Embedded newform 43.4.e.a.11.3

• Label: 43.4.e.a.11.3
• Level: $43$
• Weight: $4$
• Character: 43.11
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	43.11
Dual form			43.4.e.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29664 - 2.87990i) q^{2} +(-5.14704 + 6.45419i) q^{3} +(-1.23909 + 5.42880i) q^{4} +(11.0752 - 5.33352i) q^{5} +30.4083 q^{6} +28.5576 q^{7} +(-8.06984 + 3.88623i) q^{8} +(-9.15641 - 40.1169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29664	−	2.87990i	−0.811987	−	1.01820i	−0.999355	−	0.0359092i	\(-0.988567\pi\)
							0.187369	−	0.982290i	\(-0.440004\pi\)
\(3\)	−5.14704	+	6.45419i	−0.990549	+	1.24211i	−0.0203511	+	0.999793i	\(0.506478\pi\)
							−0.970197	+	0.242316i	\(0.922093\pi\)
\(5\)	11.0752	−	5.33352i	0.990593	−	0.477044i	0.132857	−	0.991135i	\(-0.457585\pi\)
							0.857736	+	0.514091i	\(0.171871\pi\)
\(7\)	28.5576			1.54196			0.770981	−	0.636858i	\(-0.219766\pi\)
							0.770981	+	0.636858i	\(0.219766\pi\)
\(11\)	12.4283	+	54.4519i	0.340661	+	1.49253i	0.797683	+	0.603078i	\(0.206059\pi\)
							−0.457022	+	0.889456i	\(0.651084\pi\)
\(13\)	63.9214	−	30.7829i	1.36374	−	0.656742i	0.398271	−	0.917268i	\(-0.369610\pi\)
							0.965467	+	0.260526i	\(0.0838959\pi\)
\(17\)	−18.8533	−	9.07927i	−0.268976	−	0.129532i	0.294534	−	0.955641i	\(-0.404835\pi\)
							−0.563511	+	0.826109i	\(0.690550\pi\)
\(19\)	−5.65942	+	24.7955i	−0.0683347	+	0.299394i	−0.997534	−	0.0701852i	\(-0.977641\pi\)
							0.929199	+	0.369579i	\(0.120498\pi\)
\(23\)	−23.0933	−	101.178i	−0.209360	−	0.917267i	−0.964994	−	0.262273i	\(-0.915528\pi\)
							0.755634	−	0.654995i	\(-0.227329\pi\)
\(29\)	58.6604	+	73.5578i	0.375620	+	0.471012i	0.933328	−	0.359025i	\(-0.116891\pi\)
							−0.557708	+	0.830037i	\(0.688319\pi\)
\(31\)	10.0380	+	12.5872i	0.0581573	+	0.0729270i	0.810059	−	0.586348i	\(-0.199435\pi\)
							−0.751902	+	0.659275i	\(0.770863\pi\)
\(37\)	−23.8504			−0.105972			−0.0529862	−	0.998595i	\(-0.516874\pi\)
							−0.0529862	+	0.998595i	\(0.516874\pi\)
\(41\)	−47.1291	−	59.0980i	−0.179520	−	0.225111i	0.683927	−	0.729550i	\(-0.260271\pi\)
							−0.863447	+	0.504439i	\(0.831699\pi\)
\(43\)	279.591	+	36.5531i	0.991562	+	0.129635i
							\(47\)	1.23751	−	5.42187i	0.00384061	−	0.0168268i	−0.972971	−	0.230926i	\(-0.925824\pi\)
0.976812	+	0.214099i	\(0.0686816\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.11.3	yes	60
43.2	odd	14		1849.4.a.g.1.24		30
43.4	even	7	inner	43.4.e.a.4.3	✓	60
43.41	even	7		1849.4.a.h.1.7		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.3	✓	60	43.4	even	7	inner
43.4.e.a.11.3	yes	60	1.1	even	1	trivial
1849.4.a.g.1.24		30	43.2	odd	14
1849.4.a.h.1.7		30	43.41	even	7