Embedded newform 43.3.h.a.34.2

• Label: 43.3.h.a.34.2
• Level: $43$
• Weight: $3$
• Character: 43.34
• 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			34.2
Character	\(\chi\)	\(=\)	43.34
Dual form			43.3.h.a.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24704 + 1.79196i) q^{2} +(0.540566 - 3.58642i) q^{3} +(0.948008 - 4.15349i) q^{4} +(3.03177 - 4.44679i) q^{5} +(5.21203 + 9.02750i) q^{6} +(5.12110 + 2.95667i) q^{7} +(0.324610 + 0.674059i) q^{8} +(-3.97003 - 1.22459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{23}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24704	+	1.79196i	−1.12352	+	0.895978i	−0.995402	−	0.0957805i	\(-0.969465\pi\)
							−0.128119	+	0.991759i	\(0.540894\pi\)
\(3\)	0.540566	−	3.58642i	0.180189	−	1.19547i	−0.697622	−	0.716466i	\(-0.745759\pi\)
							0.877811	−	0.479007i	\(-0.159003\pi\)
\(5\)	3.03177	−	4.44679i	0.606353	−	0.889357i	−0.393193	−	0.919456i	\(-0.628630\pi\)
							0.999547	+	0.0300986i	\(0.00958213\pi\)
\(7\)	5.12110	+	2.95667i	0.731585	+	0.422381i	0.819002	−	0.573791i	\(-0.194528\pi\)
							−0.0874167	+	0.996172i	\(0.527861\pi\)
\(11\)	−1.83758	−	8.05095i	−0.167052	−	0.731904i	−0.987165	−	0.159702i	\(-0.948947\pi\)
							0.820113	−	0.572202i	\(-0.193911\pi\)
\(13\)	0.572156	−	7.63489i	0.0440120	−	0.587299i	−0.930982	−	0.365066i	\(-0.881046\pi\)
							0.974994	−	0.222233i	\(-0.0713346\pi\)
\(17\)	−27.9603	+	19.0630i	−1.64472	+	1.12135i	−0.767847	+	0.640633i	\(0.778672\pi\)
							−0.876875	+	0.480719i	\(0.840376\pi\)
\(19\)	7.85764	+	25.4738i	0.413560	+	1.34073i	0.887758	+	0.460310i	\(0.152262\pi\)
							−0.474198	+	0.880418i	\(0.657262\pi\)
\(23\)	19.6703	−	18.2514i	0.855231	−	0.793538i	−0.124842	−	0.992177i	\(-0.539842\pi\)
							0.980073	+	0.198638i	\(0.0636519\pi\)
\(29\)	6.81483	+	45.2135i	0.234994	+	1.55908i	0.725856	+	0.687847i	\(0.241444\pi\)
							−0.490861	+	0.871238i	\(0.663318\pi\)
\(31\)	−5.75187	+	14.6555i	−0.185544	+	0.472759i	−0.993134	−	0.116986i	\(-0.962677\pi\)
							0.807589	+	0.589745i	\(0.200772\pi\)
\(37\)	−28.6872	+	16.5625i	−0.775329	+	0.447636i	−0.834772	−	0.550595i	\(-0.814401\pi\)
							0.0594435	+	0.998232i	\(0.481067\pi\)
\(41\)	4.06154	+	5.09301i	0.0990619	+	0.124220i	0.828891	−	0.559409i	\(-0.188972\pi\)
							−0.729830	+	0.683629i	\(0.760401\pi\)
\(43\)	30.2717	−	30.5389i	0.703994	−	0.710206i
							\(47\)	6.13400	−	26.8748i	0.130511	−	0.571805i	−0.866809	−	0.498640i	\(-0.833833\pi\)
0.997320	−	0.0731647i	\(-0.0233099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.34.2	yes	72
3.2	odd	2		387.3.bn.b.163.5		72
43.19	odd	42	inner	43.3.h.a.19.2	✓	72
129.62	even	42		387.3.bn.b.19.5		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.19.2	✓	72	43.19	odd	42	inner
43.3.h.a.34.2	yes	72	1.1	even	1	trivial
387.3.bn.b.19.5		72	129.62	even	42
387.3.bn.b.163.5		72	3.2	odd	2