Embedded newform 43.4.e.a.35.6

• Label: 43.4.e.a.35.6
• Level: $43$
• Weight: $4$
• Character: 43.35
• 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			35.6
Character	\(\chi\)	\(=\)	43.35
Dual form			43.4.e.a.16.6

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

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

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	−0.896059	−	0.443935i	\(-0.853582\pi\)
							0.999937	−	0.0111863i	\(-0.00356079\pi\)
\(3\)	0.969067	+	4.24576i	0.186497	+	0.817097i	0.978445	+	0.206508i	\(0.0662101\pi\)
							−0.791948	+	0.610589i	\(0.790933\pi\)
\(5\)	−9.73333	+	12.2052i	−0.870576	+	1.09167i	0.124468	+	0.992224i	\(0.460278\pi\)
							−0.995043	+	0.0994436i	\(0.968294\pi\)
\(7\)	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	−0.479421	−	0.877585i	\(-0.659153\pi\)
							0.387210	+	0.921992i	\(0.373439\pi\)
\(13\)	21.6372	−	27.1322i	0.461621	−	0.578855i	−0.495476	−	0.868622i	\(-0.665006\pi\)
							0.957097	+	0.289767i	\(0.0935778\pi\)
\(17\)	−56.2292	−	70.5092i	−0.802212	−	1.00594i	−0.999671	−	0.0256305i	\(-0.991841\pi\)
							0.197460	−	0.980311i	\(-0.436731\pi\)
\(19\)	−17.3565	−	8.35846i	−0.209571	−	0.100924i	0.326154	−	0.945317i	\(-0.394247\pi\)
							−0.535725	+	0.844393i	\(0.679962\pi\)
\(23\)	−30.8381	+	14.8508i	−0.279573	+	0.134635i	−0.568416	−	0.822741i	\(-0.692444\pi\)
							0.288843	+	0.957376i	\(0.406729\pi\)
\(29\)	6.51423	−	28.5407i	0.0417125	−	0.182754i	−0.949780	−	0.312919i	\(-0.898693\pi\)
							0.991492	+	0.130164i	\(0.0415504\pi\)
\(31\)	49.9335	−	218.773i	0.289300	−	1.26751i	−0.596187	−	0.802845i	\(-0.703318\pi\)
							0.885488	−	0.464663i	\(-0.153824\pi\)
\(37\)	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	0.155289	+	0.987869i	\(0.450369\pi\)
							−0.568531	+	0.822662i	\(0.692488\pi\)
\(43\)	−18.4028	−	281.369i	−0.0652651	−	0.997868i
							\(47\)	103.449	+	49.8182i	0.321054	+	0.154611i	0.587468	−	0.809247i	\(-0.300125\pi\)
−0.266415	+	0.963859i	\(0.585839\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.16.6	✓	60	43.16	even	7	inner
43.4.e.a.35.6	yes	60	1.1	even	1	trivial
1849.4.a.g.1.18		30	43.39	odd	14
1849.4.a.h.1.13		30	43.4	even	7