Embedded newform 21.30.e.a.4.18

• Label: 21.30.e.a.4.18
• Level: $21$
• Weight: $30$
• Character: 21.4
• Analytic conductor: $111.884$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(111.883889004\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.18
Character	\(\chi\)	\(=\)	21.4
Dual form			21.30.e.a.16.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(20330.6 + 35213.6i) q^{2} +(2.39148e6 - 4.14217e6i) q^{3} +(-5.58229e8 + 9.66881e8i) q^{4} +(9.23100e9 + 1.59886e10i) q^{5} +1.94481e11 q^{6} +(-1.57660e12 - 8.56870e11i) q^{7} -2.35667e13 q^{8} +(-1.14384e13 - 1.98119e13i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 25025 q^{2} + 90876411 q^{3} - 5448583733 q^{4} - 2101591778 q^{5} + 239387598450 q^{6} - 1147384421099 q^{7} - 53840580588510 q^{8} - 434659056644259 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	20330.6	+	35213.6i	0.877434	+	1.51976i	0.854147	+	0.520032i	\(0.174080\pi\)
							0.0232878	+	0.999729i	\(0.492587\pi\)
\(3\)	2.39148e6	−	4.14217e6i	0.288675	−	0.500000i
							\(5\)	9.23100e9	+	1.59886e10i	0.676369	+	1.17151i	0.976067	+	0.217472i	\(0.0697810\pi\)
−0.299697	+	0.954034i	\(0.596886\pi\)
\(7\)	−1.57660e12	−	8.56870e11i	−0.878620	−	0.477522i
							\(11\)	9.06994e14	−	1.57096e15i	0.720129	−	1.24730i	−0.240818	−	0.970570i	\(-0.577416\pi\)
0.960948	−	0.276731i	\(-0.0892510\pi\)
\(13\)	2.11635e16			1.49076			0.745381	−	0.666639i	\(-0.232268\pi\)
							0.745381	+	0.666639i	\(0.232268\pi\)
\(17\)	5.75857e17	−	9.97414e17i	0.829479	−	1.43670i	−0.0689677	−	0.997619i	\(-0.521971\pi\)
							0.898447	−	0.439082i	\(-0.144696\pi\)
\(19\)	2.82058e18	+	4.88539e18i	0.809863	+	1.40272i	0.912958	+	0.408053i	\(0.133792\pi\)
							−0.103095	+	0.994672i	\(0.532875\pi\)
\(23\)	1.41544e19	+	2.45161e19i	0.254587	+	0.440958i	0.964783	−	0.263046i	\(-0.0847270\pi\)
							−0.710196	+	0.704004i	\(0.751394\pi\)
\(29\)	6.27320e20			0.391488			0.195744	−	0.980655i	\(-0.437288\pi\)
							0.195744	+	0.980655i	\(0.437288\pi\)
\(31\)	2.07819e21	−	3.59953e21i	0.493106	−	0.854085i	−0.506862	−	0.862027i	\(-0.669195\pi\)
							0.999968	+	0.00794192i	\(0.00252802\pi\)
\(37\)	3.15446e22	+	5.46368e22i	0.575439	+	0.996690i	0.995994	+	0.0894227i	\(0.0285022\pi\)
							−0.420555	+	0.907267i	\(0.638164\pi\)
\(41\)	−2.01356e23			−0.829083			−0.414542	−	0.910030i	\(-0.636058\pi\)
							−0.414542	+	0.910030i	\(0.636058\pi\)
\(43\)	8.48922e22			0.175217			0.0876083	−	0.996155i	\(-0.472078\pi\)
							0.0876083	+	0.996155i	\(0.472078\pi\)
\(47\)	−1.30764e24	−	2.26490e24i	−0.743138	−	1.28715i	−0.951059	−	0.309008i	\(-0.900003\pi\)
							0.207921	−	0.978146i	\(-0.433330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.30.e.a.4.18	✓	38
7.2	even	3	inner	21.30.e.a.16.18	yes	38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.30.e.a.4.18	✓	38	1.1	even	1	trivial
21.30.e.a.16.18	yes	38	7.2	even	3	inner