Embedded newform 43.10.g.a.9.16

• Label: 43.10.g.a.9.16
• Level: $43$
• Weight: $10$
• Character: 43.9
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(32\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.16
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.71937 - 4.66394i) q^{2} +(263.417 + 39.7037i) q^{3} +(106.012 - 464.469i) q^{4} +(1636.60 + 1115.81i) q^{5} +(-794.568 - 1376.23i) q^{6} +(3931.83 - 6810.12i) q^{7} +(-5312.37 + 2558.30i) q^{8} +(49003.4 + 15115.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.71937	−	4.66394i	−0.164374	−	0.206119i	0.692822	−	0.721109i	\(-0.256367\pi\)
							−0.857196	+	0.514990i	\(0.827796\pi\)
\(3\)	263.417	+	39.7037i	1.87758	+	0.282999i	0.985601	−	0.169085i	\(-0.0540814\pi\)
							0.891975	+	0.452085i	\(0.149319\pi\)
\(5\)	1636.60	+	1115.81i	1.17105	+	0.798411i	0.982974	−	0.183743i	\(-0.0588215\pi\)
							0.188079	+	0.982154i	\(0.439774\pi\)
\(7\)	3931.83	−	6810.12i	0.618946	−	1.07205i	−0.370732	−	0.928740i	\(-0.620893\pi\)
							0.989678	−	0.143307i	\(-0.0457735\pi\)
\(11\)	−7627.38	−	33417.7i	−0.157075	−	0.688192i	−0.990723	−	0.135896i	\(-0.956609\pi\)
							0.833648	−	0.552297i	\(-0.186248\pi\)
\(13\)	−1316.81	+	17571.6i	−0.0127873	+	0.170634i	0.987154	+	0.159773i	\(0.0510763\pi\)
							−0.999941	+	0.0108609i	\(0.996543\pi\)
\(17\)	−321648.	+	219296.i	−0.934030	+	0.636811i	−0.931717	−	0.363186i	\(-0.881689\pi\)
							−0.00231384	+	0.999997i	\(0.500737\pi\)
\(19\)	−511241.	+	157697.i	−0.899983	+	0.277608i	−0.710019	−	0.704183i	\(-0.751314\pi\)
							−0.189965	+	0.981791i	\(0.560837\pi\)
\(23\)	−1.84656e6	+	1.71335e6i	−1.37590	+	1.27665i	−0.453053	+	0.891484i	\(0.649665\pi\)
							−0.922848	+	0.385165i	\(0.874144\pi\)
\(29\)	2.09856e6	−	316307.i	0.550973	−	0.0830459i	0.132345	−	0.991204i	\(-0.457749\pi\)
							0.418628	+	0.908158i	\(0.362511\pi\)
\(31\)	2.43461e6	−	6.20328e6i	0.473480	−	1.20641i	−0.472254	−	0.881463i	\(-0.656559\pi\)
							0.945733	−	0.324944i	\(-0.105345\pi\)
\(37\)	−1.62733e6	−	2.81862e6i	−0.142747	−	0.247245i	0.785783	−	0.618502i	\(-0.212260\pi\)
							−0.928530	+	0.371257i	\(0.878927\pi\)
\(41\)	2.88454e6	+	3.61710e6i	0.159422	+	0.199909i	0.855127	−	0.518419i	\(-0.173479\pi\)
							−0.695705	+	0.718328i	\(0.744908\pi\)
\(43\)	9.91424e6	−	2.01072e7i	0.442233	−	0.896900i
							\(47\)	−5.82225e6	+	2.55090e7i	−0.174041	+	0.762522i	0.810267	+	0.586061i	\(0.199322\pi\)
−0.984308	+	0.176461i	\(0.943535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.16	✓	384
43.24	even	21	inner	43.10.g.a.24.16	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.16	✓	384	1.1	even	1	trivial
43.10.g.a.24.16	yes	384	43.24	even	21	inner