Embedded newform 43.10.e.a.4.1

• Label: 43.10.e.a.4.1
• Level: $43$
• Weight: $10$
• Character: 43.4
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.2907 + 32.9675i) q^{2} +(-118.868 - 149.056i) q^{3} +(-281.723 - 1234.31i) q^{4} +(2176.31 + 1048.06i) q^{5} +8039.13 q^{6} -8174.81 q^{7} +(28647.4 + 13795.8i) q^{8} +(-3708.17 + 16246.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.2907	+	32.9675i	−1.16189	+	1.45697i	−0.297099	+	0.954847i	\(0.596019\pi\)
							−0.864796	+	0.502123i	\(0.832552\pi\)
\(3\)	−118.868	−	149.056i	−0.847267	−	1.06244i	−0.997276	−	0.0737542i	\(-0.976502\pi\)
							0.150010	−	0.988685i	\(-0.452069\pi\)
\(5\)	2176.31	+	1048.06i	1.55724	+	0.749928i	0.996926	−	0.0783522i	\(-0.0249659\pi\)
							0.560315	+	0.828280i	\(0.310680\pi\)
\(7\)	−8174.81			−1.28687			−0.643437	−	0.765499i	\(-0.722492\pi\)
							−0.643437	+	0.765499i	\(0.722492\pi\)
\(11\)	4198.32	−	18394.1i	0.0864587	−	0.378800i	−0.913124	−	0.407682i	\(-0.866337\pi\)
							0.999583	+	0.0288812i	\(0.00919445\pi\)
\(13\)	105841.	+	50970.1i	1.02780	+	0.494960i	0.870281	−	0.492556i	\(-0.163937\pi\)
							0.157515	+	0.987517i	\(0.449652\pi\)
\(17\)	−520661.	+	250737.i	−1.51194	+	0.728113i	−0.992018	−	0.126100i	\(-0.959754\pi\)
							−0.519924	+	0.854213i	\(0.674040\pi\)
\(19\)	−19081.2	−	83600.4i	−0.0335904	−	0.147169i	0.955352	−	0.295471i	\(-0.0954765\pi\)
							−0.988942	+	0.148302i	\(0.952619\pi\)
\(23\)	−77382.2	+	339034.i	−0.0576589	+	0.252620i	−0.995540	−	0.0943355i	\(-0.969927\pi\)
							0.937882	+	0.346955i	\(0.112785\pi\)
\(29\)	3.48846e6	−	4.37439e6i	0.915889	−	1.14849i	−0.0726249	−	0.997359i	\(-0.523138\pi\)
							0.988514	−	0.151129i	\(-0.0482910\pi\)
\(31\)	−1.05936e6	+	1.32839e6i	−0.206023	+	0.258345i	−0.874098	−	0.485749i	\(-0.838547\pi\)
							0.668075	+	0.744094i	\(0.267118\pi\)
\(37\)	1.51904e7			1.33248			0.666241	−	0.745736i	\(-0.267902\pi\)
							0.666241	+	0.745736i	\(0.267902\pi\)
\(41\)	1.17159e7	−	1.46913e7i	0.647515	−	0.811958i	−0.344405	−	0.938821i	\(-0.611919\pi\)
							0.991920	+	0.126863i	\(0.0404908\pi\)
\(43\)	−1.50788e7	−	1.65899e7i	−0.672601	−	0.740005i
							\(47\)	−1.09204e7	−	4.78456e7i	−0.326437	−	1.43022i	−0.825869	−	0.563862i	\(-0.809315\pi\)
0.499431	−	0.866353i	\(-0.333542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.1	✓	192
43.11	even	7	inner	43.10.e.a.11.1	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.1	✓	192	1.1	even	1	trivial
43.10.e.a.11.1	yes	192	43.11	even	7	inner