Embedded newform 43.10.g.a.10.32

• Label: 43.10.g.a.10.32
• 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			10.32
Character	\(\chi\)	\(=\)	43.10
Dual form			43.10.g.a.13.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.69863 + 42.4925i) q^{2} +(-28.1854 - 26.1522i) q^{3} +(-1250.25 + 602.089i) q^{4} +(461.591 - 69.5736i) q^{5} +(837.913 - 1451.31i) q^{6} +(1972.96 + 3417.27i) q^{7} +(-23796.4 - 29839.7i) q^{8} +(-1360.43 - 18153.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.69863	+	42.4925i	0.428623	+	1.87792i	0.476684	+	0.879075i	\(0.341839\pi\)
							−0.0480608	+	0.998844i	\(0.515304\pi\)
\(3\)	−28.1854	−	26.1522i	−0.200899	−	0.186407i	0.573307	−	0.819341i	\(-0.305660\pi\)
							−0.774206	+	0.632933i	\(0.781851\pi\)
\(5\)	461.591	−	69.5736i	0.330288	−	0.0497828i	0.0181948	−	0.999834i	\(-0.494208\pi\)
							0.312093	+	0.950052i	\(0.398970\pi\)
\(7\)	1972.96	+	3417.27i	0.310583	+	0.537946i	0.978489	−	0.206300i	\(-0.0661423\pi\)
							−0.667905	+	0.744246i	\(0.732809\pi\)
\(11\)	−70140.0	−	33777.6i	−1.44444	−	0.695604i	−0.462817	−	0.886454i	\(-0.653161\pi\)
							−0.981619	+	0.190850i	\(0.938876\pi\)
\(13\)	30257.9	−	77095.7i	0.293828	−	0.748661i	−0.705379	−	0.708831i	\(-0.749223\pi\)
							0.999206	−	0.0398305i	\(-0.0126818\pi\)
\(17\)	523176.	+	78856.1i	1.51924	+	0.228989i	0.855040	−	0.518562i	\(-0.173532\pi\)
							0.664204	+	0.747551i	\(0.268771\pi\)
\(19\)	−13335.4	+	177949.i	−0.0234755	+	0.313259i	0.973088	+	0.230432i	\(0.0740139\pi\)
							−0.996564	+	0.0828270i	\(0.973605\pi\)
\(23\)	−1.21419e6	+	827819.i	−0.904712	+	0.616823i	−0.923750	−	0.382996i	\(-0.874892\pi\)
							0.0190374	+	0.999819i	\(0.493940\pi\)
\(29\)	3.59348e6	−	3.33426e6i	0.943461	−	0.875404i	−0.0488961	−	0.998804i	\(-0.515570\pi\)
							0.992357	+	0.123400i	\(0.0393798\pi\)
\(31\)	−4.61179e6	−	1.42255e6i	−0.896895	−	0.276655i	−0.188167	−	0.982137i	\(-0.560255\pi\)
							−0.708728	+	0.705482i	\(0.750731\pi\)
\(37\)	3.78228e6	−	6.55110e6i	0.331777	−	0.574654i	−0.651083	−	0.759006i	\(-0.725685\pi\)
							0.982860	+	0.184352i	\(0.0590186\pi\)
\(41\)	−3.72724e6	−	1.63301e7i	−0.205997	−	0.902531i	−0.967200	−	0.254017i	\(-0.918248\pi\)
							0.761203	−	0.648514i	\(-0.224609\pi\)
\(43\)	−2.93364e6	−	2.22258e7i	−0.130858	−	0.991401i
							\(47\)	−2.24782e7	+	1.08249e7i	−0.671926	+	0.323583i	−0.738553	−	0.674195i	\(-0.764491\pi\)
0.0666268	+	0.997778i	\(0.478776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.10.32	✓	384
43.13	even	21	inner	43.10.g.a.13.32	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.32	✓	384	1.1	even	1	trivial
43.10.g.a.13.32	yes	384	43.13	even	21	inner