Embedded newform 43.10.g.a.13.17

• Label: 43.10.g.a.13.17
• Level: $43$
• Weight: $10$
• Character: 43.13
• 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			13.17
Character	\(\chi\)	\(=\)	43.13
Dual form			43.10.g.a.10.17

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

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