Embedded newform 43.9.h.a.12.27

• Label: 43.9.h.a.12.27
• 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			12.27
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.3913 + 25.7307i) q^{2} +(-64.7466 - 94.9658i) q^{3} +(-348.913 + 437.523i) q^{4} +(-494.372 + 532.807i) q^{5} +(1641.25 - 2842.72i) q^{6} +(1032.20 - 595.938i) q^{7} +(-8453.48 - 1929.45i) q^{8} +(-2429.38 + 6189.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.3913	+	25.7307i	0.774454	+	1.60817i	0.793665	+	0.608355i	\(0.208170\pi\)
							−0.0192116	+	0.999815i	\(0.506116\pi\)
\(3\)	−64.7466	−	94.9658i	−0.799340	−	1.17242i	−0.981933	−	0.189228i	\(-0.939401\pi\)
							0.182593	−	0.983189i	\(-0.441551\pi\)
\(5\)	−494.372	+	532.807i	−0.790996	+	0.852491i	−0.991887	−	0.127123i	\(-0.959426\pi\)
							0.200891	+	0.979614i	\(0.435616\pi\)
\(7\)	1032.20	−	595.938i	0.429902	−	0.248204i	−0.269403	−	0.963028i	\(-0.586826\pi\)
							0.699305	+	0.714823i	\(0.253493\pi\)
\(11\)	−11197.8	−	14041.5i	−0.764822	−	0.959057i	0.235095	−	0.971973i	\(-0.424460\pi\)
							−0.999917	+	0.0129160i	\(0.995889\pi\)
\(13\)	35811.1	−	11046.3i	1.25385	−	0.386761i	0.404472	−	0.914550i	\(-0.367455\pi\)
							0.849375	+	0.527790i	\(0.176979\pi\)
\(17\)	65019.0	−	60328.8i	0.778475	−	0.722319i	−0.187129	−	0.982335i	\(-0.559918\pi\)
							0.965604	+	0.260016i	\(0.0837279\pi\)
\(19\)	56054.0	−	21999.6i	0.430123	−	0.168811i	−0.140396	−	0.990095i	\(-0.544837\pi\)
							0.570519	+	0.821285i	\(0.306742\pi\)
\(23\)	−257348.	+	38789.0i	−0.919622	+	0.138611i	−0.591760	−	0.806115i	\(-0.701566\pi\)
							−0.327863	+	0.944725i	\(0.606328\pi\)
\(29\)	480904.	−	705357.i	0.679934	−	0.997280i	−0.318841	−	0.947808i	\(-0.603294\pi\)
							0.998776	−	0.0494721i	\(-0.0157539\pi\)
\(31\)	83428.0	−	1.11327e6i	0.0903369	−	1.20546i	−0.749269	−	0.662266i	\(-0.769595\pi\)
							0.839606	−	0.543196i	\(-0.182786\pi\)
\(37\)	−3.03816e6	−	1.75408e6i	−1.62108	−	0.935929i	−0.986633	−	0.162961i	\(-0.947896\pi\)
							−0.634444	−	0.772969i	\(-0.718771\pi\)
\(41\)	−3.32179e6	+	1.59969e6i	−1.17554	+	0.566109i	−0.916608	−	0.399787i	\(-0.869084\pi\)
							−0.258929	+	0.965896i	\(0.583370\pi\)
\(43\)	3.36966e6	+	577556.i	0.985627	+	0.168935i
							\(47\)	4.17280e6	−	5.23253e6i	0.855138	−	1.07231i	−0.141464	−	0.989943i	\(-0.545181\pi\)
0.996602	−	0.0823659i	\(-0.0262476\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.27	✓	336
43.18	odd	42	inner	43.9.h.a.18.27	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.27	✓	336	1.1	even	1	trivial
43.9.h.a.18.27	yes	336	43.18	odd	42	inner