Embedded newform 43.4.e.a.16.6

• Label: 43.4.e.a.16.6
• Level: $43$
• Weight: $4$
• Character: 43.16
• 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			16.6
Character	\(\chi\)	\(=\)	43.16
Dual form			43.4.e.a.35.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.293813 + 1.28728i) q^{2} +(0.969067 - 4.24576i) q^{3} +(5.63699 - 2.71463i) q^{4} +(-9.73333 - 12.2052i) q^{5} +5.75020 q^{6} +4.97789 q^{7} +(11.7367 + 14.7173i) q^{8} +(7.23876 + 3.48600i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.293813	+	1.28728i	0.103878	+	0.455121i	0.999937	+	0.0111863i	\(0.00356079\pi\)
							−0.896059	+	0.443935i	\(0.853582\pi\)
\(3\)	0.969067	−	4.24576i	0.186497	−	0.817097i	−0.791948	−	0.610589i	\(-0.790933\pi\)
							0.978445	−	0.206508i	\(-0.0662101\pi\)
\(5\)	−9.73333	−	12.2052i	−0.870576	−	1.09167i	−0.995043	−	0.0994436i	\(-0.968294\pi\)
							0.124468	−	0.992224i	\(-0.460278\pi\)
\(7\)	4.97789			0.268781			0.134390	−	0.990928i	\(-0.457092\pi\)
							0.134390	+	0.990928i	\(0.457092\pi\)
\(11\)	−3.36410	−	1.62007i	−0.0922105	−	0.0444062i	0.387210	−	0.921992i	\(-0.373439\pi\)
							−0.479421	+	0.877585i	\(0.659153\pi\)
\(13\)	21.6372	+	27.1322i	0.461621	+	0.578855i	0.957097	−	0.289767i	\(-0.0935778\pi\)
							−0.495476	+	0.868622i	\(0.665006\pi\)
\(17\)	−56.2292	+	70.5092i	−0.802212	+	1.00594i	0.197460	+	0.980311i	\(0.436731\pi\)
							−0.999671	+	0.0256305i	\(0.991841\pi\)
\(19\)	−17.3565	+	8.35846i	−0.209571	+	0.100924i	−0.535725	−	0.844393i	\(-0.679962\pi\)
							0.326154	+	0.945317i	\(0.394247\pi\)
\(23\)	−30.8381	−	14.8508i	−0.279573	−	0.134635i	0.288843	−	0.957376i	\(-0.406729\pi\)
							−0.568416	+	0.822741i	\(0.692444\pi\)
\(29\)	6.51423	+	28.5407i	0.0417125	+	0.182754i	0.991492	−	0.130164i	\(-0.0415504\pi\)
							−0.949780	+	0.312919i	\(0.898693\pi\)
\(31\)	49.9335	+	218.773i	0.289300	+	1.26751i	0.885488	+	0.464663i	\(0.153824\pi\)
							−0.596187	+	0.802845i	\(0.703318\pi\)
\(37\)	214.118			0.951371			0.475685	−	0.879615i	\(-0.342200\pi\)
							0.475685	+	0.879615i	\(0.342200\pi\)
\(41\)	−108.488	−	475.315i	−0.413242	−	1.81053i	−0.568531	−	0.822662i	\(-0.692488\pi\)
							0.155289	−	0.987869i	\(-0.450369\pi\)
\(43\)	−18.4028	+	281.369i	−0.0652651	+	0.997868i
							\(47\)	103.449	−	49.8182i	0.321054	−	0.154611i	−0.266415	−	0.963859i	\(-0.585839\pi\)
0.587468	+	0.809247i	\(0.300125\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.16.6	✓	60
43.11	even	7		1849.4.a.h.1.13		30
43.32	odd	14		1849.4.a.g.1.18		30
43.35	even	7	inner	43.4.e.a.35.6	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.16.6	✓	60	1.1	even	1	trivial
43.4.e.a.35.6	yes	60	43.35	even	7	inner
1849.4.a.g.1.18		30	43.32	odd	14
1849.4.a.h.1.13		30	43.11	even	7