Embedded newform 43.9.h.a.3.8

• Label: 43.9.h.a.3.8
• Level: $43$
• Weight: $9$
• Character: 43.3
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.23745 + 15.0287i) q^{2} +(-0.134245 - 0.0100603i) q^{3} +(-13.8685 - 17.3905i) q^{4} +(-60.7674 + 197.003i) q^{5} +(1.12279 - 1.94472i) q^{6} +(4135.93 - 2387.88i) q^{7} +(-3801.45 + 867.655i) q^{8} +(-6487.70 - 977.864i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.23745	+	15.0287i	−0.452341	+	0.939295i	0.542709	+	0.839921i	\(0.317399\pi\)
							−0.995050	+	0.0993747i	\(0.968316\pi\)
\(3\)	−0.134245	−	0.0100603i	−0.00165735	−	0.000124201i	0.0739014	−	0.997266i	\(-0.476455\pi\)
							−0.0755587	+	0.997141i	\(0.524074\pi\)
\(5\)	−60.7674	+	197.003i	−0.0972278	+	0.315205i	−0.991100	−	0.133116i	\(-0.957502\pi\)
							0.893873	+	0.448321i	\(0.147978\pi\)
\(7\)	4135.93	−	2387.88i	1.72259	−	0.994536i	0.809097	−	0.587675i	\(-0.199957\pi\)
							0.913490	−	0.406861i	\(-0.133377\pi\)
\(11\)	10791.8	−	13532.4i	0.737092	−	0.924284i	−0.262077	−	0.965047i	\(-0.584407\pi\)
							0.999169	+	0.0407628i	\(0.0129788\pi\)
\(13\)	29912.9	−	27755.1i	1.04733	−	0.971784i	0.0477003	−	0.998862i	\(-0.484811\pi\)
							0.999633	+	0.0270781i	\(0.00862029\pi\)
\(17\)	−29504.5	+	9100.93i	−0.353258	+	0.108966i	−0.466305	−	0.884624i	\(-0.654415\pi\)
							0.113047	+	0.993590i	\(0.463939\pi\)
\(19\)	−17789.4	−	118025.i	−0.136504	−	0.905646i	−0.947307	−	0.320327i	\(-0.896207\pi\)
							0.810803	−	0.585319i	\(-0.199031\pi\)
\(23\)	154107.	+	392658.i	0.550694	+	1.40315i	0.887165	+	0.461452i	\(0.152671\pi\)
							−0.336471	+	0.941694i	\(0.609233\pi\)
\(29\)	655731.	−	49140.2i	0.927115	−	0.0694777i	0.397393	−	0.917649i	\(-0.369915\pi\)
							0.529722	+	0.848171i	\(0.322296\pi\)
\(31\)	−433276.	+	295402.i	−0.469156	+	0.319865i	−0.774718	−	0.632307i	\(-0.782108\pi\)
							0.305562	+	0.952172i	\(0.401156\pi\)
\(37\)	1.05065e6	+	606592.i	0.560596	+	0.323660i	0.753385	−	0.657580i	\(-0.228420\pi\)
							−0.192789	+	0.981240i	\(0.561753\pi\)
\(41\)	944968.	+	455072.i	0.334412	+	0.161044i	0.593552	−	0.804795i	\(-0.297725\pi\)
							−0.259141	+	0.965840i	\(0.583439\pi\)
\(43\)	−2.53362e6	−	2.29542e6i	−0.741085	−	0.671411i
							\(47\)	−1.70759e6	−	2.14125e6i	−0.349939	−	0.438810i	0.575445	−	0.817840i	\(-0.304829\pi\)
−0.925384	+	0.379031i	\(0.876257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.8	✓	336
43.29	odd	42	inner	43.9.h.a.29.8	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.8	✓	336	1.1	even	1	trivial
43.9.h.a.29.8	yes	336	43.29	odd	42	inner