Embedded newform 43.8.g.a.10.16

• Label: 43.8.g.a.10.16
• Level: $43$
• Weight: $8$
• Character: 43.10
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.16
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04456 + 4.57650i) q^{2} +(18.6406 + 17.2959i) q^{3} +(95.4708 - 45.9763i) q^{4} +(-284.562 + 42.8908i) q^{5} +(-59.6836 + 103.375i) q^{6} +(-811.776 - 1406.04i) q^{7} +(684.763 + 858.665i) q^{8} +(-115.113 - 1536.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04456	+	4.57650i	0.0923265	+	0.404509i	0.999881	−	0.0154254i	\(-0.00491025\pi\)
							−0.907554	+	0.419934i	\(0.862053\pi\)
\(3\)	18.6406	+	17.2959i	0.398598	+	0.369845i	0.853851	−	0.520517i	\(-0.174261\pi\)
							−0.455253	+	0.890362i	\(0.650451\pi\)
\(5\)	−284.562	+	42.8908i	−1.01808	+	0.153451i	−0.636817	−	0.771015i	\(-0.719749\pi\)
							−0.381264	+	0.924466i	\(0.624511\pi\)
\(7\)	−811.776	−	1406.04i	−0.894526	−	1.54936i	−0.834390	−	0.551174i	\(-0.814180\pi\)
							−0.0601360	−	0.998190i	\(-0.519153\pi\)
\(11\)	−3640.76	−	1753.30i	−0.824742	−	0.397175i	−0.0266012	−	0.999646i	\(-0.508468\pi\)
							−0.798140	+	0.602472i	\(0.794183\pi\)
\(13\)	−4270.55	+	10881.2i	−0.539116	+	1.37365i	0.358773	+	0.933425i	\(0.383195\pi\)
							−0.897889	+	0.440221i	\(0.854900\pi\)
\(17\)	−7774.03	−	1171.75i	−0.383774	−	0.0578446i	−0.0456777	−	0.998956i	\(-0.514545\pi\)
							−0.338096	+	0.941112i	\(0.609783\pi\)
\(19\)	3754.26	−	50097.2i	0.125570	−	1.67562i	−0.478038	−	0.878339i	\(-0.658652\pi\)
							0.603608	−	0.797281i	\(-0.293729\pi\)
\(23\)	70095.1	−	47790.0i	1.20127	−	0.819011i	0.213808	−	0.976876i	\(-0.431413\pi\)
							0.987461	+	0.157865i	\(0.0504610\pi\)
\(29\)	−149073.	+	138320.i	−1.13503	+	1.05315i	−0.136990	+	0.990572i	\(0.543743\pi\)
							−0.998040	+	0.0625810i	\(0.980067\pi\)
\(31\)	−9586.28	−	2956.98i	−0.0577942	−	0.0178271i	0.265723	−	0.964049i	\(-0.414389\pi\)
							−0.323517	+	0.946222i	\(0.604865\pi\)
\(37\)	113182.	−	196037.i	0.367342	−	0.636255i	−0.621807	−	0.783171i	\(-0.713601\pi\)
							0.989149	+	0.146915i	\(0.0469345\pi\)
\(41\)	66056.3	+	289411.i	0.149682	+	0.655801i	0.992973	+	0.118343i	\(0.0377583\pi\)
							−0.843290	+	0.537458i	\(0.819385\pi\)
\(43\)	−520938.	−	21023.7i	−0.999187	−	0.0403246i
							\(47\)	550628.	−	265169.i	0.773599	−	0.372546i	−0.00506473	−	0.999987i	\(-0.501612\pi\)
0.778664	+	0.627442i	\(0.215898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.16	✓	300
43.13	even	21	inner	43.8.g.a.13.16	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.16	✓	300	1.1	even	1	trivial
43.8.g.a.13.16	yes	300	43.13	even	21	inner