Embedded newform 21.6.g.c.5.4

• Label: 21.6.g.c.5.4
• Level: $21$
• Weight: $6$
• Character: 21.5
• Analytic conductor: $3.368$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.36806021607\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 171*x^14 + 21495*x^12 - 1128902*x^10 + 42970860*x^8 - 655075344*x^6 + 7244325760*x^4 - 29387167488*x^2 + 90230547456
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.4
Root			\(-1.84692 - 1.06632i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.5
Dual form			21.6.g.c.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84692 - 1.06632i) q^{2} +(-6.34074 + 14.2406i) q^{3} +(-13.7259 - 23.7740i) q^{4} +(47.5262 - 82.3179i) q^{5} +(26.8959 - 19.5400i) q^{6} +(-95.0499 - 88.1619i) q^{7} +126.790i q^{8} +(-162.590 - 180.592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 18 q^{3} + 86 q^{4} - 616 q^{7} - 606 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84692	−	1.06632i	−0.326493	−	0.188501i	0.327790	−	0.944751i	\(-0.393696\pi\)
							−0.654283	+	0.756250i	\(0.727029\pi\)
\(3\)	−6.34074	+	14.2406i	−0.406759	+	0.913536i
							\(5\)	47.5262	−	82.3179i	0.850175	−	1.47255i	−0.0308744	−	0.999523i	\(-0.509829\pi\)
0.881050	−	0.473024i	\(-0.156837\pi\)
\(7\)	−95.0499	−	88.1619i	−0.733173	−	0.680042i
							\(11\)	113.171	−	65.3395i	0.282004	−	0.162815i	−0.352327	−	0.935877i	\(-0.614609\pi\)
0.634330	+	0.773062i	\(0.281276\pi\)
\(13\)			14.6710i			0.0240769i	0.999928	+	0.0120385i	\(0.00383205\pi\)
							−0.999928	+	0.0120385i	\(0.996168\pi\)
\(17\)	−320.246	−	554.683i	−0.268758	−	0.465503i	0.699783	−	0.714355i	\(-0.253280\pi\)
							−0.968541	+	0.248852i	\(0.919947\pi\)
\(19\)	−469.972	−	271.339i	−0.298668	−	0.172436i	0.343177	−	0.939271i	\(-0.388497\pi\)
							−0.641844	+	0.766835i	\(0.721830\pi\)
\(23\)	3545.78	+	2047.16i	1.39763	+	0.806922i	0.994144	−	0.108065i	\(-0.0344653\pi\)
							0.403485	+	0.914986i	\(0.367799\pi\)
\(29\)		−	3594.70i		−	0.793721i	−0.917879	−	0.396861i	\(-0.870100\pi\)
							0.917879	−	0.396861i	\(-0.129900\pi\)
\(31\)	−2512.28	+	1450.47i	−0.469531	+	0.271084i	−0.716043	−	0.698056i	\(-0.754049\pi\)
							0.246512	+	0.969140i	\(0.420715\pi\)
\(37\)	2171.02	−	3760.32i	0.260712	−	0.451566i	−0.705720	−	0.708491i	\(-0.749376\pi\)
							0.966431	+	0.256926i	\(0.0827095\pi\)
\(41\)	10945.3			1.01687			0.508436	−	0.861100i	\(-0.330224\pi\)
							0.508436	+	0.861100i	\(0.330224\pi\)
\(43\)	20541.6			1.69419			0.847096	−	0.531440i	\(-0.178349\pi\)
							0.847096	+	0.531440i	\(0.178349\pi\)
\(47\)	−4408.02	+	7634.91i	−0.291071	+	0.504149i	−0.974063	−	0.226276i	\(-0.927345\pi\)
							0.682992	+	0.730425i	\(0.260678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.6.g.c.5.4	✓	16
3.2	odd	2	inner	21.6.g.c.5.5	yes	16
7.2	even	3		147.6.c.c.146.9		16
7.3	odd	6	inner	21.6.g.c.17.5	yes	16
7.5	odd	6		147.6.c.c.146.10		16
21.2	odd	6		147.6.c.c.146.8		16
21.5	even	6		147.6.c.c.146.7		16
21.17	even	6	inner	21.6.g.c.17.4	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.g.c.5.4	✓	16	1.1	even	1	trivial
21.6.g.c.5.5	yes	16	3.2	odd	2	inner
21.6.g.c.17.4	yes	16	21.17	even	6	inner
21.6.g.c.17.5	yes	16	7.3	odd	6	inner
147.6.c.c.146.7		16	21.5	even	6
147.6.c.c.146.8		16	21.2	odd	6
147.6.c.c.146.9		16	7.2	even	3
147.6.c.c.146.10		16	7.5	odd	6