Embedded newform 43.10.g.a.9.18

• Label: 43.10.g.a.9.18
• Level: $43$
• Weight: $10$
• Character: 43.9
• 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			9.18
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.74995 + 4.70229i) q^{2} +(29.2285 + 4.40549i) q^{3} +(105.881 - 463.896i) q^{4} +(328.917 + 224.252i) q^{5} +(88.8898 + 153.962i) q^{6} +(2917.98 - 5054.09i) q^{7} +(5352.87 - 2577.81i) q^{8} +(-17973.6 - 5544.13i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.74995	+	4.70229i	0.165726	+	0.207814i	0.857759	−	0.514052i	\(-0.171856\pi\)
							−0.692033	+	0.721866i	\(0.743285\pi\)
\(3\)	29.2285	+	4.40549i	0.208335	+	0.0314014i	0.252380	−	0.967628i	\(-0.418787\pi\)
							−0.0440457	+	0.999030i	\(0.514025\pi\)
\(5\)	328.917	+	224.252i	0.235354	+	0.160462i	0.675254	−	0.737585i	\(-0.264034\pi\)
							−0.439900	+	0.898047i	\(0.644986\pi\)
\(7\)	2917.98	−	5054.09i	0.459347	−	0.795612i	−0.539580	−	0.841935i	\(-0.681417\pi\)
							0.998927	+	0.0463224i	\(0.0147501\pi\)
\(11\)	11485.7	+	50322.0i	0.236531	+	1.03631i	0.944098	+	0.329665i	\(0.106936\pi\)
							−0.707566	+	0.706647i	\(0.750207\pi\)
\(13\)	10237.6	−	136611.i	0.0994149	−	1.32660i	−0.694906	−	0.719101i	\(-0.744554\pi\)
							0.794321	−	0.607499i	\(-0.207827\pi\)
\(17\)	−245586.	+	167437.i	−0.713153	+	0.486219i	−0.864737	−	0.502225i	\(-0.832515\pi\)
							0.151584	+	0.988444i	\(0.451563\pi\)
\(19\)	−476246.	+	146903.i	−0.838379	+	0.258606i	−0.684070	−	0.729417i	\(-0.739792\pi\)
							−0.154310	+	0.988023i	\(0.549315\pi\)
\(23\)	1.37504e6	−	1.27585e6i	1.02457	−	0.950658i	0.0257669	−	0.999668i	\(-0.491797\pi\)
							0.998798	+	0.0490105i	\(0.0156068\pi\)
\(29\)	2.86487e6	−	431810.i	0.752167	−	0.113371i	0.238242	−	0.971206i	\(-0.423429\pi\)
							0.513925	+	0.857835i	\(0.328191\pi\)
\(31\)	397917.	−	1.01388e6i	0.0773865	−	0.197178i	−0.886945	−	0.461875i	\(-0.847177\pi\)
							0.964332	+	0.264697i	\(0.0852721\pi\)
\(37\)	−1.11916e7	−	1.93844e7i	−0.981712	−	1.70038i	−0.655723	−	0.755002i	\(-0.727636\pi\)
							−0.325989	−	0.945374i	\(-0.605697\pi\)
\(41\)	−1.69923e7	−	2.13076e7i	−0.939127	−	1.17763i	−0.983916	−	0.178632i	\(-0.942833\pi\)
							0.0447889	−	0.998996i	\(-0.485738\pi\)
\(43\)	2.20579e7	+	4.00520e6i	0.983912	+	0.178655i
							\(47\)	−3.75046e6	+	1.64318e7i	−0.112110	+	0.491185i	0.887433	+	0.460938i	\(0.152487\pi\)
−0.999542	+	0.0302479i	\(0.990370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.18	✓	384
43.24	even	21	inner	43.10.g.a.24.18	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.18	✓	384	1.1	even	1	trivial
43.10.g.a.24.18	yes	384	43.24	even	21	inner