Embedded newform 43.3.h.a.28.3

• Label: 43.3.h.a.28.3
• Level: $43$
• Weight: $3$
• Character: 43.28
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $72$
• 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			28.3
Character	\(\chi\)	\(=\)	43.28
Dual form			43.3.h.a.20.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.507473 - 0.404696i) q^{2} +(1.27856 + 0.501799i) q^{3} +(-0.796334 - 3.48897i) q^{4} +(4.78317 - 0.358449i) q^{5} +(-0.445760 - 0.772080i) q^{6} +(4.33114 + 2.50058i) q^{7} +(-2.13436 + 4.43204i) q^{8} +(-5.21454 - 4.83839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.507473	−	0.404696i	−0.253737	−	0.202348i	0.488358	−	0.872643i	\(-0.337596\pi\)
							−0.742095	+	0.670295i	\(0.766167\pi\)
\(3\)	1.27856	+	0.501799i	0.426188	+	0.167266i	0.568734	−	0.822522i	\(-0.307434\pi\)
							−0.142546	+	0.989788i	\(0.545529\pi\)
\(5\)	4.78317	−	0.358449i	0.956633	−	0.0716898i	0.412744	−	0.910847i	\(-0.364570\pi\)
							0.543889	+	0.839157i	\(0.316951\pi\)
\(7\)	4.33114	+	2.50058i	0.618734	+	0.357226i	0.776376	−	0.630270i	\(-0.217056\pi\)
							−0.157642	+	0.987496i	\(0.550389\pi\)
\(11\)	−0.858881	+	3.76300i	−0.0780801	+	0.342091i	−0.998846	−	0.0480239i	\(-0.984708\pi\)
							0.920766	+	0.390115i	\(0.127565\pi\)
\(13\)	−12.1004	+	8.24988i	−0.930797	+	0.634606i	−0.930853	−	0.365394i	\(-0.880934\pi\)
							5.64889e−5		1.00000i	\(0.499982\pi\)
\(17\)	−1.48199	+	19.7757i	−0.0871757	+	1.16328i	0.766419	+	0.642341i	\(0.222037\pi\)
							−0.853595	+	0.520938i	\(0.825582\pi\)
\(19\)	−5.57713	−	6.01072i	−0.293533	−	0.316354i	0.569016	−	0.822326i	\(-0.307324\pi\)
							−0.862550	+	0.505973i	\(0.831134\pi\)
\(23\)	25.9072	−	7.99132i	1.12640	−	0.347449i	0.325082	−	0.945686i	\(-0.394608\pi\)
							0.801319	+	0.598237i	\(0.204132\pi\)
\(29\)	42.8177	−	16.8047i	1.47647	−	0.579473i	0.515691	−	0.856775i	\(-0.327535\pi\)
							0.960783	+	0.277302i	\(0.0894401\pi\)
\(31\)	−19.2472	−	2.90105i	−0.620878	−	0.0935823i	−0.168932	−	0.985628i	\(-0.554032\pi\)
							−0.451946	+	0.892045i	\(0.649270\pi\)
\(37\)	−33.7389	+	19.4791i	−0.911861	+	0.526463i	−0.881029	−	0.473061i	\(-0.843149\pi\)
							−0.0308316	+	0.999525i	\(0.509816\pi\)
\(41\)	−15.4932	+	19.4279i	−0.377883	+	0.473850i	−0.934010	−	0.357247i	\(-0.883715\pi\)
							0.556127	+	0.831097i	\(0.312287\pi\)
\(43\)	37.8719	−	20.3647i	0.880742	−	0.473597i
							\(47\)	−8.41358	−	36.8623i	−0.179012	−	0.784304i	−0.982087	−	0.188426i	\(-0.939661\pi\)
0.803075	−	0.595878i	\(-0.203196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.28.3	yes	72
3.2	odd	2		387.3.bn.b.28.4		72
43.20	odd	42	inner	43.3.h.a.20.3	✓	72
129.20	even	42		387.3.bn.b.235.4		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.20.3	✓	72	43.20	odd	42	inner
43.3.h.a.28.3	yes	72	1.1	even	1	trivial
387.3.bn.b.28.4		72	3.2	odd	2
387.3.bn.b.235.4		72	129.20	even	42