Embedded newform 43.5.h.a.29.13

• Label: 43.5.h.a.29.13
• Level: $43$
• Weight: $5$
• Character: 43.29
• 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			29.13
Character	\(\chi\)	\(=\)	43.29
Dual form			43.5.h.a.3.13

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.13	✓	168	43.3	odd	42	inner
43.5.h.a.29.13	yes	168	1.1	even	1	trivial