Embedded newform 43.7.f.a.2.16

• Label: 43.7.f.a.2.16
• Level: $43$
• Weight: $7$
• Character: 43.2
• 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			2.16
Character	\(\chi\)	\(=\)	43.2
Dual form			43.7.f.a.22.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.53371 + 7.33783i) q^{2} +(-9.42884 + 19.5792i) q^{3} +(-1.45324 + 1.82231i) q^{4} +(197.653 + 45.1129i) q^{5} -176.987 q^{6} +222.322i q^{7} +(489.664 + 111.763i) q^{8} +(160.083 + 200.737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.53371	+	7.33783i	0.441714	+	0.917228i	0.996368	+	0.0851476i	\(0.0271362\pi\)
							−0.554654	+	0.832081i	\(0.687150\pi\)
\(3\)	−9.42884	+	19.5792i	−0.349216	+	0.725155i	−0.999400	−	0.0346215i	\(-0.988977\pi\)
							0.650184	+	0.759777i	\(0.274692\pi\)
\(5\)	197.653	+	45.1129i	1.58122	+	0.360903i	0.920813	−	0.390004i	\(-0.127527\pi\)
							0.660407	+	0.750908i	\(0.270384\pi\)
\(7\)			222.322i			0.648168i	0.946028	+	0.324084i	\(0.105056\pi\)
							−0.946028	+	0.324084i	\(0.894944\pi\)
\(11\)	−1447.06	−	1814.56i	−1.08720	−	1.36330i	−0.926497	−	0.376301i	\(-0.877196\pi\)
							−0.160701	−	0.987003i	\(-0.551376\pi\)
\(13\)	−628.819	+	2755.03i	−0.286217	+	1.25400i	0.603455	+	0.797397i	\(0.293790\pi\)
							−0.889672	+	0.456601i	\(0.849067\pi\)
\(17\)	−1364.17	−	5976.80i	−0.277665	−	1.21653i	−0.900737	−	0.434364i	\(-0.856973\pi\)
							0.623073	−	0.782164i	\(-0.285884\pi\)
\(19\)	−3375.98	−	2692.26i	−0.492198	−	0.392515i	0.345697	−	0.938346i	\(-0.387642\pi\)
							−0.837895	+	0.545832i	\(0.816214\pi\)
\(23\)	6836.84	+	8573.12i	0.561917	+	0.704621i	0.978911	−	0.204288i	\(-0.0654879\pi\)
							−0.416994	+	0.908909i	\(0.636917\pi\)
\(29\)	10412.9	+	21622.6i	0.426951	+	0.886573i	0.997849	+	0.0655521i	\(0.0208808\pi\)
							−0.570898	+	0.821021i	\(0.693405\pi\)
\(31\)	20667.9	−	9953.15i	0.693764	−	0.334099i	−0.0535546	−	0.998565i	\(-0.517055\pi\)
							0.747319	+	0.664466i	\(0.231341\pi\)
\(37\)		−	59124.7i		−	1.16725i	−0.812023	−	0.583625i	\(-0.801634\pi\)
							0.812023	−	0.583625i	\(-0.198366\pi\)
\(41\)	7414.10	−	3570.44i	0.107574	−	0.0518048i	−0.379323	−	0.925264i	\(-0.623843\pi\)
							0.486897	+	0.873459i	\(0.338129\pi\)
\(43\)	−70848.5	−	36081.2i	−0.891097	−	0.453812i
							\(47\)	112779.	−	141421.i	1.08627	−	1.36213i	0.159194	−	0.987247i	\(-0.449110\pi\)
0.927071	−	0.374886i	\(-0.122318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.2.16	✓	126
43.22	odd	14	inner	43.7.f.a.22.16	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.16	✓	126	1.1	even	1	trivial
43.7.f.a.22.16	yes	126	43.22	odd	14	inner