Embedded newform 43.10.g.a.13.18

• Label: 43.10.g.a.13.18
• Level: $43$
• Weight: $10$
• Character: 43.13
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• 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			13.18
Character	\(\chi\)	\(=\)	43.13
Dual form			43.10.g.a.10.18

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

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