Embedded newform 33.3.g.a.28.3

• Label: 33.3.g.a.28.3
• 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			28.3
Root			\(0.988132 - 0.846795i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.28
Dual form			33.3.g.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28981 - 0.419086i) q^{2} +(1.40126 + 1.01807i) q^{3} +(-1.74808 + 1.27006i) q^{4} +(0.708979 - 2.18201i) q^{5} +(2.23402 + 0.725878i) q^{6} +(-5.74346 - 7.90520i) q^{7} +(-4.91103 + 6.75946i) q^{8} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28981	−	0.419086i	0.644907	−	0.209543i	0.0317398	−	0.999496i	\(-0.489895\pi\)
							0.613167	+	0.789953i	\(0.289895\pi\)
\(3\)	1.40126	+	1.01807i	0.467086	+	0.339358i
							\(5\)	0.708979	−	2.18201i	0.141796	−	0.436402i	−0.854789	−	0.518975i	\(-0.826314\pi\)
0.996585	+	0.0825729i	\(0.0263137\pi\)
\(7\)	−5.74346	−	7.90520i	−0.820495	−	1.12931i	−0.989618	−	0.143720i	\(-0.954094\pi\)
							0.169123	−	0.985595i	\(-0.445906\pi\)
\(11\)	10.3340	+	3.76954i	0.939450	+	0.342686i
							\(13\)	−14.6363	+	4.75561i	−1.12587	+	0.365816i	−0.812004	−	0.583652i	\(-0.801623\pi\)
−0.313862	+	0.949468i	\(0.601623\pi\)
\(17\)	11.2386	+	3.65165i	0.661096	+	0.214803i	0.620300	−	0.784364i	\(-0.287011\pi\)
							0.0407956	+	0.999168i	\(0.487011\pi\)
\(19\)	7.10329	−	9.77683i	0.373857	−	0.514570i	−0.580087	−	0.814555i	\(-0.696981\pi\)
							0.953944	+	0.299984i	\(0.0969814\pi\)
\(23\)	16.6610			0.724390			0.362195	−	0.932102i	\(-0.382027\pi\)
							0.362195	+	0.932102i	\(0.382027\pi\)
\(29\)	−15.6013	−	21.4734i	−0.537977	−	0.740462i	0.450343	−	0.892856i	\(-0.351302\pi\)
							−0.988320	+	0.152394i	\(0.951302\pi\)
\(31\)	1.28594	+	3.95770i	0.0414818	+	0.127668i	0.969653	−	0.244486i	\(-0.0786193\pi\)
							−0.928171	+	0.372154i	\(0.878619\pi\)
\(37\)	−54.4646	+	39.5709i	−1.47202	+	1.06948i	−0.491994	+	0.870599i	\(0.663732\pi\)
							−0.980023	+	0.198884i	\(0.936268\pi\)
\(41\)	−10.6080	+	14.6006i	−0.258731	+	0.356113i	−0.918545	−	0.395316i	\(-0.870635\pi\)
							0.659814	+	0.751429i	\(0.270635\pi\)
\(43\)		−	46.3735i		−	1.07845i	−0.842160	−	0.539227i	\(-0.818716\pi\)
							0.842160	−	0.539227i	\(-0.181284\pi\)
\(47\)	−49.2812	−	35.8049i	−1.04854	−	0.761806i	−0.0766030	−	0.997062i	\(-0.524407\pi\)
							−0.971934	+	0.235255i	\(0.924407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.28.3	yes	16
3.2	odd	2		99.3.k.c.28.2		16
4.3	odd	2		528.3.bf.b.193.2		16
11.2	odd	10	inner	33.3.g.a.13.3	✓	16
11.3	even	5		363.3.c.e.241.6		16
11.4	even	5		363.3.g.a.118.3		16
11.5	even	5		363.3.g.g.40.2		16
11.6	odd	10		363.3.g.a.40.3		16
11.7	odd	10		363.3.g.g.118.2		16
11.8	odd	10		363.3.c.e.241.11		16
11.9	even	5		363.3.g.f.112.2		16
11.10	odd	2		363.3.g.f.94.2		16
33.2	even	10		99.3.k.c.46.2		16
33.8	even	10		1089.3.c.m.604.6		16
33.14	odd	10		1089.3.c.m.604.11		16
44.35	even	10		528.3.bf.b.145.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.3	✓	16	11.2	odd	10	inner
33.3.g.a.28.3	yes	16	1.1	even	1	trivial
99.3.k.c.28.2		16	3.2	odd	2
99.3.k.c.46.2		16	33.2	even	10
363.3.c.e.241.6		16	11.3	even	5
363.3.c.e.241.11		16	11.8	odd	10
363.3.g.a.40.3		16	11.6	odd	10
363.3.g.a.118.3		16	11.4	even	5
363.3.g.f.94.2		16	11.10	odd	2
363.3.g.f.112.2		16	11.9	even	5
363.3.g.g.40.2		16	11.5	even	5
363.3.g.g.118.2		16	11.7	odd	10
528.3.bf.b.145.2		16	44.35	even	10
528.3.bf.b.193.2		16	4.3	odd	2
1089.3.c.m.604.6		16	33.8	even	10
1089.3.c.m.604.11		16	33.14	odd	10