Embedded newform 43.10.g.a.13.10

• Label: 43.10.g.a.13.10
• Level: $43$
• Weight: $10$
• Character: 43.13
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• 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			13.10
Character	\(\chi\)	\(=\)	43.13
Dual form			43.10.g.a.10.10

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

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