Embedded newform 43.2.g.a.13.2

• Label: 43.2.g.a.13.2
• Level: $43$
• Weight: $2$
• Character: 43.13
• 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			13.2
Character	\(\chi\)	\(=\)	43.13
Dual form			43.2.g.a.10.2

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.10.2	✓	36	43.10	even	21	inner
43.2.g.a.13.2	yes	36	1.1	even	1	trivial
387.2.y.c.10.2		36	129.53	odd	42
387.2.y.c.271.2		36	3.2	odd	2
688.2.bg.c.225.1		36	172.139	odd	42
688.2.bg.c.529.1		36	4.3	odd	2
1849.2.a.n.1.9		18	43.15	even	21
1849.2.a.o.1.10		18	43.28	odd	42