Embedded newform 43.6.g.a.10.17

• Label: 43.6.g.a.10.17
• Level: $43$
• Weight: $6$
• Character: 43.10
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.17
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.46409 + 10.7959i) q^{2} +(15.0719 + 13.9847i) q^{3} +(-81.6481 + 39.3197i) q^{4} +(69.8097 - 10.5221i) q^{5} +(-113.838 + 197.174i) q^{6} +(10.5536 + 18.2793i) q^{7} +(-404.743 - 507.532i) q^{8} +(13.4315 + 179.231i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46409	+	10.7959i	0.435593	+	1.90846i	0.417624	+	0.908620i	\(0.362863\pi\)
							0.0179692	+	0.999839i	\(0.494280\pi\)
\(3\)	15.0719	+	13.9847i	0.966863	+	0.897118i	0.994734	−	0.102493i	\(-0.0326821\pi\)
							−0.0278703	+	0.999612i	\(0.508873\pi\)
\(5\)	69.8097	−	10.5221i	1.24879	−	0.188226i	0.508827	−	0.860869i	\(-0.330079\pi\)
							0.739967	+	0.672643i	\(0.234841\pi\)
\(7\)	10.5536	+	18.2793i	0.0814056	+	0.140999i	0.903854	−	0.427841i	\(-0.140726\pi\)
							−0.822448	+	0.568840i	\(0.807392\pi\)
\(11\)	342.496	+	164.937i	0.853442	+	0.410996i	0.808854	−	0.588009i	\(-0.200088\pi\)
							0.0445880	+	0.999005i	\(0.485802\pi\)
\(13\)	402.936	−	1026.67i	0.661269	−	1.68489i	−0.0653651	−	0.997861i	\(-0.520821\pi\)
							0.726634	−	0.687025i	\(-0.241084\pi\)
\(17\)	−761.691	−	114.806i	−0.639229	−	0.0963482i	−0.178572	−	0.983927i	\(-0.557148\pi\)
							−0.460656	+	0.887579i	\(0.652386\pi\)
\(19\)	13.9129	−	185.654i	0.00884164	−	0.117983i	−0.991037	−	0.133591i	\(-0.957349\pi\)
							0.999878	+	0.0156075i	\(0.00496822\pi\)
\(23\)	−2324.65	+	1584.92i	−0.916301	+	0.624724i	−0.926936	−	0.375220i	\(-0.877567\pi\)
							0.0106347	+	0.999943i	\(0.496615\pi\)
\(29\)	−3538.95	+	3283.67i	−0.781411	+	0.725043i	−0.966216	−	0.257732i	\(-0.917025\pi\)
							0.184806	+	0.982775i	\(0.440834\pi\)
\(31\)	−313.583	−	96.7275i	−0.0586068	−	0.0180778i	0.265313	−	0.964162i	\(-0.414525\pi\)
							−0.323920	+	0.946084i	\(0.605001\pi\)
\(37\)	−749.931	+	1298.92i	−0.0900569	+	0.155983i	−0.907535	−	0.419977i	\(-0.862038\pi\)
							0.817478	+	0.575960i	\(0.195372\pi\)
\(41\)	533.726	+	2338.40i	0.0495859	+	0.217250i	0.993651	−	0.112510i	\(-0.0358892\pi\)
							−0.944065	+	0.329760i	\(0.893032\pi\)
\(43\)	2196.08	+	11924.2i	0.181125	+	0.983460i
							\(47\)	−8348.05	+	4020.21i	−0.551240	+	0.265463i	−0.688703	−	0.725044i	\(-0.741819\pi\)
0.137463	+	0.990507i	\(0.456105\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.17	✓	204
43.13	even	21	inner	43.6.g.a.13.17	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.17	✓	204	1.1	even	1	trivial
43.6.g.a.13.17	yes	204	43.13	even	21	inner