Embedded newform 43.5.h.a.3.4

• Label: 43.5.h.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.02166 + 4.19802i) q^{2} +(-0.581783 - 0.0435986i) q^{3} +(-3.56045 - 4.46466i) q^{4} +(-5.63046 + 18.2535i) q^{5} +(1.35920 - 2.35420i) q^{6} +(-16.3870 + 9.46103i) q^{7} +(-46.7413 + 10.6684i) q^{8} +(-79.7587 - 12.0217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.02166	+	4.19802i	−0.505415	+	1.04951i	0.479671	+	0.877448i	\(0.340756\pi\)
							−0.985087	+	0.172058i	\(0.944958\pi\)
\(3\)	−0.581783	−	0.0435986i	−0.0646425	−	0.00484429i	0.0423696	−	0.999102i	\(-0.486509\pi\)
							−0.107012	+	0.994258i	\(0.534128\pi\)
\(5\)	−5.63046	+	18.2535i	−0.225218	+	0.730140i	0.770478	+	0.637466i	\(0.220017\pi\)
							−0.995697	+	0.0926734i	\(0.970459\pi\)
\(7\)	−16.3870	+	9.46103i	−0.334428	+	0.193082i	−0.657805	−	0.753188i	\(-0.728515\pi\)
							0.323377	+	0.946270i	\(0.395182\pi\)
\(11\)	141.410	−	177.323i	1.16868	−	1.46548i	0.311662	−	0.950193i	\(-0.399114\pi\)
							0.857018	−	0.515286i	\(-0.172314\pi\)
\(13\)	−146.541	+	135.970i	−0.867106	+	0.804556i	−0.982012	−	0.188820i	\(-0.939534\pi\)
							0.114906	+	0.993376i	\(0.463343\pi\)
\(17\)	−325.668	+	100.455i	−1.12688	+	0.347596i	−0.801504	−	0.597989i	\(-0.795967\pi\)
							−0.325375	+	0.945585i	\(0.605491\pi\)
\(19\)	97.3714	+	646.017i	0.269727	+	1.78952i	0.555804	+	0.831314i	\(0.312411\pi\)
							−0.286077	+	0.958207i	\(0.592351\pi\)
\(23\)	102.744	+	261.788i	0.194223	+	0.494873i	0.994497	−	0.104761i	\(-0.0334079\pi\)
							−0.800274	+	0.599634i	\(0.795313\pi\)
\(29\)	1203.65	−	90.2007i	1.43121	−	0.107254i	0.663558	−	0.748125i	\(-0.269046\pi\)
							0.767649	+	0.640871i	\(0.221426\pi\)
\(31\)	−899.830	+	613.494i	−0.936347	+	0.638391i	−0.932333	−	0.361601i	\(-0.882230\pi\)
							−0.00401446	+	0.999992i	\(0.501278\pi\)
\(37\)	−405.138	−	233.907i	−0.295938	−	0.170860i	0.344679	−	0.938721i	\(-0.387988\pi\)
							−0.640616	+	0.767861i	\(0.721321\pi\)
\(41\)	1143.44	+	550.654i	0.680217	+	0.327575i	0.741890	−	0.670521i	\(-0.233929\pi\)
							−0.0616735	+	0.998096i	\(0.519644\pi\)
\(43\)	1093.46	+	1491.02i	0.591382	+	0.806392i
							\(47\)	−1834.64	−	2300.57i	−0.830531	−	1.04145i	−0.998450	−	0.0556526i	\(-0.982276\pi\)
0.167919	−	0.985801i	\(-0.446295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.4	✓	168
43.29	odd	42	inner	43.5.h.a.29.4	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.4	✓	168	1.1	even	1	trivial
43.5.h.a.29.4	yes	168	43.29	odd	42	inner