Embedded newform 43.3.h.a.3.4

• Label: 43.3.h.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	43.3
Dual form			43.3.h.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.304460 - 0.632217i) q^{2} +(0.576841 + 0.0432282i) q^{3} +(2.18696 + 2.74236i) q^{4} +(1.20897 - 3.91938i) q^{5} +(0.202954 - 0.351527i) q^{6} +(-0.834967 + 0.482068i) q^{7} +(5.13606 - 1.17227i) q^{8} +(-8.56860 - 1.29151i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.304460	−	0.632217i	0.152230	−	0.316109i	−0.810882	−	0.585210i	\(-0.801012\pi\)
							0.963112	+	0.269101i	\(0.0867265\pi\)
\(3\)	0.576841	+	0.0432282i	0.192280	+	0.0144094i	0.170522	−	0.985354i	\(-0.445455\pi\)
							0.0217581	+	0.999763i	\(0.493074\pi\)
\(5\)	1.20897	−	3.91938i	0.241794	−	0.783877i	−0.750598	−	0.660759i	\(-0.770235\pi\)
							0.992392	−	0.123118i	\(-0.0392893\pi\)
\(7\)	−0.834967	+	0.482068i	−0.119281	+	0.0688669i	−0.558453	−	0.829536i	\(-0.688605\pi\)
							0.439172	+	0.898403i	\(0.355272\pi\)
\(11\)	−0.320988	+	0.402507i	−0.0291807	+	0.0365915i	−0.796208	−	0.605023i	\(-0.793164\pi\)
							0.767027	+	0.641615i	\(0.221735\pi\)
\(13\)	−14.0797	+	13.0641i	−1.08306	+	1.00493i	−0.0830928	+	0.996542i	\(0.526480\pi\)
							−0.999965	+	0.00838875i	\(0.997330\pi\)
\(17\)	−2.26423	+	0.698422i	−0.133190	+	0.0410836i	−0.360633	−	0.932708i	\(-0.617439\pi\)
							0.227444	+	0.973791i	\(0.426963\pi\)
\(19\)	−3.61315	−	23.9717i	−0.190166	−	1.26167i	−0.856552	−	0.516061i	\(-0.827398\pi\)
							0.666386	−	0.745607i	\(-0.267840\pi\)
\(23\)	−2.36760	−	6.03255i	−0.102939	−	0.262285i	0.870120	−	0.492840i	\(-0.164041\pi\)
							−0.973059	+	0.230555i	\(0.925946\pi\)
\(29\)	34.1781	−	2.56130i	1.17856	−	0.0883206i	0.528992	−	0.848627i	\(-0.322570\pi\)
							0.649565	+	0.760306i	\(0.274951\pi\)
\(31\)	22.8894	−	15.6057i	0.738368	−	0.503411i	−0.134774	−	0.990876i	\(-0.543031\pi\)
							0.873142	+	0.487466i	\(0.162079\pi\)
\(37\)	−19.1372	−	11.0489i	−0.517223	−	0.298619i	0.218575	−	0.975820i	\(-0.429859\pi\)
							−0.735798	+	0.677202i	\(0.763193\pi\)
\(41\)	42.7236	+	20.5746i	1.04204	+	0.501820i	0.874996	−	0.484130i	\(-0.160864\pi\)
							0.167043	+	0.985950i	\(0.446578\pi\)
\(43\)	0.714886	+	42.9941i	0.0166252	+	0.999862i
							\(47\)	−37.5523	−	47.0892i	−0.798986	−	1.00190i	−0.999752	−	0.0222577i	\(-0.992915\pi\)
0.200766	−	0.979639i	\(-0.435657\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.3.4	✓	72
3.2	odd	2		387.3.bn.b.46.3		72
43.29	odd	42	inner	43.3.h.a.29.4	yes	72
129.29	even	42		387.3.bn.b.244.3		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.3.4	✓	72	1.1	even	1	trivial
43.3.h.a.29.4	yes	72	43.29	odd	42	inner
387.3.bn.b.46.3		72	3.2	odd	2
387.3.bn.b.244.3		72	129.29	even	42