Embedded newform 43.8.g.a.10.14

• Label: 43.8.g.a.10.14
• 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			10.14
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.252699 + 1.10715i) q^{2} +(-51.5382 - 47.8205i) q^{3} +(114.162 - 54.9776i) q^{4} +(396.846 - 59.8149i) q^{5} +(39.9206 - 69.1446i) q^{6} +(-274.041 - 474.654i) q^{7} +(180.347 + 226.148i) q^{8} +(205.955 + 2748.28i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.252699	+	1.10715i	0.0223357	+	0.0978589i	0.984867	−	0.173311i	\(-0.0554466\pi\)
							−0.962531	+	0.271170i	\(0.912589\pi\)
\(3\)	−51.5382	−	47.8205i	−1.10206	−	1.02256i	−0.999613	−	0.0278231i	\(-0.991142\pi\)
							−0.102446	−	0.994739i	\(-0.532667\pi\)
\(5\)	396.846	−	59.8149i	1.41980	−	0.214000i	0.606147	−	0.795353i	\(-0.292714\pi\)
							0.813652	+	0.581353i	\(0.197476\pi\)
\(7\)	−274.041	−	474.654i	−0.301976	−	0.523039i	0.674607	−	0.738177i	\(-0.264313\pi\)
							−0.976584	+	0.215138i	\(0.930980\pi\)
\(11\)	6639.06	+	3197.20i	1.50395	+	0.724262i	0.990963	−	0.134137i	\(-0.0428262\pi\)
							0.512983	+	0.858399i	\(0.328540\pi\)
\(13\)	2294.41	−	5846.05i	0.289647	−	0.738008i	−0.709789	−	0.704415i	\(-0.751210\pi\)
							0.999436	−	0.0335934i	\(-0.0106951\pi\)
\(17\)	−32693.3	−	4927.73i	−1.61394	−	0.243263i	−0.720835	−	0.693106i	\(-0.756242\pi\)
							−0.893107	+	0.449843i	\(0.851480\pi\)
\(19\)	3467.62	−	46272.2i	0.115983	−	1.54769i	−0.570962	−	0.820977i	\(-0.693430\pi\)
							0.686945	−	0.726709i	\(-0.258951\pi\)
\(23\)	−13334.6	+	9091.41i	−0.228525	+	0.155806i	−0.672170	−	0.740397i	\(-0.734638\pi\)
							0.443645	+	0.896203i	\(0.353685\pi\)
\(29\)	−139955.	+	129859.i	−1.06560	+	0.988736i	−0.999959	−	0.00910735i	\(-0.997101\pi\)
							−0.0656451	+	0.997843i	\(0.520911\pi\)
\(31\)	162179.	+	50025.7i	0.977754	+	0.301597i	0.742137	−	0.670249i	\(-0.233813\pi\)
							0.235618	+	0.971846i	\(0.424289\pi\)
\(37\)	−52821.4	+	91489.4i	−0.171437	+	0.296937i	−0.938922	−	0.344129i	\(-0.888174\pi\)
							0.767486	+	0.641066i	\(0.221508\pi\)
\(41\)	8263.52	+	36204.8i	0.0187250	+	0.0820395i	0.983427	−	0.181304i	\(-0.0580318\pi\)
							−0.964702	+	0.263344i	\(0.915175\pi\)
\(43\)	−514539.	−	84070.5i	−0.986913	−	0.161252i
							\(47\)	303554.	−	146184.i	0.426474	−	0.205379i	−0.208319	−	0.978061i	\(-0.566799\pi\)
0.634794	+	0.772682i	\(0.281085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.14	✓	300
43.13	even	21	inner	43.8.g.a.13.14	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.14	✓	300	1.1	even	1	trivial
43.8.g.a.13.14	yes	300	43.13	even	21	inner