Embedded newform 43.9.h.a.12.23

• Label: 43.9.h.a.12.23
• 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			12.23
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.78211 + 20.3128i) q^{2} +(-69.5391 - 101.995i) q^{3} +(-157.305 + 197.254i) q^{4} +(400.306 - 431.427i) q^{5} +(1391.56 - 2410.26i) q^{6} +(-2344.25 + 1353.45i) q^{7} +(81.3730 + 18.5729i) q^{8} +(-3170.32 + 8077.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.78211	+	20.3128i	0.611382	+	1.26955i	0.945071	+	0.326864i	\(0.105992\pi\)
							−0.333689	+	0.942683i	\(0.608294\pi\)
\(3\)	−69.5391	−	101.995i	−0.858507	−	1.25920i	−0.964300	−	0.264811i	\(-0.914690\pi\)
							0.105793	−	0.994388i	\(-0.466262\pi\)
\(5\)	400.306	−	431.427i	0.640490	−	0.690284i	−0.326472	−	0.945207i	\(-0.605860\pi\)
							0.966961	+	0.254923i	\(0.0820503\pi\)
\(7\)	−2344.25	+	1353.45i	−0.976364	+	0.563704i	−0.901170	−	0.433465i	\(-0.857291\pi\)
							−0.0751937	+	0.997169i	\(0.523958\pi\)
\(11\)	2310.60	+	2897.40i	0.157817	+	0.197896i	0.854453	−	0.519528i	\(-0.173892\pi\)
							−0.696636	+	0.717424i	\(0.745321\pi\)
\(13\)	−33344.9	+	10285.5i	−1.16750	+	0.360125i	−0.817111	−	0.576480i	\(-0.804426\pi\)
							−0.350386	+	0.936605i	\(0.613950\pi\)
\(17\)	−57146.4	+	53024.1i	−0.684216	+	0.634860i	−0.943667	−	0.330897i	\(-0.892649\pi\)
							0.259451	+	0.965756i	\(0.416458\pi\)
\(19\)	−22321.5	+	8760.54i	−0.171281	+	0.0672227i	−0.449432	−	0.893315i	\(-0.648374\pi\)
							0.278151	+	0.960537i	\(0.410278\pi\)
\(23\)	−258026.	+	38891.2i	−0.922045	+	0.138976i	−0.592875	−	0.805295i	\(-0.702007\pi\)
							−0.329170	+	0.944271i	\(0.606769\pi\)
\(29\)	342292.	−	502050.i	0.483955	−	0.709832i	−0.504149	−	0.863617i	\(-0.668194\pi\)
							0.988104	+	0.153785i	\(0.0491462\pi\)
\(31\)	22491.3	−	300126.i	0.0243539	−	0.324980i	−0.971699	−	0.236221i	\(-0.924091\pi\)
							0.996053	−	0.0887588i	\(-0.0282900\pi\)
\(37\)	2.63243e6	+	1.51984e6i	1.40459	+	0.810942i	0.994860	−	0.101264i	\(-0.0322885\pi\)
							0.409733	+	0.912205i	\(0.365622\pi\)
\(41\)	−3.29393e6	+	1.58627e6i	−1.16568	+	0.561362i	−0.913707	−	0.406373i	\(-0.866793\pi\)
							−0.251972	+	0.967734i	\(0.581079\pi\)
\(43\)	−3.39716e6	+	384033.i	−0.993671	+	0.112330i
							\(47\)	−2.81246e6	+	3.52672e6i	−0.576362	+	0.722735i	−0.981488	−	0.191526i	\(-0.938656\pi\)
0.405125	+	0.914261i	\(0.367228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.23	✓	336
43.18	odd	42	inner	43.9.h.a.18.23	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.23	✓	336	1.1	even	1	trivial
43.9.h.a.18.23	yes	336	43.18	odd	42	inner