Embedded newform 43.9.h.a.12.19

• Label: 43.9.h.a.12.19
• 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			12.19
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.47419 + 9.29075i) q^{2} +(53.8421 + 78.9719i) q^{3} +(93.3137 - 117.012i) q^{4} +(-87.0303 + 93.7963i) q^{5} +(-492.808 + 853.569i) q^{6} +(2152.57 - 1242.79i) q^{7} +(4078.30 + 930.846i) q^{8} +(-940.580 + 2396.56i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.47419	+	9.29075i	0.279637	+	0.580672i	0.992725	−	0.120402i	\(-0.0384184\pi\)
							−0.713088	+	0.701074i	\(0.752704\pi\)
\(3\)	53.8421	+	78.9719i	0.664717	+	0.974961i	0.999467	+	0.0326540i	\(0.0103959\pi\)
							−0.334749	+	0.942307i	\(0.608652\pi\)
\(5\)	−87.0303	+	93.7963i	−0.139248	+	0.150074i	−0.798825	−	0.601563i	\(-0.794545\pi\)
							0.659577	+	0.751637i	\(0.270735\pi\)
\(7\)	2152.57	−	1242.79i	0.896532	−	0.517613i	0.0204589	−	0.999791i	\(-0.493487\pi\)
							0.876074	+	0.482177i	\(0.160154\pi\)
\(11\)	11902.9	+	14925.8i	0.812987	+	1.01945i	0.999317	+	0.0369664i	\(0.0117694\pi\)
							−0.186329	+	0.982487i	\(0.559659\pi\)
\(13\)	2240.82	−	691.200i	0.0784572	−	0.0242008i	−0.255278	−	0.966868i	\(-0.582167\pi\)
							0.333735	+	0.942667i	\(0.391691\pi\)
\(17\)	88865.8	−	82455.4i	1.06399	−	0.987241i	0.0640502	−	0.997947i	\(-0.479598\pi\)
							0.999943	+	0.0107055i	\(0.00340773\pi\)
\(19\)	−196043.	+	76941.1i	−1.50431	+	0.590397i	−0.967402	−	0.253246i	\(-0.918502\pi\)
							−0.536905	+	0.843643i	\(0.680407\pi\)
\(23\)	−375322.	+	56570.7i	−1.34120	+	0.202153i	−0.780130	−	0.625617i	\(-0.784847\pi\)
							−0.561067	+	0.827770i	\(0.689609\pi\)
\(29\)	131790.	−	193300.i	0.186333	−	0.273300i	−0.721694	−	0.692213i	\(-0.756636\pi\)
							0.908027	+	0.418912i	\(0.137588\pi\)
\(31\)	−73201.4	+	976805.i	−0.0792634	+	1.05770i	0.805788	+	0.592204i	\(0.201742\pi\)
							−0.885052	+	0.465493i	\(0.845877\pi\)
\(37\)	370255.	+	213767.i	0.197558	+	0.114060i	0.595516	−	0.803344i	\(-0.296948\pi\)
							−0.397958	+	0.917404i	\(0.630281\pi\)
\(41\)	−3.90490e6	+	1.88050e6i	−1.38189	+	0.665485i	−0.969402	−	0.245477i	\(-0.921055\pi\)
							−0.412491	+	0.910962i	\(0.635341\pi\)
\(43\)	377954.	−	3.39785e6i	0.110552	−	0.993870i
							\(47\)	4.53755e6	−	5.68991e6i	0.929887	−	1.16604i	−0.0559661	−	0.998433i	\(-0.517824\pi\)
0.985854	−	0.167609i	\(-0.0536047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.19	✓	336
43.18	odd	42	inner	43.9.h.a.18.19	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.19	✓	336	1.1	even	1	trivial
43.9.h.a.18.19	yes	336	43.18	odd	42	inner