Embedded newform 43.8.e.a.4.11

• Label: 43.8.e.a.4.11
• Level: $43$
• Weight: $8$
• Character: 43.4
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.11
Character	\(\chi\)	\(=\)	43.4
Dual form			43.8.e.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71133 + 2.14594i) q^{2} +(-42.9085 - 53.8056i) q^{3} +(26.8063 + 117.446i) q^{4} +(-41.1887 - 19.8354i) q^{5} +188.894 q^{6} +1550.98 q^{7} +(-614.443 - 295.900i) q^{8} +(-567.247 + 2485.27i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71133	+	2.14594i	−0.151262	+	0.189676i	−0.851689	−	0.524048i	\(-0.824421\pi\)
							0.700427	+	0.713724i	\(0.252993\pi\)
\(3\)	−42.9085	−	53.8056i	−0.917528	−	1.15054i	−0.988219	−	0.153045i	\(-0.951092\pi\)
							0.0706911	−	0.997498i	\(-0.477480\pi\)
\(5\)	−41.1887	−	19.8354i	−0.147361	−	0.0709653i	0.358750	−	0.933434i	\(-0.383203\pi\)
							−0.506111	+	0.862468i	\(0.668917\pi\)
\(7\)	1550.98			1.70908			0.854542	−	0.519382i	\(-0.173838\pi\)
							0.854542	+	0.519382i	\(0.173838\pi\)
\(11\)	706.355	−	3094.74i	0.160011	−	0.701052i	−0.829729	−	0.558167i	\(-0.811505\pi\)
							0.989739	−	0.142885i	\(-0.0456379\pi\)
\(13\)	−11661.0	−	5615.62i	−1.47208	−	0.708918i	−0.485814	−	0.874062i	\(-0.661477\pi\)
							−0.986269	+	0.165144i	\(0.947191\pi\)
\(17\)	−19493.1	+	9387.39i	−0.962299	+	0.463419i	−0.847982	−	0.530025i	\(-0.822182\pi\)
							−0.114317	+	0.993444i	\(0.536468\pi\)
\(19\)	1546.09	+	6773.88i	0.0517129	+	0.226569i	0.994180	−	0.107729i	\(-0.0343579\pi\)
							−0.942467	+	0.334298i	\(0.891501\pi\)
\(23\)	16099.5	−	70536.6i	0.275909	−	1.20884i	−0.627006	−	0.779015i	\(-0.715720\pi\)
							0.902914	−	0.429820i	\(-0.141423\pi\)
\(29\)	130652.	−	163833.i	0.994772	−	1.24740i	0.0259453	−	0.999663i	\(-0.491740\pi\)
							0.968826	−	0.247741i	\(-0.0796881\pi\)
\(31\)	14771.7	−	18523.1i	0.0890560	−	0.111673i	−0.735307	−	0.677734i	\(-0.762962\pi\)
							0.824363	+	0.566061i	\(0.191533\pi\)
\(37\)	−407345.			−1.32208			−0.661038	−	0.750353i	\(-0.729884\pi\)
							−0.661038	+	0.750353i	\(0.729884\pi\)
\(41\)	181779.	−	227944.i	0.411909	−	0.516517i	−0.531991	−	0.846750i	\(-0.678556\pi\)
							0.943899	+	0.330233i	\(0.107127\pi\)
\(43\)	−425198.	+	301704.i	−0.815552	+	0.578684i
							\(47\)	−62095.2	−	272057.i	−0.0872400	−	0.382223i	0.912393	−	0.409315i	\(-0.134232\pi\)
−0.999633	+	0.0270921i	\(0.991375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.e.a.4.11	✓	144
43.11	even	7	inner	43.8.e.a.11.11	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.e.a.4.11	✓	144	1.1	even	1	trivial
43.8.e.a.11.11	yes	144	43.11	even	7	inner