Embedded newform 21.6.e.c.4.2

• Label: 21.6.e.c.4.2
• Level: $21$
• Weight: $6$
• Character: 21.4
• Analytic conductor: $3.368$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.36806021607\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.2
Root			\(-0.874091 - 1.51397i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.4
Dual form			21.6.e.c.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37409 - 2.37999i) q^{2} +(-4.50000 + 7.79423i) q^{3} +(12.2237 - 21.1722i) q^{4} +(-29.1836 - 50.5475i) q^{5} +24.7336 q^{6} +(-21.4366 - 127.857i) q^{7} -155.128 q^{8} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} - 36 q^{3} - 69 q^{4} + 54 q^{6} + 258 q^{7} + 246 q^{8} - 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37409	−	2.37999i	−0.242907	−	0.420728i	0.718634	−	0.695389i	\(-0.244768\pi\)
							−0.961541	+	0.274661i	\(0.911434\pi\)
\(3\)	−4.50000	+	7.79423i	−0.288675	+	0.500000i
							\(5\)	−29.1836	−	50.5475i	−0.522053	−	0.904222i	−0.999671	−	0.0256544i	\(-0.991833\pi\)
0.477618	−	0.878568i	\(-0.341500\pi\)
\(7\)	−21.4366	−	127.857i	−0.165352	−	0.986235i
							\(11\)	−8.71205	+	15.0897i	−0.0217089	+	0.0376010i	−0.876676	−	0.481082i	\(-0.840244\pi\)
0.854967	+	0.518683i	\(0.173577\pi\)
\(13\)	889.933			1.46049			0.730246	−	0.683185i	\(-0.239406\pi\)
							0.730246	+	0.683185i	\(0.239406\pi\)
\(17\)	−513.318	+	889.092i	−0.430788	+	0.746147i	−0.996941	−	0.0781529i	\(-0.975098\pi\)
							0.566153	+	0.824300i	\(0.308431\pi\)
\(19\)	869.702	+	1506.37i	0.552696	+	0.957297i	0.998079	+	0.0619572i	\(0.0197342\pi\)
							−0.445383	+	0.895340i	\(0.646932\pi\)
\(23\)	−1968.11	−	3408.87i	−0.775764	−	1.34366i	−0.934364	−	0.356320i	\(-0.884031\pi\)
							0.158599	−	0.987343i	\(-0.449302\pi\)
\(29\)	5633.53			1.24390			0.621950	−	0.783057i	\(-0.286341\pi\)
							0.621950	+	0.783057i	\(0.286341\pi\)
\(31\)	1548.27	−	2681.68i	0.289362	−	0.501190i	−0.684296	−	0.729205i	\(-0.739890\pi\)
							0.973658	+	0.228015i	\(0.0732236\pi\)
\(37\)	−2513.43	−	4353.39i	−0.301830	−	0.522785i	0.674720	−	0.738073i	\(-0.264264\pi\)
							−0.976551	+	0.215288i	\(0.930931\pi\)
\(41\)	18367.0			1.70639			0.853195	−	0.521592i	\(-0.174662\pi\)
							0.853195	+	0.521592i	\(0.174662\pi\)
\(43\)	−1630.91			−0.134511			−0.0672557	−	0.997736i	\(-0.521424\pi\)
							−0.0672557	+	0.997736i	\(0.521424\pi\)
\(47\)	4802.62	+	8318.38i	0.317127	+	0.549280i	0.979887	−	0.199551i	\(-0.0639485\pi\)
							−0.662760	+	0.748832i	\(0.730615\pi\)