Embedded newform 43.8.e.a.4.1

• Label: 43.8.e.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	43.4
Dual form			43.8.e.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.6930 + 17.1704i) q^{2} +(16.1961 + 20.3092i) q^{3} +(-78.8438 - 345.437i) q^{4} +(-261.650 - 126.004i) q^{5} -570.491 q^{6} -1281.81 q^{7} +(4478.19 + 2156.58i) q^{8} +(336.501 - 1474.31i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.6930	+	17.1704i	−1.21030	+	1.51767i	−0.416913	+	0.908946i	\(0.636888\pi\)
							−0.793385	+	0.608720i	\(0.791683\pi\)
\(3\)	16.1961	+	20.3092i	0.346326	+	0.434280i	0.924236	−	0.381821i	\(-0.124703\pi\)
							−0.577910	+	0.816101i	\(0.696131\pi\)
\(5\)	−261.650	−	126.004i	−0.936108	−	0.450806i	−0.0973135	−	0.995254i	\(-0.531025\pi\)
							−0.838795	+	0.544448i	\(0.816739\pi\)
\(7\)	−1281.81			−1.41247			−0.706236	−	0.707976i	\(-0.749608\pi\)
							−0.706236	+	0.707976i	\(0.749608\pi\)
\(11\)	−704.803	+	3087.94i	−0.159659	+	0.699511i	0.830201	+	0.557464i	\(0.188226\pi\)
							−0.989860	+	0.142047i	\(0.954632\pi\)
\(13\)	5781.47	+	2784.21i	0.729855	+	0.351480i	0.761627	−	0.648016i	\(-0.224401\pi\)
							−0.0317723	+	0.999495i	\(0.510115\pi\)
\(17\)	4098.94	−	1973.94i	0.202349	−	0.0974459i	−0.329965	−	0.943993i	\(-0.607037\pi\)
							0.532314	+	0.846547i	\(0.321323\pi\)
\(19\)	7801.50	+	34180.6i	0.260940	+	1.14325i	0.920235	+	0.391367i	\(0.127998\pi\)
							−0.659295	+	0.751885i	\(0.729145\pi\)
\(23\)	8621.95	−	37775.2i	0.147760	−	0.647381i	−0.845744	−	0.533589i	\(-0.820843\pi\)
							0.993505	−	0.113792i	\(-0.0362998\pi\)
\(29\)	108607.	−	136189.i	0.826921	−	1.03693i	−0.171738	−	0.985143i	\(-0.554938\pi\)
							0.998658	−	0.0517829i	\(-0.0164904\pi\)
\(31\)	−148799.	+	186589.i	−0.897089	+	1.12491i	0.0945049	+	0.995524i	\(0.469873\pi\)
							−0.991594	+	0.129390i	\(0.958698\pi\)
\(37\)	179653.			0.583080			0.291540	−	0.956559i	\(-0.405832\pi\)
							0.291540	+	0.956559i	\(0.405832\pi\)
\(41\)	334870.	−	419914.i	0.758809	−	0.951517i	−0.241010	−	0.970523i	\(-0.577479\pi\)
							0.999819	+	0.0190058i	\(0.00605011\pi\)
\(43\)	186771.	−	486760.i	0.358237	−	0.933631i
							\(47\)	259892.	+	1.13866e6i	0.365132	+	1.59975i	0.739957	+	0.672654i	\(0.234846\pi\)
−0.374824	+	0.927096i	\(0.622297\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.e.a.4.1	✓	144
43.11	even	7	inner	43.8.e.a.11.1	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.e.a.4.1	✓	144	1.1	even	1	trivial
43.8.e.a.11.1	yes	144	43.11	even	7	inner