Embedded newform 43.4.g.a.9.4

• Label: 43.4.g.a.9.4
• Level: $43$
• Weight: $4$
• Character: 43.9
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.4
Character	\(\chi\)	\(=\)	43.9
Dual form			43.4.g.a.24.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62068 - 2.03226i) q^{2} +(-6.93204 - 1.04484i) q^{3} +(0.276661 - 1.21213i) q^{4} +(6.69761 + 4.56636i) q^{5} +(9.11121 + 15.7811i) q^{6} +(-16.5418 + 28.6512i) q^{7} +(-21.6473 + 10.4248i) q^{8} +(21.1610 + 6.52730i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62068	−	2.03226i	−0.572996	−	0.718514i	0.407905	−	0.913025i	\(-0.366260\pi\)
							−0.980900	+	0.194510i	\(0.937688\pi\)
\(3\)	−6.93204	−	1.04484i	−1.33407	−	0.201079i	−0.557011	−	0.830505i	\(-0.688052\pi\)
							−0.777060	+	0.629426i	\(0.783290\pi\)
\(5\)	6.69761	+	4.56636i	0.599053	+	0.408427i	0.824528	−	0.565822i	\(-0.191441\pi\)
							−0.225475	+	0.974249i	\(0.572393\pi\)
\(7\)	−16.5418	+	28.6512i	−0.893173	+	1.54702i	−0.0571241	+	0.998367i	\(0.518193\pi\)
							−0.836049	+	0.548654i	\(0.815140\pi\)
\(11\)	0.540068	+	2.36619i	0.0148033	+	0.0648576i	0.981790	−	0.189969i	\(-0.0608387\pi\)
							−0.966987	+	0.254827i	\(0.917982\pi\)
\(13\)	4.87033	−	64.9901i	0.103907	−	1.38654i	−0.664840	−	0.746985i	\(-0.731500\pi\)
							0.768747	−	0.639553i	\(-0.220881\pi\)
\(17\)	−78.9585	+	53.8330i	−1.12649	+	0.768025i	−0.975388	−	0.220494i	\(-0.929233\pi\)
							−0.151097	+	0.988519i	\(0.548281\pi\)
\(19\)	−98.5398	+	30.3955i	−1.18982	+	0.367011i	−0.825575	−	0.564292i	\(-0.809149\pi\)
							−0.364245	+	0.931303i	\(0.618673\pi\)
\(23\)	−71.4375	+	66.2843i	−0.647641	+	0.600923i	−0.934032	−	0.357190i	\(-0.883735\pi\)
							0.286391	+	0.958113i	\(0.407544\pi\)
\(29\)	5.06972	−	0.764137i	0.0324628	−	0.00489299i	−0.132791	−	0.991144i	\(-0.542394\pi\)
							0.165254	+	0.986251i	\(0.447156\pi\)
\(31\)	41.3729	−	105.416i	0.239703	−	0.610753i	−0.759507	−	0.650499i	\(-0.774560\pi\)
							0.999210	+	0.0397463i	\(0.0126550\pi\)
\(37\)	−145.884	−	252.679i	−0.648195	−	1.12271i	−0.983554	−	0.180617i	\(-0.942191\pi\)
							0.335358	−	0.942091i	\(-0.391143\pi\)
\(41\)	136.332	+	170.955i	0.519304	+	0.651187i	0.970461	−	0.241258i	\(-0.0775600\pi\)
							−0.451157	+	0.892445i	\(0.648989\pi\)
\(43\)	259.101	−	111.238i	0.918895	−	0.394502i
							\(47\)	−26.9107	+	117.903i	−0.0835176	+	0.365915i	−0.999366	−	0.0356103i	\(-0.988663\pi\)
0.915848	+	0.401525i	\(0.131520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.9.4	✓	120
43.14	even	21		1849.4.a.l.1.16		60
43.24	even	21	inner	43.4.g.a.24.4	yes	120
43.29	odd	42		1849.4.a.k.1.45		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.9.4	✓	120	1.1	even	1	trivial
43.4.g.a.24.4	yes	120	43.24	even	21	inner
1849.4.a.k.1.45		60	43.29	odd	42
1849.4.a.l.1.16		60	43.14	even	21