Embedded newform 43.10.g.a.10.17

• Label: 43.10.g.a.10.17
• 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			10.17
Character	\(\chi\)	\(=\)	43.10
Dual form			43.10.g.a.13.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.721278 + 3.16013i) q^{2} +(-124.032 - 115.085i) q^{3} +(451.830 - 217.590i) q^{4} +(2339.37 - 352.603i) q^{5} +(274.222 - 474.966i) q^{6} +(3672.08 + 6360.22i) q^{7} +(2048.25 + 2568.42i) q^{8} +(668.503 + 8920.55i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.721278	+	3.16013i	0.0318763	+	0.139659i	0.988362	−	0.152120i	\(-0.0486100\pi\)
							−0.956486	+	0.291779i	\(0.905753\pi\)
\(3\)	−124.032	−	115.085i	−0.884075	−	0.820301i	0.100560	−	0.994931i	\(-0.467937\pi\)
							−0.984634	+	0.174630i	\(0.944127\pi\)
\(5\)	2339.37	−	352.603i	1.67391	−	0.252302i	0.757588	−	0.652733i	\(-0.226378\pi\)
							0.916327	+	0.400431i	\(0.131140\pi\)
\(7\)	3672.08	+	6360.22i	0.578057	+	1.00122i	0.995702	+	0.0926138i	\(0.0295222\pi\)
							−0.417645	+	0.908610i	\(0.637144\pi\)
\(11\)	27734.8	+	13356.4i	0.571160	+	0.275056i	0.697091	−	0.716983i	\(-0.254477\pi\)
							−0.125931	+	0.992039i	\(0.540192\pi\)
\(13\)	−54570.3	+	139043.i	−0.529921	+	1.35022i	0.375941	+	0.926644i	\(0.377320\pi\)
							−0.905862	+	0.423573i	\(0.860776\pi\)
\(17\)	366967.	+	55311.4i	1.06563	+	0.160618i	0.658371	−	0.752694i	\(-0.271246\pi\)
							0.407261	+	0.913312i	\(0.366484\pi\)
\(19\)	−73082.5	+	975219.i	−0.128654	+	1.71677i	0.443211	+	0.896417i	\(0.353839\pi\)
							−0.571865	+	0.820348i	\(0.693780\pi\)
\(23\)	687467.	−	468707.i	0.512244	−	0.349242i	−0.279450	−	0.960160i	\(-0.590152\pi\)
							0.791694	+	0.610918i	\(0.209200\pi\)
\(29\)	143261.	−	132927.i	0.0376130	−	0.0348998i	−0.661142	−	0.750260i	\(-0.729928\pi\)
							0.698755	+	0.715361i	\(0.253738\pi\)
\(31\)	−5.07165e6	−	1.56440e6i	−0.986328	−	0.304242i	−0.240697	−	0.970600i	\(-0.577376\pi\)
							−0.745632	+	0.666359i	\(0.767852\pi\)
\(37\)	−1.86131e6	+	3.22388e6i	−0.163272	+	0.282795i	−0.936040	−	0.351893i	\(-0.885538\pi\)
							0.772769	+	0.634688i	\(0.218871\pi\)
\(41\)	−6.49588e6	−	2.84603e7i	−0.359014	−	1.57294i	−0.755657	−	0.654968i	\(-0.772682\pi\)
							0.396643	−	0.917973i	\(-0.370175\pi\)
\(43\)	2.89653e6	−	2.22307e7i	0.129202	−	0.991618i
							\(47\)	3.30199e7	−	1.59016e7i	0.987042	−	0.475334i	0.130520	−	0.991446i	\(-0.458335\pi\)
0.856522	+	0.516111i	\(0.172621\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.10.17	✓	384
43.13	even	21	inner	43.10.g.a.13.17	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.17	✓	384	1.1	even	1	trivial
43.10.g.a.13.17	yes	384	43.13	even	21	inner