Embedded newform 43.10.g.a.10.10

• Label: 43.10.g.a.10.10
• Level: $43$
• Weight: $10$
• Character: 43.10
• 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			10.10
Character	\(\chi\)	\(=\)	43.10
Dual form			43.10.g.a.13.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.88303 - 21.3940i) q^{2} +(-49.8210 - 46.2272i) q^{3} +(27.4387 - 13.2138i) q^{4} +(614.410 - 92.6074i) q^{5} +(-745.705 + 1291.60i) q^{6} +(2279.68 + 3948.52i) q^{7} +(-7421.84 - 9306.69i) q^{8} +(-1125.73 - 15021.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.88303	−	21.3940i	−0.215802	−	0.945488i	−0.960542	−	0.278134i	\(-0.910284\pi\)
							0.744741	−	0.667354i	\(-0.232573\pi\)
\(3\)	−49.8210	−	46.2272i	−0.355114	−	0.329497i	0.482362	−	0.875972i	\(-0.339779\pi\)
							−0.837476	+	0.546475i	\(0.815969\pi\)
\(5\)	614.410	−	92.6074i	0.439636	−	0.0662645i	0.0745050	−	0.997221i	\(-0.476262\pi\)
							0.365131	+	0.930956i	\(0.381024\pi\)
\(7\)	2279.68	+	3948.52i	0.358866	+	0.621574i	0.987772	−	0.155908i	\(-0.0498302\pi\)
							−0.628906	+	0.777481i	\(0.716497\pi\)
\(11\)	77402.7	+	37275.2i	1.59400	+	0.767631i	0.999339	−	0.0363434i	\(-0.0115710\pi\)
							0.594663	+	0.803975i	\(0.297285\pi\)
\(13\)	57919.3	−	147576.i	0.562442	−	1.43308i	−0.312998	−	0.949754i	\(-0.601333\pi\)
							0.875440	−	0.483326i	\(-0.160571\pi\)
\(17\)	105474.	+	15897.7i	0.306285	+	0.0461650i	0.300386	−	0.953818i	\(-0.402884\pi\)
							0.00589849	+	0.999983i	\(0.498122\pi\)
\(19\)	42629.5	−	568852.i	0.0750446	−	1.00140i	−0.824867	−	0.565327i	\(-0.808750\pi\)
							0.899911	−	0.436073i	\(-0.143631\pi\)
\(23\)	70822.9	−	48286.3i	0.0527714	−	0.0359789i	−0.536647	−	0.843807i	\(-0.680309\pi\)
							0.589418	+	0.807828i	\(0.299357\pi\)
\(29\)	−414514.	+	384613.i	−0.108830	+	0.100979i	−0.732711	−	0.680540i	\(-0.761745\pi\)
							0.623881	+	0.781519i	\(0.285555\pi\)
\(31\)	−2.79920e6	−	863440.i	−0.544386	−	0.167921i	0.0103512	−	0.999946i	\(-0.496705\pi\)
							−0.554737	+	0.832026i	\(0.687181\pi\)
\(37\)	−4.79088e6	+	8.29805e6i	−0.420250	+	0.727894i	−0.995964	−	0.0897573i	\(-0.971391\pi\)
							0.575714	+	0.817651i	\(0.304724\pi\)
\(41\)	−7.74242e6	−	3.39217e7i	−0.427907	−	1.87478i	−0.481876	−	0.876239i	\(-0.660045\pi\)
							0.0539696	−	0.998543i	\(-0.482813\pi\)
\(43\)	1.20011e7	+	1.89358e7i	0.535321	+	0.844649i
							\(47\)	−1.05846e7	+	5.09728e6i	−0.316399	+	0.152370i	−0.585342	−	0.810786i	\(-0.699040\pi\)
0.268943	+	0.963156i	\(0.413326\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.10.10	✓	384
43.13	even	21	inner	43.10.g.a.13.10	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.10	✓	384	1.1	even	1	trivial
43.10.g.a.13.10	yes	384	43.13	even	21	inner