Embedded newform 33.3.g.a.13.2

• Label: 33.3.g.a.13.2
• Level: $33$
• Weight: $3$
• Character: 33.13
• 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			13.2
Root			\(-1.29715 + 0.104262i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.13
Dual form			33.3.g.a.28.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40785 - 0.782357i) q^{2} +(1.40126 - 1.01807i) q^{3} +(1.94958 + 1.41645i) q^{4} +(-2.61024 - 8.03348i) q^{5} +(-4.17052 + 1.35508i) q^{6} +(1.43445 - 1.97435i) q^{7} +(2.36641 + 3.25708i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40785	−	0.782357i	−1.20392	−	0.391179i	−0.362721	−	0.931898i	\(-0.618152\pi\)
							−0.841203	+	0.540719i	\(0.818152\pi\)
\(3\)	1.40126	−	1.01807i	0.467086	−	0.339358i
							\(5\)	−2.61024	−	8.03348i	−0.522047	−	1.60670i	−0.770082	−	0.637945i	\(-0.779785\pi\)
0.248034	−	0.968751i	\(-0.420215\pi\)
\(7\)	1.43445	−	1.97435i	0.204921	−	0.282050i	−0.694170	−	0.719811i	\(-0.744228\pi\)
							0.899091	+	0.437761i	\(0.144228\pi\)
\(11\)	2.51703	+	10.7082i	0.228821	+	0.973469i
							\(13\)	11.1993	+	3.63888i	0.861485	+	0.279914i	0.706249	−	0.707964i	\(-0.250386\pi\)
0.155236	+	0.987877i	\(0.450386\pi\)
\(17\)	−1.93053	+	0.627268i	−0.113561	+	0.0368981i	−0.365246	−	0.930911i	\(-0.619015\pi\)
							0.251685	+	0.967809i	\(0.419015\pi\)
\(19\)	−4.97861	−	6.85246i	−0.262032	−	0.360656i	0.657648	−	0.753326i	\(-0.271552\pi\)
							−0.919680	+	0.392670i	\(0.871552\pi\)
\(23\)	41.9571			1.82422			0.912110	−	0.409946i	\(-0.134452\pi\)
							0.912110	+	0.409946i	\(0.134452\pi\)
\(29\)	−14.4679	+	19.9133i	−0.498893	+	0.686667i	−0.981997	−	0.188896i	\(-0.939509\pi\)
							0.483105	+	0.875563i	\(0.339509\pi\)
\(31\)	6.74074	−	20.7459i	0.217443	−	0.669222i	−0.781528	−	0.623870i	\(-0.785559\pi\)
							0.998971	−	0.0453514i	\(-0.0144407\pi\)
\(37\)	12.9478	+	9.40714i	0.349941	+	0.254247i	0.748844	−	0.662746i	\(-0.230609\pi\)
							−0.398903	+	0.916993i	\(0.630609\pi\)
\(41\)	−30.9272	−	42.5677i	−0.754323	−	1.03824i	−0.997665	−	0.0682958i	\(-0.978244\pi\)
							0.243342	−	0.969940i	\(-0.421756\pi\)
\(43\)			42.3507i			0.984900i	0.870341	+	0.492450i	\(0.163899\pi\)
							−0.870341	+	0.492450i	\(0.836101\pi\)
\(47\)	13.0910	−	9.51117i	0.278532	−	0.202365i	−0.439745	−	0.898123i	\(-0.644931\pi\)
							0.718277	+	0.695757i	\(0.244931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.13.2	✓	16
3.2	odd	2		99.3.k.c.46.3		16
4.3	odd	2		528.3.bf.b.145.1		16
11.2	odd	10		363.3.g.a.118.2		16
11.3	even	5		363.3.g.a.40.2		16
11.4	even	5		363.3.c.e.241.3		16
11.5	even	5		363.3.g.f.94.3		16
11.6	odd	10	inner	33.3.g.a.28.2	yes	16
11.7	odd	10		363.3.c.e.241.14		16
11.8	odd	10		363.3.g.g.40.3		16
11.9	even	5		363.3.g.g.118.3		16
11.10	odd	2		363.3.g.f.112.3		16
33.17	even	10		99.3.k.c.28.3		16
33.26	odd	10		1089.3.c.m.604.14		16
33.29	even	10		1089.3.c.m.604.3		16
44.39	even	10		528.3.bf.b.193.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.2	✓	16	1.1	even	1	trivial
33.3.g.a.28.2	yes	16	11.6	odd	10	inner
99.3.k.c.28.3		16	33.17	even	10
99.3.k.c.46.3		16	3.2	odd	2
363.3.c.e.241.3		16	11.4	even	5
363.3.c.e.241.14		16	11.7	odd	10
363.3.g.a.40.2		16	11.3	even	5
363.3.g.a.118.2		16	11.2	odd	10
363.3.g.f.94.3		16	11.5	even	5
363.3.g.f.112.3		16	11.10	odd	2
363.3.g.g.40.3		16	11.8	odd	10
363.3.g.g.118.3		16	11.9	even	5
528.3.bf.b.145.1		16	4.3	odd	2
528.3.bf.b.193.1		16	44.39	even	10
1089.3.c.m.604.3		16	33.29	even	10
1089.3.c.m.604.14		16	33.26	odd	10