Embedded newform 43.3.h.a.3.2

• Label: 43.3.h.a.3.2
• Level: $43$
• Weight: $3$
• Character: 43.3
• 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			3.2
Character	\(\chi\)	\(=\)	43.3
Dual form			43.3.h.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03664 + 2.15260i) q^{2} +(3.30208 + 0.247456i) q^{3} +(-1.06511 - 1.33560i) q^{4} +(0.257183 - 0.833768i) q^{5} +(-3.95573 + 6.85153i) q^{6} +(-1.23199 + 0.711287i) q^{7} +(-5.33806 + 1.21838i) q^{8} +(1.94300 + 0.292860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03664	+	2.15260i	−0.518319	+	1.07630i	0.463433	+	0.886132i	\(0.346617\pi\)
							−0.981752	+	0.190168i	\(0.939097\pi\)
\(3\)	3.30208	+	0.247456i	1.10069	+	0.0824855i	0.612665	−	0.790343i	\(-0.290098\pi\)
							0.488027	+	0.872828i	\(0.337717\pi\)
\(5\)	0.257183	−	0.833768i	0.0514367	−	0.166754i	−0.926210	−	0.377008i	\(-0.876953\pi\)
							0.977647	+	0.210254i	\(0.0674292\pi\)
\(7\)	−1.23199	+	0.711287i	−0.175998	+	0.101612i	−0.585411	−	0.810737i	\(-0.699067\pi\)
							0.409413	+	0.912349i	\(0.365734\pi\)
\(11\)	2.14873	−	2.69443i	0.195339	−	0.244948i	−0.674509	−	0.738266i	\(-0.735645\pi\)
							0.869849	+	0.493318i	\(0.164216\pi\)
\(13\)	15.7571	−	14.6204i	1.21208	−	1.12465i	0.223382	−	0.974731i	\(-0.428290\pi\)
							0.988699	−	0.149916i	\(-0.0479003\pi\)
\(17\)	−14.8515	+	4.58108i	−0.873618	+	0.269476i	−0.698966	−	0.715155i	\(-0.746356\pi\)
							−0.174652	+	0.984630i	\(0.555880\pi\)
\(19\)	−2.01375	−	13.3603i	−0.105987	−	0.703175i	−0.977208	−	0.212285i	\(-0.931910\pi\)
							0.871221	−	0.490891i	\(-0.163329\pi\)
\(23\)	6.84036	+	17.4289i	0.297407	+	0.757780i	0.998979	+	0.0451816i	\(0.0143866\pi\)
							−0.701572	+	0.712599i	\(0.747518\pi\)
\(29\)	−14.2014	+	1.06425i	−0.489705	+	0.0366983i	−0.317295	−	0.948327i	\(-0.602774\pi\)
							−0.172410	+	0.985025i	\(0.555155\pi\)
\(31\)	−21.8628	+	14.9058i	−0.705252	+	0.480833i	−0.862067	−	0.506795i	\(-0.830830\pi\)
							0.156814	+	0.987628i	\(0.449878\pi\)
\(37\)	−15.8174	−	9.13216i	−0.427497	−	0.246815i	0.270783	−	0.962640i	\(-0.412717\pi\)
							−0.698280	+	0.715825i	\(0.746051\pi\)
\(41\)	−59.5259	−	28.6661i	−1.45185	−	0.699174i	−0.468935	−	0.883233i	\(-0.655362\pi\)
							−0.982915	+	0.184058i	\(0.941076\pi\)
\(43\)	42.8319	−	3.79817i	0.996091	−	0.0883295i
							\(47\)	50.7411	+	63.6273i	1.07960	+	1.35377i	0.931060	+	0.364867i	\(0.118885\pi\)
0.148538	+	0.988907i	\(0.452543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.3.2	✓	72
3.2	odd	2		387.3.bn.b.46.5		72
43.29	odd	42	inner	43.3.h.a.29.2	yes	72
129.29	even	42		387.3.bn.b.244.5		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.3.2	✓	72	1.1	even	1	trivial
43.3.h.a.29.2	yes	72	43.29	odd	42	inner
387.3.bn.b.46.5		72	3.2	odd	2
387.3.bn.b.244.5		72	129.29	even	42