Embedded newform 21.4.g.a.5.3

• Label: 21.4.g.a.5.3
• Level: $21$
• Weight: $4$
• Character: 21.5
• Analytic conductor: $1.239$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.23904011012\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.3
Root			\(-2.23014 + 2.00661i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.5
Dual form			21.4.g.a.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65310 - 0.954416i) q^{2} +(-3.47555 - 3.86271i) q^{3} +(-2.17818 - 3.77272i) q^{4} +(0.623706 - 1.08029i) q^{5} +(2.05878 + 9.70256i) q^{6} +(10.0808 - 15.5363i) q^{7} +23.5862i q^{8} +(-2.84113 + 26.8501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 14 q^{4} - 56 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65310	−	0.954416i	−0.584458	−	0.337437i	0.178445	−	0.983950i	\(-0.442893\pi\)
							−0.762903	+	0.646513i	\(0.776227\pi\)
\(3\)	−3.47555	−	3.86271i	−0.668870	−	0.743380i
							\(5\)	0.623706	−	1.08029i	0.0557859	−	0.0966240i	−0.836784	−	0.547533i	\(-0.815567\pi\)
0.892570	+	0.450909i	\(0.148900\pi\)
\(7\)	10.0808	−	15.5363i	0.544314	−	0.838881i
							\(11\)	35.2392	−	20.3453i	0.965910	−	0.557668i	0.0679230	−	0.997691i	\(-0.478363\pi\)
0.897987	+	0.440022i	\(0.145029\pi\)
\(13\)		−	19.5973i		−	0.418101i	−0.977905	−	0.209050i	\(-0.932963\pi\)
							0.977905	−	0.209050i	\(-0.0670373\pi\)
\(17\)	−52.3592	−	90.6889i	−0.746999	−	1.29384i	−0.949255	−	0.314507i	\(-0.898161\pi\)
							0.202256	−	0.979333i	\(-0.435173\pi\)
\(19\)	35.0345	+	20.2272i	0.423025	+	0.244234i	0.696371	−	0.717682i	\(-0.254797\pi\)
							−0.273346	+	0.961916i	\(0.588130\pi\)
\(23\)	69.6324	+	40.2023i	0.631276	+	0.364467i	0.781246	−	0.624223i	\(-0.214584\pi\)
							−0.149970	+	0.988691i	\(0.547918\pi\)
\(29\)			211.712i			1.35565i	0.735222	+	0.677827i	\(0.237078\pi\)
							−0.735222	+	0.677827i	\(0.762922\pi\)
\(31\)	−86.6242	+	50.0125i	−0.501876	+	0.289758i	−0.729488	−	0.683994i	\(-0.760242\pi\)
							0.227612	+	0.973752i	\(0.426908\pi\)
\(37\)	94.9875	−	164.523i	0.422050	−	0.731012i	−0.574090	−	0.818792i	\(-0.694644\pi\)
							0.996140	+	0.0877801i	\(0.0279773\pi\)
\(41\)	−186.753			−0.711362			−0.355681	−	0.934607i	\(-0.615751\pi\)
							−0.355681	+	0.934607i	\(0.615751\pi\)
\(43\)	158.618			0.562536			0.281268	−	0.959629i	\(-0.409245\pi\)
							0.281268	+	0.959629i	\(0.409245\pi\)
\(47\)	179.034	−	310.097i	0.555635	−	0.962388i	−0.442219	−	0.896907i	\(-0.645809\pi\)
							0.997854	−	0.0654808i	\(-0.0208581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.4.g.a.5.3	✓	12
3.2	odd	2	inner	21.4.g.a.5.4	yes	12
4.3	odd	2		336.4.bc.d.257.5		12
7.2	even	3		147.4.c.a.146.8		12
7.3	odd	6	inner	21.4.g.a.17.4	yes	12
7.4	even	3		147.4.g.d.80.4		12
7.5	odd	6		147.4.c.a.146.7		12
7.6	odd	2		147.4.g.d.68.3		12
12.11	even	2		336.4.bc.d.257.3		12
21.2	odd	6		147.4.c.a.146.5		12
21.5	even	6		147.4.c.a.146.6		12
21.11	odd	6		147.4.g.d.80.3		12
21.17	even	6	inner	21.4.g.a.17.3	yes	12
21.20	even	2		147.4.g.d.68.4		12
28.3	even	6		336.4.bc.d.17.3		12
84.59	odd	6		336.4.bc.d.17.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.g.a.5.3	✓	12	1.1	even	1	trivial
21.4.g.a.5.4	yes	12	3.2	odd	2	inner
21.4.g.a.17.3	yes	12	21.17	even	6	inner
21.4.g.a.17.4	yes	12	7.3	odd	6	inner
147.4.c.a.146.5		12	21.2	odd	6
147.4.c.a.146.6		12	21.5	even	6
147.4.c.a.146.7		12	7.5	odd	6
147.4.c.a.146.8		12	7.2	even	3
147.4.g.d.68.3		12	7.6	odd	2
147.4.g.d.68.4		12	21.20	even	2
147.4.g.d.80.3		12	21.11	odd	6
147.4.g.d.80.4		12	7.4	even	3
336.4.bc.d.17.3		12	28.3	even	6
336.4.bc.d.17.5		12	84.59	odd	6
336.4.bc.d.257.3		12	12.11	even	2
336.4.bc.d.257.5		12	4.3	odd	2