Embedded newform 43.10.e.a.4.8

• Label: 43.10.e.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-17.2044 + 21.5737i) q^{2} +(-168.740 - 211.593i) q^{3} +(-55.5006 - 243.164i) q^{4} +(-169.052 - 81.4112i) q^{5} +7467.91 q^{6} +4297.73 q^{7} +(-6528.11 - 3143.77i) q^{8} +(-11918.6 + 52218.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−17.2044	+	21.5737i	−0.760336	+	0.953432i	−0.999848	−	0.0174620i	\(-0.994441\pi\)
							0.239511	+	0.970894i	\(0.423013\pi\)
\(3\)	−168.740	−	211.593i	−1.20274	−	1.50819i	−0.807775	−	0.589491i	\(-0.799329\pi\)
							−0.394964	−	0.918697i	\(-0.629243\pi\)
\(5\)	−169.052	−	81.4112i	−0.120964	−	0.0582531i	0.372423	−	0.928063i	\(-0.378527\pi\)
							−0.493387	+	0.869810i	\(0.664241\pi\)
\(7\)	4297.73			0.676547			0.338274	−	0.941048i	\(-0.390157\pi\)
							0.338274	+	0.941048i	\(0.390157\pi\)
\(11\)	−12357.7	+	54142.7i	−0.254490	+	1.11499i	0.672556	+	0.740046i	\(0.265197\pi\)
							−0.927046	+	0.374948i	\(0.877661\pi\)
\(13\)	−28623.8	−	13784.5i	−0.277960	−	0.133859i	0.289710	−	0.957114i	\(-0.406441\pi\)
							−0.567671	+	0.823256i	\(0.692155\pi\)
\(17\)	139906.	−	67375.2i	0.406271	−	0.195650i	−0.219576	−	0.975595i	\(-0.570467\pi\)
							0.625848	+	0.779945i	\(0.284753\pi\)
\(19\)	94140.6	+	412457.i	0.165724	+	0.726085i	0.987674	+	0.156525i	\(0.0500292\pi\)
							−0.821950	+	0.569560i	\(0.807114\pi\)
\(23\)	−328465.	+	1.43910e6i	−0.244745	+	1.07230i	0.691893	+	0.722000i	\(0.256777\pi\)
							−0.936638	+	0.350298i	\(0.886080\pi\)
\(29\)	1.51893e6	−	1.90468e6i	0.398792	−	0.500070i	−0.541376	−	0.840781i	\(-0.682096\pi\)
							0.940168	+	0.340711i	\(0.110668\pi\)
\(31\)	2.06459e6	−	2.58892e6i	0.401520	−	0.503490i	−0.539433	−	0.842029i	\(-0.681361\pi\)
							0.940952	+	0.338539i	\(0.109933\pi\)
\(37\)	761372.			0.0667866			0.0333933	−	0.999442i	\(-0.489369\pi\)
							0.0333933	+	0.999442i	\(0.489369\pi\)
\(41\)	−6.52062e6	+	8.17660e6i	−0.360381	+	0.451903i	−0.928660	−	0.370933i	\(-0.879038\pi\)
							0.568279	+	0.822836i	\(0.307610\pi\)
\(43\)	−8.97597e6	−	2.05432e7i	−0.400381	−	0.916349i
							\(47\)	−1.03474e7	−	4.53347e7i	−0.309306	−	1.35516i	−0.855631	−	0.517586i	\(-0.826831\pi\)
0.546325	−	0.837574i	\(-0.316027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.8	✓	192
43.11	even	7	inner	43.10.e.a.11.8	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.8	✓	192	1.1	even	1	trivial
43.10.e.a.11.8	yes	192	43.11	even	7	inner