Embedded newform 43.2.g.a.10.2

• Label: 43.2.g.a.10.2
• Level: $43$
• Weight: $2$
• Character: 43.10
• 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			10.2
Character	\(\chi\)	\(=\)	43.10
Dual form			43.2.g.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.188565 + 0.826155i) q^{2} +(-0.0129300 - 0.0119973i) q^{3} +(1.15496 - 0.556200i) q^{4} +(-3.39819 + 0.512194i) q^{5} +(0.00747346 - 0.0129444i) q^{6} +(-0.134521 - 0.232998i) q^{7} +(1.73398 + 2.17435i) q^{8} +(-0.224167 - 2.99130i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 10 q^{2} - 16 q^{3} - 18 q^{4} - 17 q^{5} - 4 q^{6} + 6 q^{7} + 18 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.188565	+	0.826155i	0.133335	+	0.584180i	0.996812	+	0.0797907i	\(0.0254252\pi\)
							−0.863476	+	0.504389i	\(0.831718\pi\)
\(3\)	−0.0129300	−	0.0119973i	−0.00746512	−	0.00692662i	0.676431	−	0.736506i	\(-0.263526\pi\)
							−0.683896	+	0.729579i	\(0.739716\pi\)
\(5\)	−3.39819	+	0.512194i	−1.51972	+	0.229060i	−0.855233	−	0.518243i	\(-0.826586\pi\)
							−0.664483	+	0.747304i	\(0.731348\pi\)
\(7\)	−0.134521	−	0.232998i	−0.0508443	−	0.0880650i	0.839483	−	0.543386i	\(-0.182858\pi\)
							−0.890327	+	0.455321i	\(0.849525\pi\)
\(11\)	−2.96782	−	1.42923i	−0.894833	−	0.430929i	−0.0708129	−	0.997490i	\(-0.522559\pi\)
							−0.824020	+	0.566561i	\(0.808274\pi\)
\(13\)	−0.736485	+	1.87653i	−0.204264	+	0.520457i	−0.995892	−	0.0905467i	\(-0.971139\pi\)
							0.791628	+	0.611003i	\(0.209234\pi\)
\(17\)	6.37398	+	0.960723i	1.54592	+	0.233010i	0.865911	−	0.500197i	\(-0.166739\pi\)
							0.680006	+	0.733207i	\(0.261977\pi\)
\(19\)	−0.449267	+	5.99506i	−0.103069	+	1.37536i	0.670623	+	0.741798i	\(0.266027\pi\)
							−0.773692	+	0.633562i	\(0.781592\pi\)
\(23\)	−1.83546	+	1.25139i	−0.382719	+	0.260934i	−0.739363	−	0.673307i	\(-0.764873\pi\)
							0.356644	+	0.934240i	\(0.383921\pi\)
\(29\)	1.75989	−	1.63294i	0.326803	−	0.303229i	−0.499658	−	0.866223i	\(-0.666541\pi\)
							0.826461	+	0.562994i	\(0.190351\pi\)
\(31\)	−4.93996	−	1.52378i	−0.887243	−	0.273678i	−0.182555	−	0.983196i	\(-0.558437\pi\)
							−0.704688	+	0.709517i	\(0.748913\pi\)
\(37\)	2.63757	−	4.56841i	0.433615	−	0.751042i	−0.563567	−	0.826070i	\(-0.690571\pi\)
							0.997181	+	0.0750281i	\(0.0239046\pi\)
\(41\)	−0.643386	−	2.81886i	−0.100480	−	0.440232i	−0.999994	−	0.00334069i	\(-0.998937\pi\)
							0.899514	−	0.436891i	\(-0.143921\pi\)
\(43\)	−4.01778	−	5.18242i	−0.612705	−	0.790311i
							\(47\)	−5.31835	+	2.56118i	−0.775761	+	0.373587i	−0.779497	−	0.626406i	\(-0.784525\pi\)
0.00373559	+	0.999993i	\(0.498811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.2.g.a.10.2	✓	36
3.2	odd	2		387.2.y.c.10.2		36
4.3	odd	2		688.2.bg.c.225.1		36
43.13	even	21	inner	43.2.g.a.13.2	yes	36
43.20	odd	42		1849.2.a.o.1.10		18
43.23	even	21		1849.2.a.n.1.9		18
129.56	odd	42		387.2.y.c.271.2		36
172.99	odd	42		688.2.bg.c.529.1		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.10.2	✓	36	1.1	even	1	trivial
43.2.g.a.13.2	yes	36	43.13	even	21	inner
387.2.y.c.10.2		36	3.2	odd	2
387.2.y.c.271.2		36	129.56	odd	42
688.2.bg.c.225.1		36	4.3	odd	2
688.2.bg.c.529.1		36	172.99	odd	42
1849.2.a.n.1.9		18	43.23	even	21
1849.2.a.o.1.10		18	43.20	odd	42