Embedded newform 43.6.g.a.10.15

• Label: 43.6.g.a.10.15
• 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			10.15
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.70260 + 7.45959i) q^{2} +(-18.8231 - 17.4653i) q^{3} +(-23.9156 + 11.5171i) q^{4} +(55.4727 - 8.36117i) q^{5} +(98.2354 - 170.149i) q^{6} +(41.1456 + 71.2662i) q^{7} +(26.0268 + 32.6366i) q^{8} +(31.1133 + 415.179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.70260	+	7.45959i	0.300980	+	1.31868i	0.868651	+	0.495425i	\(0.164988\pi\)
							−0.567670	+	0.823256i	\(0.692155\pi\)
\(3\)	−18.8231	−	17.4653i	−1.20750	−	1.12040i	−0.989513	−	0.144445i	\(-0.953860\pi\)
							−0.217987	−	0.975952i	\(-0.569949\pi\)
\(5\)	55.4727	−	8.36117i	0.992326	−	0.149569i	0.367241	−	0.930126i	\(-0.380302\pi\)
							0.625085	+	0.780557i	\(0.285064\pi\)
\(7\)	41.1456	+	71.2662i	0.317379	+	0.549716i	0.979940	−	0.199291i	\(-0.0638639\pi\)
							−0.662561	+	0.749008i	\(0.730531\pi\)
\(11\)	278.041	+	133.898i	0.692832	+	0.333650i	0.746946	−	0.664885i	\(-0.231519\pi\)
							−0.0541145	+	0.998535i	\(0.517234\pi\)
\(13\)	333.157	−	848.869i	0.546752	−	1.39310i	−0.344154	−	0.938913i	\(-0.611834\pi\)
							0.890906	−	0.454188i	\(-0.150070\pi\)
\(17\)	2054.97	+	309.737i	1.72458	+	0.259939i	0.935240	−	0.354015i	\(-0.115184\pi\)
							0.789342	+	0.613954i	\(0.210422\pi\)
\(19\)	−56.5383	+	754.452i	−0.0359301	+	0.479454i	0.949995	+	0.312266i	\(0.101088\pi\)
							−0.985925	+	0.167189i	\(0.946531\pi\)
\(23\)	1133.04	−	772.492i	0.446606	−	0.304491i	−0.319054	−	0.947737i	\(-0.603365\pi\)
							0.765660	+	0.643246i	\(0.222413\pi\)
\(29\)	−263.332	+	244.337i	−0.0581446	+	0.0539503i	−0.708710	−	0.705500i	\(-0.750722\pi\)
							0.650565	+	0.759450i	\(0.274532\pi\)
\(31\)	1780.35	+	549.166i	0.332738	+	0.102636i	0.456622	−	0.889661i	\(-0.349059\pi\)
							−0.123885	+	0.992297i	\(0.539535\pi\)
\(37\)	−1472.85	+	2551.06i	−0.176870	+	0.306348i	−0.940807	−	0.338943i	\(-0.889931\pi\)
							0.763937	+	0.645291i	\(0.223264\pi\)
\(41\)	−4580.11	−	20066.8i	−0.425517	−	1.86431i	−0.498365	−	0.866967i	\(-0.666066\pi\)
							0.0728480	−	0.997343i	\(-0.476791\pi\)
\(43\)	−10846.1	−	5419.54i	−0.894542	−	0.446983i
							\(47\)	−14470.3	+	6968.54i	−0.955506	+	0.460147i	−0.845613	−	0.533796i	\(-0.820765\pi\)
−0.109893	+	0.993943i	\(0.535051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.15	✓	204
43.13	even	21	inner	43.6.g.a.13.15	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.15	✓	204	1.1	even	1	trivial
43.6.g.a.13.15	yes	204	43.13	even	21	inner