Embedded newform 43.9.h.a.12.28

• Label: 43.9.h.a.12.28
• 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			12.28
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.6024 + 28.2457i) q^{2} +(14.1222 + 20.7135i) q^{3} +(-453.180 + 568.270i) q^{4} +(115.579 - 124.565i) q^{5} +(-392.970 + 680.645i) q^{6} +(-1979.18 + 1142.68i) q^{7} +(-14391.0 - 3284.66i) q^{8} +(2167.39 - 5522.42i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.6024	+	28.2457i	0.850150	+	1.76536i	0.605699	+	0.795694i	\(0.292894\pi\)
							0.244452	+	0.969661i	\(0.421392\pi\)
\(3\)	14.1222	+	20.7135i	0.174348	+	0.255722i	0.903460	−	0.428673i	\(-0.141019\pi\)
							−0.729111	+	0.684395i	\(0.760066\pi\)
\(5\)	115.579	−	124.565i	0.184927	−	0.199304i	−0.633783	−	0.773511i	\(-0.718499\pi\)
							0.818710	+	0.574207i	\(0.194689\pi\)
\(7\)	−1979.18	+	1142.68i	−0.824317	+	0.475920i	−0.851903	−	0.523700i	\(-0.824551\pi\)
							0.0275859	+	0.999619i	\(0.491218\pi\)
\(11\)	8478.82	+	10632.1i	0.579115	+	0.726187i	0.981962	−	0.189081i	\(-0.0605507\pi\)
							−0.402847	+	0.915267i	\(0.631979\pi\)
\(13\)	−8940.20	+	2757.69i	−0.313021	+	0.0965542i	−0.447285	−	0.894391i	\(-0.647609\pi\)
							0.134264	+	0.990946i	\(0.457133\pi\)
\(17\)	−3317.11	+	3077.83i	−0.0397158	+	0.0368509i	−0.699781	−	0.714357i	\(-0.746719\pi\)
							0.660065	+	0.751208i	\(0.270529\pi\)
\(19\)	−183040.	+	71838.1i	−1.40453	+	0.551239i	−0.942075	−	0.335401i	\(-0.891128\pi\)
							−0.462459	+	0.886640i	\(0.653033\pi\)
\(23\)	255586.	−	38523.4i	0.913326	−	0.137662i	0.324466	−	0.945897i	\(-0.394815\pi\)
							0.588859	+	0.808236i	\(0.299577\pi\)
\(29\)	207794.	−	304778.i	0.293793	−	0.430915i	−0.650756	−	0.759287i	\(-0.725548\pi\)
							0.944548	+	0.328372i	\(0.106500\pi\)
\(31\)	−89630.3	+	1.19603e6i	−0.0970528	+	1.29508i	0.709832	+	0.704371i	\(0.248771\pi\)
							−0.806885	+	0.590709i	\(0.798848\pi\)
\(37\)	−2.52020e6	−	1.45504e6i	−1.34471	−	0.776367i	−0.357213	−	0.934023i	\(-0.616273\pi\)
							−0.987494	+	0.157656i	\(0.949606\pi\)
\(41\)	−25143.5	+	12108.5i	−0.00889797	+	0.00428504i	−0.438327	−	0.898815i	\(-0.644429\pi\)
							0.429429	+	0.903100i	\(0.358715\pi\)
\(43\)	44543.0	+	3.41851e6i	0.0130288	+	0.999915i
							\(47\)	−2.18168e6	+	2.73574e6i	−0.447094	+	0.560638i	−0.953397	−	0.301717i	\(-0.902440\pi\)
0.506303	+	0.862355i	\(0.331012\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.28	✓	336
43.18	odd	42	inner	43.9.h.a.18.28	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.28	✓	336	1.1	even	1	trivial
43.9.h.a.18.28	yes	336	43.18	odd	42	inner