Embedded newform 43.10.g.a.9.11

• Label: 43.10.g.a.9.11
• Level: $43$
• Weight: $10$
• Character: 43.9
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(32\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.11
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.1893 - 12.7770i) q^{2} +(228.731 + 34.4757i) q^{3} +(54.5013 - 238.786i) q^{4} +(-780.841 - 532.368i) q^{5} +(-1890.12 - 3273.78i) q^{6} +(-4940.14 + 8556.57i) q^{7} +(-11145.0 + 5367.13i) q^{8} +(32320.8 + 9969.64i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.1893	−	12.7770i	−0.450308	−	0.564669i	0.503919	−	0.863751i	\(-0.331891\pi\)
							−0.954227	+	0.299082i	\(0.903320\pi\)
\(3\)	228.731	+	34.4757i	1.63034	+	0.245735i	0.899536	−	0.436847i	\(-0.143905\pi\)
							0.730809	+	0.682582i	\(0.239143\pi\)
\(5\)	−780.841	−	532.368i	−0.558724	−	0.380932i	0.250781	−	0.968044i	\(-0.419312\pi\)
							−0.809506	+	0.587112i	\(0.800265\pi\)
\(7\)	−4940.14	+	8556.57i	−0.777674	+	1.34697i	0.155605	+	0.987819i	\(0.450267\pi\)
							−0.933279	+	0.359152i	\(0.883066\pi\)
\(11\)	16662.9	+	73004.9i	0.343150	+	1.50344i	0.792384	+	0.610023i	\(0.208840\pi\)
							−0.449235	+	0.893414i	\(0.648303\pi\)
\(13\)	−6851.27	+	91423.9i	−0.0665313	+	0.887799i	0.859689	+	0.510817i	\(0.170657\pi\)
							−0.926221	+	0.376982i	\(0.876962\pi\)
\(17\)	−405558.	+	276505.i	−1.17769	+	0.802938i	−0.984008	−	0.178125i	\(-0.942997\pi\)
							−0.193686	+	0.981064i	\(0.562044\pi\)
\(19\)	262999.	−	81124.5i	0.462981	−	0.142811i	−0.0544848	−	0.998515i	\(-0.517352\pi\)
							0.517466	+	0.855704i	\(0.326875\pi\)
\(23\)	1.50124e6	−	1.39295e6i	1.11860	−	1.03791i	0.119616	−	0.992820i	\(-0.461834\pi\)
							0.998983	−	0.0450884i	\(-0.0143569\pi\)
\(29\)	2.67409e6	−	403055.i	0.702078	−	0.105821i	0.211705	−	0.977334i	\(-0.432098\pi\)
							0.490373	+	0.871512i	\(0.336860\pi\)
\(31\)	−1.45604e6	+	3.70992e6i	−0.283168	+	0.721501i	0.716545	+	0.697541i	\(0.245723\pi\)
							−0.999713	+	0.0239595i	\(0.992373\pi\)
\(37\)	−608331.	−	1.05366e6i	−0.0533621	−	0.0924258i	0.838110	−	0.545501i	\(-0.183660\pi\)
							−0.891473	+	0.453075i	\(0.850327\pi\)
\(41\)	−3.86092e6	−	4.84144e6i	−0.213385	−	0.267576i	0.663607	−	0.748081i	\(-0.269025\pi\)
							−0.876992	+	0.480505i	\(0.840453\pi\)
\(43\)	−2.04186e7	−	9.25601e6i	−0.910789	−	0.412872i
							\(47\)	−1.29030e7	+	5.65317e7i	−0.385701	+	1.68987i	0.293538	+	0.955947i	\(0.405167\pi\)
−0.679238	+	0.733918i	\(0.737690\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.11	✓	384
43.24	even	21	inner	43.10.g.a.24.11	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.11	✓	384	1.1	even	1	trivial
43.10.g.a.24.11	yes	384	43.24	even	21	inner