Embedded newform 43.3.h.a.5.3

• Label: 43.3.h.a.5.3
• Level: $43$
• Weight: $3$
• Character: 43.5
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	43.5
Dual form			43.3.h.a.26.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.948568 - 0.216505i) q^{2} +(1.34540 - 4.36170i) q^{3} +(-2.75097 - 1.32480i) q^{4} +(-6.23473 + 2.44695i) q^{5} +(-2.22054 + 3.84608i) q^{6} +(9.41041 - 5.43310i) q^{7} +(5.36543 + 4.27879i) q^{8} +(-9.77813 - 6.66661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.948568	−	0.216505i	−0.474284	−	0.108252i	−0.0213043	−	0.999773i	\(-0.506782\pi\)
							−0.452980	+	0.891521i	\(0.649639\pi\)
\(3\)	1.34540	−	4.36170i	0.448468	−	1.45390i	−0.395979	−	0.918260i	\(-0.629595\pi\)
							0.844447	−	0.535639i	\(-0.179929\pi\)
\(5\)	−6.23473	+	2.44695i	−1.24695	+	0.489390i	−0.894763	−	0.446541i	\(-0.852656\pi\)
							−0.352183	+	0.935931i	\(0.614561\pi\)
\(7\)	9.41041	−	5.43310i	1.34434	−	0.776158i	0.356903	−	0.934141i	\(-0.383833\pi\)
							0.987442	+	0.157984i	\(0.0504993\pi\)
\(11\)	13.3783	−	6.44264i	1.21621	−	0.585694i	0.287954	−	0.957644i	\(-0.407025\pi\)
							0.928253	+	0.371950i	\(0.121311\pi\)
\(13\)	2.29457	+	0.345850i	0.176505	+	0.0266039i	0.236700	−	0.971583i	\(-0.423934\pi\)
							−0.0601949	+	0.998187i	\(0.519172\pi\)
\(17\)	−6.89245	+	17.5617i	−0.405438	+	1.03304i	0.571705	+	0.820459i	\(0.306282\pi\)
							−0.977144	+	0.212581i	\(0.931813\pi\)
\(19\)	−11.8385	−	17.3640i	−0.623081	−	0.913893i	0.376842	−	0.926278i	\(-0.377010\pi\)
							−0.999923	+	0.0123849i	\(0.996058\pi\)
\(23\)	−0.541149	+	7.22113i	−0.0235282	+	0.313962i	0.973006	+	0.230780i	\(0.0741276\pi\)
							−0.996534	+	0.0831829i	\(0.973491\pi\)
\(29\)	4.91518	+	15.9346i	0.169489	+	0.549470i	0.999971	−	0.00757493i	\(-0.00241120\pi\)
							−0.830482	+	0.557045i	\(0.811935\pi\)
\(31\)	13.5233	+	12.5478i	0.436235	+	0.404767i	0.867483	−	0.497466i	\(-0.165736\pi\)
							−0.431248	+	0.902233i	\(0.641927\pi\)
\(37\)	41.1524	+	23.7593i	1.11223	+	0.642144i	0.939405	−	0.342809i	\(-0.111378\pi\)
							0.172821	+	0.984953i	\(0.444712\pi\)
\(41\)	−7.41322	+	32.4795i	−0.180810	+	0.792182i	0.800435	+	0.599419i	\(0.204602\pi\)
							−0.981245	+	0.192762i	\(0.938255\pi\)
\(43\)	1.70453	−	42.9662i	0.0396401	−	0.999214i
							\(47\)	−15.6158	−	7.52016i	−0.332250	−	0.160003i	0.260319	−	0.965523i	\(-0.416172\pi\)
−0.592570	+	0.805519i	\(0.701887\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.5.3	✓	72
3.2	odd	2		387.3.bn.b.91.4		72
43.26	odd	42	inner	43.3.h.a.26.3	yes	72
129.26	even	42		387.3.bn.b.370.4		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.5.3	✓	72	1.1	even	1	trivial
43.3.h.a.26.3	yes	72	43.26	odd	42	inner
387.3.bn.b.91.4		72	3.2	odd	2
387.3.bn.b.370.4		72	129.26	even	42