Embedded newform 43.10.c.a.36.4

• Label: 43.10.c.a.36.4
• Level: $43$
• Weight: $10$
• Character: 43.36
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			36.4
Character	\(\chi\)	\(=\)	43.36
Dual form			43.10.c.a.6.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-38.7934 q^{2} +(-85.4891 - 148.071i) q^{3} +992.932 q^{4} +(642.710 + 1113.21i) q^{5} +(3316.42 + 5744.20i) q^{6} +(-2797.86 + 4846.03i) q^{7} -18657.0 q^{8} +(-4775.28 + 8271.02i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 70 q^{2} + 307 q^{3} + 16038 q^{4} + 681 q^{5} - 1281 q^{6} + 957 q^{7} - 10920 q^{8} - 255339 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−38.7934			−1.71444			−0.857222	−	0.514947i	\(-0.827812\pi\)
							−0.857222	+	0.514947i	\(0.827812\pi\)
\(3\)	−85.4891	−	148.071i	−0.609348	−	1.05542i	−0.991348	−	0.131259i	\(-0.958098\pi\)
							0.382000	−	0.924162i	\(-0.375235\pi\)
\(5\)	642.710	+	1113.21i	0.459886	+	0.796545i	0.998954	−	0.0457166i	\(-0.0145571\pi\)
							−0.539069	+	0.842262i	\(0.681224\pi\)
\(7\)	−2797.86	+	4846.03i	−0.440438	+	0.762861i	−0.997722	−	0.0674613i	\(-0.978510\pi\)
							0.557284	+	0.830322i	\(0.311843\pi\)
\(11\)	39952.5			0.822767			0.411384	−	0.911462i	\(-0.365046\pi\)
							0.411384	+	0.911462i	\(0.365046\pi\)
\(13\)	17994.0	−	31166.4i	0.174736	−	0.302651i	−0.765334	−	0.643633i	\(-0.777426\pi\)
							0.940070	+	0.340982i	\(0.110760\pi\)
\(17\)	−57239.6	+	99141.9i	−0.166217	+	0.287897i	−0.937087	−	0.349096i	\(-0.886489\pi\)
							0.770870	+	0.636993i	\(0.219822\pi\)
\(19\)	−90037.1	−	155949.i	−0.158500	−	0.274531i	0.775828	−	0.630945i	\(-0.217333\pi\)
							−0.934328	+	0.356414i	\(0.883999\pi\)
\(23\)	−1.09202e6	−	1.89144e6i	−0.813685	−	1.40934i	−0.910268	−	0.414019i	\(-0.864125\pi\)
							0.0965834	−	0.995325i	\(-0.469209\pi\)
\(29\)	−1.63261e6	+	2.82777e6i	−0.428640	+	0.742426i	−0.996753	−	0.0805250i	\(-0.974340\pi\)
							0.568113	+	0.822951i	\(0.307674\pi\)
\(31\)	1.87091e6	+	3.24050e6i	0.363852	+	0.630209i	0.988591	−	0.150624i	\(-0.0481282\pi\)
							−0.624740	+	0.780833i	\(0.714795\pi\)
\(37\)	5.71339e6	+	9.89589e6i	0.501171	+	0.868054i	0.999999	+	0.00135325i	\(0.000430752\pi\)
							−0.498828	+	0.866701i	\(0.666236\pi\)
\(41\)	1.08532e7			0.599831			0.299916	−	0.953966i	\(-0.403041\pi\)
							0.299916	+	0.953966i	\(0.403041\pi\)
\(43\)	−5.71493e6	+	2.16779e7i	−0.254920	+	0.966962i
							\(47\)	−2.27238e6			−0.0679266			−0.0339633	−	0.999423i	\(-0.510813\pi\)
−0.0339633	+	0.999423i	\(0.510813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.c.a.36.4	yes	64
43.6	even	3	inner	43.10.c.a.6.4	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.c.a.6.4	✓	64	43.6	even	3	inner
43.10.c.a.36.4	yes	64	1.1	even	1	trivial