Embedded newform 43.5.h.a.5.6

• Label: 43.5.h.a.5.6
• Level: $43$
• Weight: $5$
• Character: 43.5
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	43.5
Dual form			43.5.h.a.26.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.83316 - 0.646651i) q^{2} +(2.81726 - 9.13334i) q^{3} +(-6.80686 - 3.27801i) q^{4} +(-12.8396 + 5.03918i) q^{5} +(-13.8878 + 24.0544i) q^{6} +(-12.3907 + 7.15376i) q^{7} +(53.5175 + 42.6788i) q^{8} +(-8.55552 - 5.83306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.83316	−	0.646651i	−0.708290	−	0.161663i	−0.146824	−	0.989163i	\(-0.546905\pi\)
							−0.561466	+	0.827500i	\(0.689762\pi\)
\(3\)	2.81726	−	9.13334i	0.313029	−	1.01482i	−0.653212	−	0.757175i	\(-0.726579\pi\)
							0.966241	−	0.257640i	\(-0.0829449\pi\)
\(5\)	−12.8396	+	5.03918i	−0.513584	+	0.201567i	−0.607957	−	0.793970i	\(-0.708011\pi\)
							0.0943726	+	0.995537i	\(0.469915\pi\)
\(7\)	−12.3907	+	7.15376i	−0.252871	+	0.145995i	−0.621078	−	0.783749i	\(-0.713305\pi\)
							0.368207	+	0.929744i	\(0.379972\pi\)
\(11\)	−174.985	+	84.2683i	−1.44616	+	0.696433i	−0.981923	−	0.189280i	\(-0.939385\pi\)
							−0.464234	+	0.885713i	\(0.653670\pi\)
\(13\)	−203.134	−	30.6176i	−1.20198	−	0.181169i	−0.482633	−	0.875823i	\(-0.660320\pi\)
							−0.719344	+	0.694654i	\(0.755558\pi\)
\(17\)	76.8760	−	195.877i	0.266007	−	0.677774i	−0.733992	−	0.679158i	\(-0.762345\pi\)
							0.999999	+	0.00138345i	\(0.000440366\pi\)
\(19\)	−14.9657	−	21.9507i	−0.0414564	−	0.0608053i	0.804951	−	0.593341i	\(-0.202192\pi\)
							−0.846407	+	0.532536i	\(0.821239\pi\)
\(23\)	33.1157	−	441.898i	0.0626005	−	0.835346i	−0.874219	−	0.485532i	\(-0.838626\pi\)
							0.936819	−	0.349814i	\(-0.113755\pi\)
\(29\)	71.5183	+	231.857i	0.0850396	+	0.275692i	0.988098	−	0.153826i	\(-0.0491597\pi\)
							−0.903058	+	0.429518i	\(0.858683\pi\)
\(31\)	−913.376	−	847.489i	−0.950443	−	0.881882i	0.0426645	−	0.999089i	\(-0.486415\pi\)
							−0.993107	+	0.117207i	\(0.962606\pi\)
\(37\)	−784.147	−	452.728i	−0.572788	−	0.330700i	0.185474	−	0.982649i	\(-0.440618\pi\)
							−0.758262	+	0.651950i	\(0.773951\pi\)
\(41\)	17.4713	−	76.5468i	0.0103934	−	0.0455365i	−0.969465	−	0.245228i	\(-0.921137\pi\)
							0.979859	+	0.199692i	\(0.0639942\pi\)
\(43\)	−1714.47	−	692.391i	−0.927240	−	0.374468i
							\(47\)	2519.68	+	1213.41i	1.14064	+	0.549305i	0.906210	−	0.422828i	\(-0.138962\pi\)
0.234433	+	0.972132i	\(0.424677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.5.6	✓	168
43.26	odd	42	inner	43.5.h.a.26.6	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.5.6	✓	168	1.1	even	1	trivial
43.5.h.a.26.6	yes	168	43.26	odd	42	inner