Embedded newform 43.2.g.a.31.2

• Label: 43.2.g.a.31.2
• Level: $43$
• Weight: $2$
• Character: 43.31
• Analytic conductor: $0.343$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.343356728692\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(3\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			31.2
Character	\(\chi\)	\(=\)	43.31
Dual form			43.2.g.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.118393 - 0.0570152i) q^{2} +(1.09131 - 0.744044i) q^{3} +(-1.23621 + 1.55016i) q^{4} +(-1.30537 - 1.21121i) q^{5} +(0.0867822 - 0.150311i) q^{6} +(-0.00749281 - 0.0129779i) q^{7} +(-0.116458 + 0.510236i) q^{8} +(-0.458662 + 1.16865i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 10 q^{2} - 16 q^{3} - 18 q^{4} - 17 q^{5} - 4 q^{6} + 6 q^{7} + 18 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{17}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.118393	−	0.0570152i	0.0837167	−	0.0403158i	−0.391557	−	0.920154i	\(-0.628063\pi\)
							0.475274	+	0.879838i	\(0.342349\pi\)
\(3\)	1.09131	−	0.744044i	0.630069	−	0.429574i	−0.205730	−	0.978609i	\(-0.565957\pi\)
							0.835800	+	0.549035i	\(0.185005\pi\)
\(5\)	−1.30537	−	1.21121i	−0.583780	−	0.541669i	0.332036	−	0.943267i	\(-0.392264\pi\)
							−0.915816	+	0.401598i	\(0.868455\pi\)
\(7\)	−0.00749281	−	0.0129779i	−0.00283201	−	0.00490519i	0.864606	−	0.502451i	\(-0.167568\pi\)
							−0.867438	+	0.497545i	\(0.834235\pi\)
\(11\)	−1.29669	−	1.62599i	−0.390966	−	0.490256i	0.546927	−	0.837180i	\(-0.315797\pi\)
							−0.937893	+	0.346924i	\(0.887226\pi\)
\(13\)	−1.55224	+	0.478804i	−0.430515	+	0.132796i	−0.502437	−	0.864614i	\(-0.667563\pi\)
							0.0719219	+	0.997410i	\(0.477087\pi\)
\(17\)	4.42356	−	4.10446i	1.07287	−	0.995478i	0.0728722	−	0.997341i	\(-0.476783\pi\)
							0.999998	+	0.00186302i	\(0.000593018\pi\)
\(19\)	2.78497	+	7.09598i	0.638916	+	1.62793i	0.771107	+	0.636706i	\(0.219704\pi\)
							−0.132191	+	0.991224i	\(0.542201\pi\)
\(23\)	5.24284	−	0.790231i	1.09321	−	0.164775i	0.422403	−	0.906408i	\(-0.361187\pi\)
							0.670805	+	0.741634i	\(0.265949\pi\)
\(29\)	−5.92981	−	4.04287i	−1.10114	−	0.750743i	−0.130538	−	0.991443i	\(-0.541671\pi\)
							−0.970599	+	0.240700i	\(0.922623\pi\)
\(31\)	0.178251	−	2.37860i	0.0320149	−	0.427209i	−0.958050	−	0.286600i	\(-0.907475\pi\)
							0.990065	−	0.140609i	\(-0.0449060\pi\)
\(37\)	−2.52043	+	4.36551i	−0.414356	+	0.717685i	−0.995361	−	0.0962150i	\(-0.969326\pi\)
							0.581005	+	0.813900i	\(0.302660\pi\)
\(41\)	−3.20110	+	1.54157i	−0.499928	+	0.240753i	−0.666815	−	0.745223i	\(-0.732343\pi\)
							0.166887	+	0.985976i	\(0.446629\pi\)
\(43\)	−1.84169	+	6.29350i	−0.280856	+	0.959750i
							\(47\)	−6.24911	+	7.83614i	−0.911527	+	1.14302i	0.0777509	+	0.996973i	\(0.475226\pi\)
−0.989278	+	0.146046i	\(0.953345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.2.g.a.31.2	yes	36
3.2	odd	2		387.2.y.c.289.2		36
4.3	odd	2		688.2.bg.c.289.1		36
43.5	odd	42		1849.2.a.o.1.9		18
43.25	even	21	inner	43.2.g.a.25.2	✓	36
43.38	even	21		1849.2.a.n.1.10		18
129.68	odd	42		387.2.y.c.154.2		36
172.111	odd	42		688.2.bg.c.369.1		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.25.2	✓	36	43.25	even	21	inner
43.2.g.a.31.2	yes	36	1.1	even	1	trivial
387.2.y.c.154.2		36	129.68	odd	42
387.2.y.c.289.2		36	3.2	odd	2
688.2.bg.c.289.1		36	4.3	odd	2
688.2.bg.c.369.1		36	172.111	odd	42
1849.2.a.n.1.10		18	43.38	even	21
1849.2.a.o.1.9		18	43.5	odd	42