Embedded newform 43.10.g.a.24.9

• Label: 43.10.g.a.24.9
• Level: $43$
• Weight: $10$
• Character: 43.24
• 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			24.9
Character	\(\chi\)	\(=\)	43.24
Dual form			43.10.g.a.9.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.1357 + 20.2335i) q^{2} +(-257.419 + 38.7997i) q^{3} +(-35.1038 - 153.800i) q^{4} +(-1370.08 + 934.106i) q^{5} +(3368.58 - 5834.56i) q^{6} +(71.3754 + 123.626i) q^{7} +(-8259.84 - 3977.73i) q^{8} +(45950.8 - 14173.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.1357	+	20.2335i	−0.713103	+	0.894203i	−0.997926	−	0.0643772i	\(-0.979494\pi\)
							0.284822	+	0.958580i	\(0.408065\pi\)
\(3\)	−257.419	+	38.7997i	−1.83483	+	0.276556i	−0.973027	−	0.230690i	\(-0.925902\pi\)
							−0.861801	+	0.507246i	\(0.830664\pi\)
\(5\)	−1370.08	+	934.106i	−0.980351	+	0.668392i	−0.943654	−	0.330935i	\(-0.892636\pi\)
							−0.0366969	+	0.999326i	\(0.511684\pi\)
\(7\)	71.3754	+	123.626i	0.0112359	+	0.0194611i	0.871589	−	0.490238i	\(-0.163090\pi\)
							−0.860353	+	0.509699i	\(0.829757\pi\)
\(11\)	5122.32	−	22442.3i	0.105487	−	0.462169i	−0.894402	−	0.447264i	\(-0.852398\pi\)
							0.999889	−	0.0149048i	\(-0.00474452\pi\)
\(13\)	−614.346	−	8197.88i	−0.00596579	−	0.0796079i	0.993426	−	0.114475i	\(-0.0365184\pi\)
							−0.999392	+	0.0348666i	\(0.988899\pi\)
\(17\)	−149481.	−	101914.i	−0.434076	−	0.295948i	0.326500	−	0.945197i	\(-0.394131\pi\)
							−0.760575	+	0.649250i	\(0.775083\pi\)
\(19\)	−319162.	−	98448.4i	−0.561849	−	0.173308i	0.000805249	−	1.00000i	\(-0.499744\pi\)
							−0.562655	+	0.826692i	\(0.690220\pi\)
\(23\)	847527.	+	786391.i	0.631507	+	0.585953i	0.929594	−	0.368586i	\(-0.120158\pi\)
							−0.298086	+	0.954539i	\(0.596348\pi\)
\(29\)	−6.57910e6	−	991640.i	−1.72733	−	0.260353i	−0.791087	−	0.611704i	\(-0.790484\pi\)
							−0.936244	+	0.351351i	\(0.885722\pi\)
\(31\)	−2.85498e6	−	7.27436e6i	−0.555233	−	1.41471i	−0.882740	−	0.469861i	\(-0.844304\pi\)
							0.327508	−	0.944848i	\(-0.393791\pi\)
\(37\)	−4.64512e6	+	8.04558e6i	−0.407464	+	0.705748i	−0.994605	−	0.103737i	\(-0.966920\pi\)
							0.587141	+	0.809485i	\(0.300253\pi\)
\(41\)	4.17621e6	−	5.23681e6i	0.230810	−	0.289427i	−0.652917	−	0.757430i	\(-0.726455\pi\)
							0.883727	+	0.468003i	\(0.155026\pi\)
\(43\)	2.19983e7	−	4.32039e6i	0.981255	−	0.192715i
							\(47\)	1.12269e7	+	4.91883e7i	0.335598	+	1.47035i	0.808112	+	0.589029i	\(0.200490\pi\)
−0.472514	+	0.881323i	\(0.656653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.24.9	yes	384
43.9	even	21	inner	43.10.g.a.9.9	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.9	✓	384	43.9	even	21	inner
43.10.g.a.24.9	yes	384	1.1	even	1	trivial