Embedded newform 21.6.e.c.4.1

• Label: 21.6.e.c.4.1
• 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			4.1
Root			\(-4.61193 - 7.98809i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.4
Dual form			21.6.e.c.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.11193 - 8.85412i) q^{2} +(-4.50000 + 7.79423i) q^{3} +(-36.2636 + 62.8104i) q^{4} +(-11.8764 - 20.5705i) q^{5} +92.0147 q^{6} +(30.6840 + 125.958i) q^{7} +414.344 q^{8} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} - 36 q^{3} - 69 q^{4} + 54 q^{6} + 258 q^{7} + 246 q^{8} - 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.11193	−	8.85412i	−0.903669	−	1.56520i	−0.822693	−	0.568485i	\(-0.807530\pi\)
							−0.0809760	−	0.996716i	\(-0.525804\pi\)
\(3\)	−4.50000	+	7.79423i	−0.288675	+	0.500000i
							\(5\)	−11.8764	−	20.5705i	−0.212452	−	0.367977i	0.740030	−	0.672574i	\(-0.234811\pi\)
−0.952481	+	0.304597i	\(0.901478\pi\)
\(7\)	30.6840	+	125.958i	0.236683	+	0.971587i
							\(11\)	−232.763	+	403.157i	−0.580006	+	1.00460i	0.415472	+	0.909606i	\(0.363616\pi\)
−0.995478	+	0.0949934i	\(0.969717\pi\)
\(13\)	−1019.30			−1.67280			−0.836399	−	0.548121i	\(-0.815343\pi\)
							−0.836399	+	0.548121i	\(0.815343\pi\)
\(17\)	−280.878	+	486.496i	−0.235720	+	0.408279i	−0.959482	−	0.281771i	\(-0.909078\pi\)
							0.723762	+	0.690050i	\(0.242411\pi\)
\(19\)	693.789	+	1201.68i	0.440903	+	0.763667i	0.997757	−	0.0669438i	\(-0.0213248\pi\)
							−0.556853	+	0.830611i	\(0.687991\pi\)
\(23\)	−2056.81	−	3562.50i	−0.810727	−	1.40422i	−0.912356	−	0.409397i	\(-0.865739\pi\)
							0.101630	−	0.994822i	\(-0.467594\pi\)
\(29\)	−2381.37			−0.525814			−0.262907	−	0.964821i	\(-0.584681\pi\)
							−0.262907	+	0.964821i	\(0.584681\pi\)
\(31\)	−1475.33	+	2555.34i	−0.275730	+	0.477579i	−0.970319	−	0.241828i	\(-0.922253\pi\)
							0.694589	+	0.719407i	\(0.255586\pi\)
\(37\)	4954.48	+	8581.40i	0.594968	+	1.03051i	0.993551	+	0.113382i	\(0.0361685\pi\)
							−0.398584	+	0.917132i	\(0.630498\pi\)
\(41\)	−4477.13			−0.415949			−0.207974	−	0.978134i	\(-0.566687\pi\)
							−0.207974	+	0.978134i	\(0.566687\pi\)
\(43\)	5181.48			0.427349			0.213675	−	0.976905i	\(-0.431457\pi\)
							0.213675	+	0.976905i	\(0.431457\pi\)
\(47\)	1560.80	+	2703.38i	0.103063	+	0.178510i	0.912945	−	0.408082i	\(-0.133802\pi\)
							−0.809882	+	0.586592i	\(0.800469\pi\)