Embedded newform 43.10.g.a.9.15

• Label: 43.10.g.a.9.15
• 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			9.15
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.32399 - 5.42212i) q^{2} +(83.5118 + 12.5874i) q^{3} +(103.228 - 452.273i) q^{4} +(-697.274 - 475.394i) q^{5} +(-292.854 - 507.238i) q^{6} +(3015.21 - 5222.49i) q^{7} +(-6097.79 + 2936.54i) q^{8} +(-11992.8 - 3699.28i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.32399	−	5.42212i	−0.191095	−	0.239626i	0.677049	−	0.735938i	\(-0.263259\pi\)
							−0.868144	+	0.496312i	\(0.834687\pi\)
\(3\)	83.5118	+	12.5874i	0.595253	+	0.0897200i	0.439758	−	0.898117i	\(-0.355064\pi\)
							0.155496	+	0.987837i	\(0.450302\pi\)
\(5\)	−697.274	−	475.394i	−0.498929	−	0.340164i	0.287566	−	0.957761i	\(-0.407154\pi\)
							−0.786495	+	0.617597i	\(0.788106\pi\)
\(7\)	3015.21	−	5222.49i	0.474653	−	0.822123i	−0.524926	−	0.851148i	\(-0.675907\pi\)
							0.999579	+	0.0290251i	\(0.00924028\pi\)
\(11\)	−6452.18	−	28268.9i	−0.132874	−	0.582159i	−0.996898	−	0.0787077i	\(-0.974921\pi\)
							0.864024	−	0.503451i	\(-0.167937\pi\)
\(13\)	−10990.0	+	146652.i	−0.106722	+	1.42411i	0.644776	+	0.764371i	\(0.276951\pi\)
							−0.751498	+	0.659735i	\(0.770669\pi\)
\(17\)	−58486.9	+	39875.7i	−0.169839	+	0.115794i	−0.645257	−	0.763966i	\(-0.723250\pi\)
							0.475417	+	0.879760i	\(0.342297\pi\)
\(19\)	797406.	−	245967.i	1.40374	−	0.432998i	0.501850	−	0.864954i	\(-0.332653\pi\)
							0.901894	+	0.431956i	\(0.142177\pi\)
\(23\)	−644771.	+	598260.i	−0.480430	+	0.445774i	−0.882832	−	0.469689i	\(-0.844366\pi\)
							0.402402	+	0.915463i	\(0.368176\pi\)
\(29\)	−5.34099e6	+	805025.i	−1.40227	+	0.211358i	−0.806232	−	0.591600i	\(-0.798497\pi\)
							−0.596036	+	0.802958i	\(0.703258\pi\)
\(31\)	−3.04664e6	+	7.76271e6i	−0.592507	+	1.50968i	0.248485	+	0.968636i	\(0.420067\pi\)
							−0.840992	+	0.541047i	\(0.818028\pi\)
\(37\)	−3.49242e6	−	6.04905e6i	−0.306350	−	0.530615i	0.671211	−	0.741267i	\(-0.265775\pi\)
							−0.977561	+	0.210652i	\(0.932441\pi\)
\(41\)	−1.74301e7	−	2.18567e7i	−0.963324	−	1.20797i	−0.978112	−	0.208080i	\(-0.933279\pi\)
							0.0147877	−	0.999891i	\(-0.495293\pi\)
\(43\)	−1.64853e6	+	2.23579e7i	−0.0735341	+	0.997293i
							\(47\)	1.37646e7	−	6.03068e7i	0.411457	−	1.80271i	−0.165814	−	0.986157i	\(-0.553025\pi\)
0.577271	−	0.816553i	\(-0.304118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.15	✓	384
43.24	even	21	inner	43.10.g.a.24.15	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.15	✓	384	1.1	even	1	trivial
43.10.g.a.24.15	yes	384	43.24	even	21	inner