Embedded newform 43.10.g.a.10.18

• Label: 43.10.g.a.10.18
• Level: $43$
• Weight: $10$
• Character: 43.10
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(32\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.18
Character	\(\chi\)	\(=\)	43.10
Dual form			43.10.g.a.13.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.872962 + 3.82470i) q^{2} +(123.291 + 114.397i) q^{3} +(447.430 - 215.471i) q^{4} +(1673.28 - 252.207i) q^{5} +(-329.906 + 571.414i) q^{6} +(5961.43 + 10325.5i) q^{7} +(2467.04 + 3093.58i) q^{8} +(643.001 + 8580.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.872962	+	3.82470i	0.0385798	+	0.169029i	0.990548	−	0.137169i	\(-0.0438002\pi\)
							−0.951968	+	0.306198i	\(0.900943\pi\)
\(3\)	123.291	+	114.397i	0.878790	+	0.815398i	0.983836	−	0.179070i	\(-0.0573088\pi\)
							−0.105047	+	0.994467i	\(0.533499\pi\)
\(5\)	1673.28	−	252.207i	1.19730	−	0.180464i	0.480027	−	0.877254i	\(-0.340627\pi\)
							0.717276	+	0.696789i	\(0.245389\pi\)
\(7\)	5961.43	+	10325.5i	0.938447	+	1.62544i	0.768370	+	0.640006i	\(0.221068\pi\)
							0.170077	+	0.985431i	\(0.445598\pi\)
\(11\)	−15538.3	−	7482.86i	−0.319990	−	0.154099i	0.266992	−	0.963699i	\(-0.413970\pi\)
							−0.586983	+	0.809599i	\(0.699684\pi\)
\(13\)	60533.4	−	154237.i	0.587827	−	1.49776i	−0.258992	−	0.965879i	\(-0.583390\pi\)
							0.846820	−	0.531880i	\(-0.178514\pi\)
\(17\)	−578972.	−	87266.0i	−1.68127	−	0.253411i	−0.762154	−	0.647396i	\(-0.775858\pi\)
							−0.919117	+	0.393985i	\(0.871096\pi\)
\(19\)	9843.04	−	131346.i	0.0173276	−	0.231221i	−0.981816	−	0.189836i	\(-0.939204\pi\)
							0.999143	−	0.0413841i	\(-0.0131767\pi\)
\(23\)	−1.13169e6	+	771576.i	−0.843245	+	0.574915i	−0.906079	−	0.423109i	\(-0.860939\pi\)
							0.0628338	+	0.998024i	\(0.479986\pi\)
\(29\)	−400099.	+	371238.i	−0.105045	+	0.0974677i	−0.730947	−	0.682434i	\(-0.760921\pi\)
							0.625902	+	0.779902i	\(0.284731\pi\)
\(31\)	−3.38257e6	−	1.04339e6i	−0.657839	−	0.202917i	−0.0521764	−	0.998638i	\(-0.516616\pi\)
							−0.605663	+	0.795721i	\(0.707092\pi\)
\(37\)	−3.40450e6	+	5.89677e6i	−0.298639	+	0.517257i	−0.975825	−	0.218555i	\(-0.929866\pi\)
							0.677186	+	0.735812i	\(0.263199\pi\)
\(41\)	4.05755e6	+	1.77773e7i	0.224252	+	0.982513i	0.954237	+	0.299050i	\(0.0966698\pi\)
							−0.729985	+	0.683463i	\(0.760473\pi\)
\(43\)	−1.83772e7	−	1.28402e7i	−0.819732	−	0.572748i
							\(47\)	1.21305e7	−	5.84174e6i	0.362609	−	0.174623i	−0.243704	−	0.969850i	\(-0.578362\pi\)
0.606312	+	0.795227i	\(0.292648\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.10.18	✓	384
43.13	even	21	inner	43.10.g.a.13.18	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.18	✓	384	1.1	even	1	trivial
43.10.g.a.13.18	yes	384	43.13	even	21	inner