0.0457926	−	0.998951i	\(-0.485419\pi\)
							0.946627	+	0.322331i	\(0.104466\pi\)
\(37\)	−1168.85	−	674.835i	−0.853797	−	0.492940i	0.00813338	−	0.999967i	\(-0.497411\pi\)
							−0.861930	+	0.507027i	\(0.830744\pi\)
\(41\)	−1687.28	−	812.554i	−1.00374	−	0.483375i	−0.141533	−	0.989934i	\(-0.545203\pi\)
							−0.862206	+	0.506558i	\(0.830917\pi\)
\(43\)	1775.71	−	515.428i	0.960361	−	0.278760i
							\(47\)	−209.553	−	262.771i	−0.0948631	−	0.118955i	0.732134	−	0.681160i	\(-0.238524\pi\)
−0.826997	+	0.562206i	\(0.809953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.13	✓	168
43.29	odd	42	inner	43.5.h.a.29.13	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.13	✓	168	1.1	even	1	trivial
43.5.h.a.29.13	yes	168	43.29	odd	42	inner