Embedded newform 33.3.g.a.7.4

• Label: 33.3.g.a.7.4
• 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			7.4
Root			\(-1.43448 + 2.82504i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.7
Dual form			33.3.g.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.69557 - 2.33376i) q^{2} +(-0.535233 + 1.64728i) q^{3} +(-1.33538 - 4.10989i) q^{4} +(0.356879 - 0.259287i) q^{5} +(2.93682 + 4.04219i) q^{6} +(-10.0641 + 3.27002i) q^{7} +(-0.881730 - 0.286491i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.69557	−	2.33376i	0.847787	−	1.16688i	−0.136559	−	0.990632i	\(-0.543604\pi\)
							0.984346	−	0.176246i	\(-0.0563956\pi\)
\(3\)	−0.535233	+	1.64728i	−0.178411	+	0.549093i
							\(5\)	0.356879	−	0.259287i	0.0713757	−	0.0518575i	−0.551525	−	0.834158i	\(-0.685954\pi\)
0.622901	+	0.782301i	\(0.285954\pi\)
\(7\)	−10.0641	+	3.27002i	−1.43773	+	0.467146i	−0.921188	−	0.389118i	\(-0.872780\pi\)
							−0.516539	+	0.856264i	\(0.672780\pi\)
\(11\)	9.39298	+	5.72468i	0.853907	+	0.520425i
							\(13\)	3.78967	−	5.21603i	0.291513	−	0.401233i	−0.637992	−	0.770043i	\(-0.720235\pi\)
0.929505	+	0.368810i	\(0.120235\pi\)
\(17\)	−11.1080	−	15.2889i	−0.653414	−	0.899347i	0.345827	−	0.938298i	\(-0.387598\pi\)
							−0.999241	+	0.0389508i	\(0.987598\pi\)
\(19\)	−26.1968	−	8.51187i	−1.37878	−	0.447993i	−0.476512	−	0.879168i	\(-0.658099\pi\)
							−0.902268	+	0.431175i	\(0.858099\pi\)
\(23\)	6.29263			0.273593			0.136796	−	0.990599i	\(-0.456319\pi\)
							0.136796	+	0.990599i	\(0.456319\pi\)
\(29\)	42.0059	−	13.6485i	1.44848	−	0.470639i	0.523949	−	0.851750i	\(-0.324458\pi\)
							0.924529	+	0.381111i	\(0.124458\pi\)
\(31\)	21.9270	+	15.9309i	0.707323	+	0.513900i	0.882309	−	0.470671i	\(-0.155988\pi\)
							−0.174986	+	0.984571i	\(0.555988\pi\)
\(37\)	−0.263636	−	0.811388i	−0.00712529	−	0.0219294i	0.947431	−	0.319961i	\(-0.103670\pi\)
							−0.954556	+	0.298032i	\(0.903670\pi\)
\(41\)	−12.5843	−	4.08890i	−0.306935	−	0.0997292i	0.151500	−	0.988457i	\(-0.451590\pi\)
							−0.458435	+	0.888728i	\(0.651590\pi\)
\(43\)			68.8186i			1.60043i	0.599712	+	0.800216i	\(0.295282\pi\)
							−0.599712	+	0.800216i	\(0.704718\pi\)
\(47\)	−4.98694	+	15.3482i	−0.106105	+	0.326558i	−0.989988	−	0.141150i	\(-0.954920\pi\)
							0.883883	+	0.467708i	\(0.154920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.7.4	✓	16
3.2	odd	2		99.3.k.c.73.1		16
4.3	odd	2		528.3.bf.b.337.4		16
11.2	odd	10		363.3.g.g.94.4		16
11.3	even	5		363.3.g.f.118.1		16
11.4	even	5		363.3.g.g.112.4		16
11.5	even	5		363.3.c.e.241.15		16
11.6	odd	10		363.3.c.e.241.2		16
11.7	odd	10		363.3.g.a.112.1		16
11.8	odd	10	inner	33.3.g.a.19.4	yes	16
11.9	even	5		363.3.g.a.94.1		16
11.10	odd	2		363.3.g.f.40.1		16
33.5	odd	10		1089.3.c.m.604.2		16
33.8	even	10		99.3.k.c.19.1		16
33.17	even	10		1089.3.c.m.604.15		16
44.19	even	10		528.3.bf.b.481.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.4	✓	16	1.1	even	1	trivial
33.3.g.a.19.4	yes	16	11.8	odd	10	inner
99.3.k.c.19.1		16	33.8	even	10
99.3.k.c.73.1		16	3.2	odd	2
363.3.c.e.241.2		16	11.6	odd	10
363.3.c.e.241.15		16	11.5	even	5
363.3.g.a.94.1		16	11.9	even	5
363.3.g.a.112.1		16	11.7	odd	10
363.3.g.f.40.1		16	11.10	odd	2
363.3.g.f.118.1		16	11.3	even	5
363.3.g.g.94.4		16	11.2	odd	10
363.3.g.g.112.4		16	11.4	even	5
528.3.bf.b.337.4		16	4.3	odd	2
528.3.bf.b.481.4		16	44.19	even	10
1089.3.c.m.604.2		16	33.5	odd	10
1089.3.c.m.604.15		16	33.17	even	10