Embedded newform 43.10.g.a.10.19

• Label: 43.10.g.a.10.19
• 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			10.19
Character	\(\chi\)	\(=\)	43.10
Dual form			43.10.g.a.13.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.96392 + 8.60447i) q^{2} +(176.174 + 163.465i) q^{3} +(391.116 - 188.352i) q^{4} +(58.9610 - 8.88695i) q^{5} +(-1060.54 + 1836.92i) q^{6} +(-1467.14 - 2541.16i) q^{7} +(5206.20 + 6528.37i) q^{8} +(2845.36 + 37968.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.96392	+	8.60447i	0.0867936	+	0.380268i	0.999605	−	0.0281144i	\(-0.00895027\pi\)
							−0.912811	+	0.408382i	\(0.866093\pi\)
\(3\)	176.174	+	163.465i	1.25573	+	1.16515i	0.978925	+	0.204219i	\(0.0654655\pi\)
							0.276803	+	0.960927i	\(0.410725\pi\)
\(5\)	58.9610	−	8.88695i	0.0421891	−	0.00635898i	−0.127914	−	0.991785i	\(-0.540828\pi\)
							0.170103	+	0.985426i	\(0.445590\pi\)
\(7\)	−1467.14	−	2541.16i	−0.230957	−	0.400029i	0.727133	−	0.686496i	\(-0.240852\pi\)
							−0.958090	+	0.286468i	\(0.907519\pi\)
\(11\)	27736.5	+	13357.2i	0.571196	+	0.275073i	0.697106	−	0.716968i	\(-0.254471\pi\)
							−0.125910	+	0.992042i	\(0.540185\pi\)
\(13\)	−35043.1	+	89288.3i	−0.340296	+	0.867061i	0.653610	+	0.756832i	\(0.273254\pi\)
							−0.993906	+	0.110229i	\(0.964841\pi\)
\(17\)	197235.	+	29728.3i	0.572747	+	0.0863277i	0.429029	−	0.903291i	\(-0.358856\pi\)
							0.143718	+	0.989619i	\(0.454094\pi\)
\(19\)	6730.79	−	89816.2i	0.0118488	−	0.158111i	−0.988141	−	0.153551i	\(-0.950929\pi\)
							0.999990	−	0.00456086i	\(-0.00145177\pi\)
\(23\)	256533.	−	174901.i	0.191147	−	0.130322i	−0.463965	−	0.885853i	\(-0.653574\pi\)
							0.655112	+	0.755532i	\(0.272621\pi\)
\(29\)	759001.	−	704250.i	0.199274	−	0.184899i	−0.574223	−	0.818699i	\(-0.694696\pi\)
							0.773498	+	0.633799i	\(0.218505\pi\)
\(31\)	5.12644e6	+	1.58130e6i	0.996985	+	0.307529i	0.749962	−	0.661481i	\(-0.230072\pi\)
							0.247023	+	0.969010i	\(0.420548\pi\)
\(37\)	−3.40254e6	+	5.89338e6i	−0.298467	+	0.516959i	−0.975785	−	0.218730i	\(-0.929808\pi\)
							0.677319	+	0.735690i	\(0.263142\pi\)
\(41\)	−4.49966e6	−	1.97143e7i	−0.248686	−	1.08957i	−0.932858	−	0.360245i	\(-0.882693\pi\)
							0.684171	−	0.729321i	\(-0.260164\pi\)
\(43\)	−2.21070e7	+	3.72444e6i	−0.986104	+	0.166132i
							\(47\)	−3.87191e7	+	1.86461e7i	−1.15740	+	0.557376i	−0.911249	−	0.411855i	\(-0.864881\pi\)
−0.246153	+	0.969231i	\(0.579167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.10.19	✓	384
43.13	even	21	inner	43.10.g.a.13.19	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.19	✓	384	1.1	even	1	trivial
43.10.g.a.13.19	yes	384	43.13	even	21	inner