Embedded newform 43.9.h.a.12.4

• Label: 43.9.h.a.12.4
• Level: $43$
• Weight: $9$
• Character: 43.12
• 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			12.4
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.4624 - 23.8020i) q^{2} +(-60.3142 - 88.4646i) q^{3} +(-275.533 + 345.508i) q^{4} +(-115.926 + 124.939i) q^{5} +(-1414.29 + 2449.62i) q^{6} +(812.788 - 469.263i) q^{7} +(4788.56 + 1092.96i) q^{8} +(-1791.19 + 4563.88i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.4624	−	23.8020i	−0.716402	−	1.48762i	−0.866619	−	0.498970i	\(-0.833712\pi\)
							0.150217	−	0.988653i	\(-0.452003\pi\)
\(3\)	−60.3142	−	88.4646i	−0.744619	−	1.09216i	−0.992366	−	0.123331i	\(-0.960642\pi\)
							0.247746	−	0.968825i	\(-0.420310\pi\)
\(5\)	−115.926	+	124.939i	−0.185482	+	0.199902i	−0.818947	−	0.573869i	\(-0.805442\pi\)
							0.633465	+	0.773772i	\(0.281632\pi\)
\(7\)	812.788	−	469.263i	0.338520	−	0.195445i	−0.321097	−	0.947046i	\(-0.604052\pi\)
							0.659618	+	0.751601i	\(0.270718\pi\)
\(11\)	13419.1	+	16827.0i	0.916544	+	1.14931i	0.988397	+	0.151894i	\(0.0485372\pi\)
							−0.0718533	+	0.997415i	\(0.522891\pi\)
\(13\)	17951.5	−	5537.32i	0.628533	−	0.193877i	0.0359080	−	0.999355i	\(-0.488568\pi\)
							0.592625	+	0.805478i	\(0.298091\pi\)
\(17\)	−35068.2	+	32538.6i	−0.419873	+	0.389585i	−0.861620	−	0.507555i	\(-0.830550\pi\)
							0.441746	+	0.897140i	\(0.354359\pi\)
\(19\)	−105290.	+	41323.3i	−0.807928	+	0.317089i	−0.733111	−	0.680109i	\(-0.761933\pi\)
							−0.0748172	+	0.997197i	\(0.523837\pi\)
\(23\)	−72368.1	+	10907.7i	−0.258604	+	0.0389783i	−0.277065	−	0.960851i	\(-0.589362\pi\)
							0.0184604	+	0.999830i	\(0.494124\pi\)
\(29\)	25015.7	−	36691.3i	0.0353688	−	0.0518766i	−0.808147	−	0.588981i	\(-0.799529\pi\)
							0.843516	+	0.537104i	\(0.180482\pi\)
\(31\)	41150.3	−	549113.i	0.0445581	−	0.594587i	−0.929578	−	0.368625i	\(-0.879829\pi\)
							0.974136	−	0.225961i	\(-0.0725523\pi\)
\(37\)	745816.	+	430597.i	0.397947	+	0.229755i	0.685598	−	0.727981i	\(-0.259541\pi\)
							−0.287651	+	0.957735i	\(0.592874\pi\)
\(41\)	858394.	−	413381.i	0.303775	−	0.146290i	−0.275787	−	0.961219i	\(-0.588938\pi\)
							0.579561	+	0.814929i	\(0.303224\pi\)
\(43\)	1.44197e6	+	3.09982e6i	0.421778	+	0.906699i
							\(47\)	795251.	−	997213.i	0.162972	−	0.204360i	−0.693640	−	0.720322i	\(-0.743994\pi\)
0.856612	+	0.515962i	\(0.172565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.4	✓	336
43.18	odd	42	inner	43.9.h.a.18.4	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.4	✓	336	1.1	even	1	trivial
43.9.h.a.18.4	yes	336	43.18	odd	42	inner