Embedded newform 43.7.f.a.22.19

• Label: 43.7.f.a.22.19
• Level: $43$
• Weight: $7$
• Character: 43.22
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.19
Character	\(\chi\)	\(=\)	43.22
Dual form			43.7.f.a.2.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.62961 - 11.6900i) q^{2} +(-16.3881 - 34.0303i) q^{3} +(-65.0604 - 81.5832i) q^{4} +(15.2899 - 3.48982i) q^{5} -490.074 q^{6} -124.671i q^{7} +(-510.398 + 116.495i) q^{8} +(-434.969 + 545.434i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.62961	−	11.6900i	0.703701	−	1.46125i	−0.175341	−	0.984508i	\(-0.556103\pi\)
							0.879043	−	0.476743i	\(-0.158183\pi\)
\(3\)	−16.3881	−	34.0303i	−0.606969	−	1.26038i	−0.947381	−	0.320107i	\(-0.896281\pi\)
							0.340413	−	0.940276i	\(-0.389433\pi\)
\(5\)	15.2899	−	3.48982i	0.122319	−	0.0279186i	−0.160923	−	0.986967i	\(-0.551447\pi\)
							0.283242	+	0.959048i	\(0.408590\pi\)
\(7\)		−	124.671i		−	0.363471i	−0.983347	−	0.181736i	\(-0.941829\pi\)
							0.983347	−	0.181736i	\(-0.0581715\pi\)
\(11\)	726.508	−	911.012i	0.545836	−	0.684457i	−0.430033	−	0.902813i	\(-0.641498\pi\)
							0.975869	+	0.218356i	\(0.0700694\pi\)
\(13\)	539.169	+	2362.25i	0.245411	+	1.07522i	0.936008	+	0.351978i	\(0.114491\pi\)
							−0.690597	+	0.723240i	\(0.742652\pi\)
\(17\)	−273.205	+	1196.99i	−0.0556085	+	0.243637i	−0.995092	−	0.0989586i	\(-0.968449\pi\)
							0.939483	+	0.342595i	\(0.111306\pi\)
\(19\)	3350.05	−	2671.58i	0.488417	−	0.389500i	−0.348085	−	0.937463i	\(-0.613168\pi\)
							0.836502	+	0.547963i	\(0.184597\pi\)
\(23\)	3335.86	−	4183.03i	0.274173	−	0.343801i	−0.625613	−	0.780133i	\(-0.715151\pi\)
							0.899786	+	0.436332i	\(0.143723\pi\)
\(29\)	−4817.00	+	10002.6i	−0.197507	+	0.410128i	−0.976075	−	0.217432i	\(-0.930232\pi\)
							0.778568	+	0.627560i	\(0.215946\pi\)
\(31\)	−31915.8	−	15369.8i	−1.07132	−	0.515923i	−0.186791	−	0.982400i	\(-0.559809\pi\)
							−0.884533	+	0.466477i	\(0.845523\pi\)
\(37\)		−	52431.9i		−	1.03512i	−0.855647	−	0.517560i	\(-0.826840\pi\)
							0.855647	−	0.517560i	\(-0.173160\pi\)
\(41\)	73351.9	+	35324.4i	1.06429	+	0.512535i	0.882262	−	0.470759i	\(-0.156020\pi\)
							0.182028	+	0.983293i	\(0.441734\pi\)
\(43\)	−5628.06	−	79307.6i	−0.0707870	−	0.997491i
							\(47\)	86720.4	+	108744.i	0.835272	+	1.04740i	0.998153	+	0.0607521i	\(0.0193499\pi\)
−0.162881	+	0.986646i	\(0.552079\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.22.19	yes	126
43.2	odd	14	inner	43.7.f.a.2.19	✓	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.19	✓	126	43.2	odd	14	inner
43.7.f.a.22.19	yes	126	1.1	even	1	trivial