Embedded newform 33.3.g.a.7.3

• Label: 33.3.g.a.7.3
• Level: $33$
• Weight: $3$
• Character: 33.7
• Analytic conductor: $0.899$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.899184872389\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 3*x^14 - 4*x^13 + 77*x^12 + 88*x^11 - 577*x^10 + 578*x^9 + 1520*x^8 + 1868*x^7 - 1619*x^6 - 16804*x^5 + 32427*x^4 + 43316*x^3 - 71672*x^2 + 83521
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			7.3
Root			\(-0.797732 + 1.94863i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.7
Dual form			33.3.g.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30204 - 1.79211i) q^{2} +(0.535233 - 1.64728i) q^{3} +(-0.280267 - 0.862573i) q^{4} +(-7.03442 + 5.11081i) q^{5} +(-2.25520 - 3.10402i) q^{6} +(6.34535 - 2.06173i) q^{7} +(6.51625 + 2.11726i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30204	−	1.79211i	0.651021	−	0.896054i	−0.348122	−	0.937449i	\(-0.613180\pi\)
							0.999143	+	0.0413956i	\(0.0131804\pi\)
\(3\)	0.535233	−	1.64728i	0.178411	−	0.549093i
							\(5\)	−7.03442	+	5.11081i	−1.40688	+	1.02216i	−0.413118	+	0.910678i	\(0.635560\pi\)
−0.993766	+	0.111484i	\(0.964440\pi\)
\(7\)	6.34535	−	2.06173i	0.906478	−	0.294533i	0.181570	−	0.983378i	\(-0.441882\pi\)
							0.724908	+	0.688846i	\(0.241882\pi\)
\(11\)	−8.95368	−	6.38997i	−0.813970	−	0.580906i
							\(13\)	−6.55655	+	9.02432i	−0.504350	+	0.694178i	−0.982954	−	0.183853i	\(-0.941143\pi\)
0.478604	+	0.878031i	\(0.341143\pi\)
\(17\)	−4.60052	−	6.33207i	−0.270619	−	0.372475i	0.651980	−	0.758236i	\(-0.273939\pi\)
							−0.922599	+	0.385762i	\(0.873939\pi\)
\(19\)	−8.02898	−	2.60877i	−0.422578	−	0.137304i	0.0900055	−	0.995941i	\(-0.471312\pi\)
							−0.512584	+	0.858637i	\(0.671312\pi\)
\(23\)	9.30611			0.404613			0.202307	−	0.979322i	\(-0.435156\pi\)
							0.202307	+	0.979322i	\(0.435156\pi\)
\(29\)	6.82718	−	2.21829i	0.235420	−	0.0764926i	−0.188931	−	0.981990i	\(-0.560502\pi\)
							0.424351	+	0.905498i	\(0.360502\pi\)
\(31\)	22.1867	+	16.1196i	0.715701	+	0.519987i	0.885008	−	0.465576i	\(-0.154153\pi\)
							−0.169307	+	0.985563i	\(0.554153\pi\)
\(37\)	−16.3598	−	50.3504i	−0.442158	−	1.36082i	−0.885571	−	0.464505i	\(-0.846232\pi\)
							0.443412	−	0.896318i	\(-0.353768\pi\)
\(41\)	45.5114	+	14.7876i	1.11003	+	0.360672i	0.805956	−	0.591975i	\(-0.201652\pi\)
							0.304078	+	0.952647i	\(0.401652\pi\)
\(43\)			45.8381i			1.06600i	0.846114	+	0.533002i	\(0.178936\pi\)
							−0.846114	+	0.533002i	\(0.821064\pi\)
\(47\)	−4.96833	+	15.2909i	−0.105709	+	0.325339i	−0.989896	−	0.141794i	\(-0.954713\pi\)
							0.884187	+	0.467133i	\(0.154713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.7.3	✓	16
3.2	odd	2		99.3.k.c.73.2		16
4.3	odd	2		528.3.bf.b.337.1		16
11.2	odd	10		363.3.g.g.94.3		16
11.3	even	5		363.3.g.f.118.2		16
11.4	even	5		363.3.g.g.112.3		16
11.5	even	5		363.3.c.e.241.12		16
11.6	odd	10		363.3.c.e.241.5		16
11.7	odd	10		363.3.g.a.112.2		16
11.8	odd	10	inner	33.3.g.a.19.3	yes	16
11.9	even	5		363.3.g.a.94.2		16
11.10	odd	2		363.3.g.f.40.2		16
33.5	odd	10		1089.3.c.m.604.5		16
33.8	even	10		99.3.k.c.19.2		16
33.17	even	10		1089.3.c.m.604.12		16
44.19	even	10		528.3.bf.b.481.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.3	✓	16	1.1	even	1	trivial
33.3.g.a.19.3	yes	16	11.8	odd	10	inner
99.3.k.c.19.2		16	33.8	even	10
99.3.k.c.73.2		16	3.2	odd	2
363.3.c.e.241.5		16	11.6	odd	10
363.3.c.e.241.12		16	11.5	even	5
363.3.g.a.94.2		16	11.9	even	5
363.3.g.a.112.2		16	11.7	odd	10
363.3.g.f.40.2		16	11.10	odd	2
363.3.g.f.118.2		16	11.3	even	5
363.3.g.g.94.3		16	11.2	odd	10
363.3.g.g.112.3		16	11.4	even	5
528.3.bf.b.337.1		16	4.3	odd	2
528.3.bf.b.481.1		16	44.19	even	10
1089.3.c.m.604.5		16	33.5	odd	10
1089.3.c.m.604.12		16	33.17	even	10