Embedded newform 33.3.g.a.28.4

• Label: 33.3.g.a.28.4
• Level: $33$
• Weight: $3$
• Character: 33.28
• 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			28.4
Root			\(1.64608 - 1.06057i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.28
Dual form			33.3.g.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.35440 - 0.764990i) q^{2} +(-1.40126 - 1.01807i) q^{3} +(1.72190 - 1.25104i) q^{4} +(-0.789076 + 2.42853i) q^{5} +(-4.07793 - 1.32500i) q^{6} +(-0.100159 - 0.137856i) q^{7} +(-2.72337 + 3.74840i) q^{8} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.35440	−	0.764990i	1.17720	−	0.382495i	0.345873	−	0.938281i	\(-0.387583\pi\)
							0.831325	+	0.555786i	\(0.187583\pi\)
\(3\)	−1.40126	−	1.01807i	−0.467086	−	0.339358i
							\(5\)	−0.789076	+	2.42853i	−0.157815	+	0.485705i	−0.998435	−	0.0559199i	\(-0.982191\pi\)
0.840620	+	0.541625i	\(0.182191\pi\)
\(7\)	−0.100159	−	0.137856i	−0.0143084	−	0.0196938i	0.801803	−	0.597589i	\(-0.203874\pi\)
							−0.816111	+	0.577895i	\(0.803874\pi\)
\(11\)	−7.69573	−	7.85976i	−0.699612	−	0.714523i
							\(13\)	18.3310	−	5.95611i	1.41008	−	0.458163i	0.497643	−	0.867382i	\(-0.334199\pi\)
0.912436	+	0.409219i	\(0.134199\pi\)
\(17\)	−19.3014	−	6.27142i	−1.13538	−	0.368907i	−0.319761	−	0.947498i	\(-0.603603\pi\)
							−0.815618	+	0.578591i	\(0.803603\pi\)
\(19\)	8.57343	−	11.8003i	0.451233	−	0.621069i	−0.521429	−	0.853295i	\(-0.674601\pi\)
							0.972662	+	0.232226i	\(0.0746008\pi\)
\(23\)	7.74583			0.336775			0.168388	−	0.985721i	\(-0.446144\pi\)
							0.168388	+	0.985721i	\(0.446144\pi\)
\(29\)	22.4904	+	30.9554i	0.775532	+	1.06743i	0.995761	+	0.0919801i	\(0.0293196\pi\)
							−0.220229	+	0.975448i	\(0.570680\pi\)
\(31\)	−13.0940	−	40.2993i	−0.422389	−	1.29998i	−0.905473	−	0.424405i	\(-0.860483\pi\)
							0.483084	−	0.875574i	\(-0.339517\pi\)
\(37\)	−41.7807	+	30.3554i	−1.12921	+	0.820417i	−0.985579	−	0.169214i	\(-0.945877\pi\)
							−0.143628	+	0.989632i	\(0.545877\pi\)
\(41\)	−27.1820	+	37.4128i	−0.662976	+	0.912508i	−0.999575	−	0.0291401i	\(-0.990723\pi\)
							0.336600	+	0.941648i	\(0.390723\pi\)
\(43\)			59.7836i			1.39032i	0.718857	+	0.695158i	\(0.244666\pi\)
							−0.718857	+	0.695158i	\(0.755334\pi\)
\(47\)	−27.6137	−	20.0626i	−0.587526	−	0.426863i	0.253903	−	0.967230i	\(-0.418286\pi\)
							−0.841430	+	0.540367i	\(0.818286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.28.4	yes	16
3.2	odd	2		99.3.k.c.28.1		16
4.3	odd	2		528.3.bf.b.193.3		16
11.2	odd	10	inner	33.3.g.a.13.4	✓	16
11.3	even	5		363.3.c.e.241.4		16
11.4	even	5		363.3.g.a.118.4		16
11.5	even	5		363.3.g.g.40.1		16
11.6	odd	10		363.3.g.a.40.4		16
11.7	odd	10		363.3.g.g.118.1		16
11.8	odd	10		363.3.c.e.241.13		16
11.9	even	5		363.3.g.f.112.1		16
11.10	odd	2		363.3.g.f.94.1		16
33.2	even	10		99.3.k.c.46.1		16
33.8	even	10		1089.3.c.m.604.4		16
33.14	odd	10		1089.3.c.m.604.13		16
44.35	even	10		528.3.bf.b.145.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.4	✓	16	11.2	odd	10	inner
33.3.g.a.28.4	yes	16	1.1	even	1	trivial
99.3.k.c.28.1		16	3.2	odd	2
99.3.k.c.46.1		16	33.2	even	10
363.3.c.e.241.4		16	11.3	even	5
363.3.c.e.241.13		16	11.8	odd	10
363.3.g.a.40.4		16	11.6	odd	10
363.3.g.a.118.4		16	11.4	even	5
363.3.g.f.94.1		16	11.10	odd	2
363.3.g.f.112.1		16	11.9	even	5
363.3.g.g.40.1		16	11.5	even	5
363.3.g.g.118.1		16	11.7	odd	10
528.3.bf.b.145.3		16	44.35	even	10
528.3.bf.b.193.3		16	4.3	odd	2
1089.3.c.m.604.4		16	33.8	even	10
1089.3.c.m.604.13		16	33.14	odd	10