Embedded newform 21.6.e.c.4.4

• Label: 21.6.e.c.4.4
• 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			4.4
Root			\(5.09061 + 8.81720i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.4
Dual form			21.6.e.c.16.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.59061 + 7.95118i) q^{2} +(-4.50000 + 7.79423i) q^{3} +(-26.1475 + 45.2888i) q^{4} +(-11.0358 - 19.1146i) q^{5} -82.6311 q^{6} +(126.882 + 26.6059i) q^{7} -186.333 q^{8} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} - 36 q^{3} - 69 q^{4} + 54 q^{6} + 258 q^{7} + 246 q^{8} - 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.59061	+	7.95118i	0.811514	+	1.40558i	0.911804	+	0.410625i	\(0.134689\pi\)
							−0.100291	+	0.994958i	\(0.531977\pi\)
\(3\)	−4.50000	+	7.79423i	−0.288675	+	0.500000i
							\(5\)	−11.0358	−	19.1146i	−0.197414	−	0.341932i	0.750275	−	0.661126i	\(-0.229921\pi\)
−0.947689	+	0.319194i	\(0.896588\pi\)
\(7\)	126.882	+	26.6059i	0.978715	+	0.205226i
							\(11\)	−208.355	+	360.881i	−0.519184	+	0.899254i	0.480567	+	0.876958i	\(0.340431\pi\)
−0.999751	+	0.0222959i	\(0.992902\pi\)
\(13\)	797.918			1.30948			0.654742	−	0.755853i	\(-0.272777\pi\)
							0.654742	+	0.755853i	\(0.272777\pi\)
\(17\)	687.775	−	1191.26i	0.577197	−	0.999735i	−0.418602	−	0.908170i	\(-0.637480\pi\)
							0.995799	−	0.0915652i	\(-0.0291870\pi\)
\(19\)	−1156.51	−	2003.14i	−0.734965	−	1.27300i	−0.954739	−	0.297446i	\(-0.903865\pi\)
							0.219774	−	0.975551i	\(-0.429468\pi\)
\(23\)	477.701	+	827.402i	0.188294	+	0.326135i	0.944682	−	0.327989i	\(-0.106371\pi\)
							−0.756388	+	0.654124i	\(0.773038\pi\)
\(29\)	−7035.29			−1.55341			−0.776707	−	0.629862i	\(-0.783111\pi\)
							−0.776707	+	0.629862i	\(0.783111\pi\)
\(31\)	−630.596	+	1092.22i	−0.117855	+	0.204130i	−0.918917	−	0.394450i	\(-0.870935\pi\)
							0.801063	+	0.598581i	\(0.204268\pi\)
\(37\)	−4888.22	−	8466.65i	−0.587012	−	1.01673i	−0.994621	−	0.103578i	\(-0.966971\pi\)
							0.407610	−	0.913156i	\(-0.366362\pi\)
\(41\)	−5400.95			−0.501777			−0.250888	−	0.968016i	\(-0.580723\pi\)
							−0.250888	+	0.968016i	\(0.580723\pi\)
\(43\)	19686.6			1.62367			0.811837	−	0.583885i	\(-0.198468\pi\)
							0.811837	+	0.583885i	\(0.198468\pi\)
\(47\)	−1028.28	−	1781.04i	−0.0678997	−	0.117606i	0.830077	−	0.557649i	\(-0.188296\pi\)
							−0.897977	+	0.440043i	\(0.854963\pi\)