Embedded newform 43.6.e.a.4.16

• Label: 43.6.e.a.4.16
• Level: $43$
• Weight: $6$
• Character: 43.4
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.75093 - 7.21143i) q^{2} +(8.62525 + 10.8157i) q^{3} +(-11.8110 - 51.7472i) q^{4} +(90.5645 + 43.6136i) q^{5} +127.600 q^{6} -71.2640 q^{7} +(-175.165 - 84.3549i) q^{8} +(11.4876 - 50.3307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.75093	−	7.21143i	1.01663	−	1.27481i	0.0555755	−	0.998454i	\(-0.482301\pi\)
							0.961054	−	0.276359i	\(-0.0891279\pi\)
\(3\)	8.62525	+	10.8157i	0.553310	+	0.693829i	0.977305	−	0.211835i	\(-0.0679440\pi\)
							−0.423995	+	0.905665i	\(0.639373\pi\)
\(5\)	90.5645	+	43.6136i	1.62007	+	0.780183i	0.999991	−	0.00435811i	\(-0.00138723\pi\)
							0.620077	+	0.784541i	\(0.287102\pi\)
\(7\)	−71.2640			−0.549699			−0.274850	−	0.961487i	\(-0.588628\pi\)
							−0.274850	+	0.961487i	\(0.588628\pi\)
\(11\)	−24.3267	+	106.582i	−0.0606181	+	0.265585i	−0.996151	−	0.0876545i	\(-0.972063\pi\)
							0.935533	+	0.353240i	\(0.114920\pi\)
\(13\)	−693.245	−	333.849i	−1.13770	−	0.547888i	−0.232385	−	0.972624i	\(-0.574653\pi\)
							−0.905317	+	0.424736i	\(0.860367\pi\)
\(17\)	−1309.02	+	630.389i	−1.09856	+	0.529037i	−0.893204	−	0.449651i	\(-0.851548\pi\)
							−0.205353	+	0.978688i	\(0.565834\pi\)
\(19\)	−314.177	−	1376.50i	−0.199660	−	0.874768i	−0.971139	−	0.238513i	\(-0.923340\pi\)
							0.771479	−	0.636254i	\(-0.219517\pi\)
\(23\)	−236.868	+	1037.79i	−0.0933656	+	0.409062i	−0.999915	−	0.0130382i	\(-0.995850\pi\)
							0.906549	+	0.422100i	\(0.138707\pi\)
\(29\)	−2658.34	+	3333.45i	−0.586969	+	0.736036i	−0.983284	−	0.182079i	\(-0.941717\pi\)
							0.396315	+	0.918115i	\(0.370289\pi\)
\(31\)	−7.83439	+	9.82402i	−0.00146420	+	0.00183605i	−0.782563	−	0.622571i	\(-0.786088\pi\)
							0.781099	+	0.624407i	\(0.214659\pi\)
\(37\)	11255.0			1.35157			0.675787	−	0.737097i	\(-0.263804\pi\)
							0.675787	+	0.737097i	\(0.263804\pi\)
\(41\)	−12554.0	+	15742.2i	−1.16633	+	1.46253i	−0.306563	+	0.951850i	\(0.599179\pi\)
							−0.859769	+	0.510683i	\(0.829392\pi\)
\(43\)	−5506.17	−	10802.3i	−0.454128	−	0.890936i
							\(47\)	2118.49	+	9281.71i	0.139888	+	0.612891i	0.995458	+	0.0952032i	\(0.0303501\pi\)
−0.855569	+	0.517688i	\(0.826793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.16	✓	108
43.11	even	7	inner	43.6.e.a.11.16	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.16	✓	108	1.1	even	1	trivial
43.6.e.a.11.16	yes	108	43.11	even	7	inner