Embedded newform 43.8.g.a.9.11

• Label: 43.8.g.a.9.11
• Level: $43$
• Weight: $8$
• Character: 43.9
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.11
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.56596 - 3.21761i) q^{2} +(-90.2338 - 13.6006i) q^{3} +(24.7138 - 108.278i) q^{4} +(394.907 + 269.243i) q^{5} +(187.775 + 325.235i) q^{6} +(265.754 - 460.300i) q^{7} +(-886.424 + 426.880i) q^{8} +(5867.33 + 1809.83i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.56596	−	3.21761i	−0.226801	−	0.284399i	0.655391	−	0.755290i	\(-0.272504\pi\)
							−0.882191	+	0.470891i	\(0.843933\pi\)
\(3\)	−90.2338	−	13.6006i	−1.92950	−	0.290826i	−0.932059	−	0.362306i	\(-0.881990\pi\)
							−0.997442	+	0.0714804i	\(0.977228\pi\)
\(5\)	394.907	+	269.243i	1.41286	+	0.963273i	0.998588	+	0.0531210i	\(0.0169169\pi\)
							0.414274	+	0.910152i	\(0.364035\pi\)
\(7\)	265.754	−	460.300i	0.292845	−	0.507222i	−0.681637	−	0.731691i	\(-0.738731\pi\)
							0.974481	+	0.224469i	\(0.0720648\pi\)
\(11\)	−202.771	−	888.396i	−0.0459336	−	0.201248i	0.946754	−	0.321957i	\(-0.104341\pi\)
							−0.992688	+	0.120708i	\(0.961483\pi\)
\(13\)	395.714	−	5280.44i	0.0499551	−	0.666605i	−0.914876	−	0.403735i	\(-0.867712\pi\)
							0.964831	−	0.262870i	\(-0.0846690\pi\)
\(17\)	12435.2	−	8478.20i	0.613879	−	0.418535i	−0.216068	−	0.976378i	\(-0.569323\pi\)
							0.829947	+	0.557843i	\(0.188371\pi\)
\(19\)	−35528.4	+	10959.1i	−1.18833	+	0.366552i	−0.825014	−	0.565113i	\(-0.808833\pi\)
							−0.363319	+	0.931665i	\(0.618356\pi\)
\(23\)	54044.6	−	50146.0i	0.926200	−	0.859388i	−0.0641553	−	0.997940i	\(-0.520435\pi\)
							0.990355	+	0.138552i	\(0.0442448\pi\)
\(29\)	−50496.2	+	7611.07i	−0.384473	+	0.0579499i	−0.338435	−	0.940990i	\(-0.609898\pi\)
							−0.0460377	+	0.998940i	\(0.514659\pi\)
\(31\)	49200.8	−	125362.i	0.296624	−	0.755786i	−0.702407	−	0.711775i	\(-0.747891\pi\)
							0.999031	−	0.0440102i	\(-0.0140134\pi\)
\(37\)	−192591.	−	333577.i	−0.625071	−	1.08266i	−0.988527	−	0.151044i	\(-0.951737\pi\)
							0.363456	−	0.931611i	\(-0.381597\pi\)
\(41\)	−68716.2	−	86167.4i	−0.155710	−	0.195254i	0.697857	−	0.716237i	\(-0.254137\pi\)
							−0.853567	+	0.520983i	\(0.825566\pi\)
\(43\)	−426531.	−	299816.i	−0.818109	−	0.575063i
							\(47\)	110701.	−	485015.i	0.155529	−	0.681416i	−0.835692	−	0.549199i	\(-0.814933\pi\)
0.991221	−	0.132218i	\(-0.0422098\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.11	✓	300
43.24	even	21	inner	43.8.g.a.24.11	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.11	✓	300	1.1	even	1	trivial
43.8.g.a.24.11	yes	300	43.24	even	21	inner