Embedded newform 33.6.f.a.2.12

• Label: 33.6.f.a.2.12
• Level: $33$
• Weight: $6$
• Character: 33.2
• Analytic conductor: $5.293$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29266605383\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.12695 - 3.46841i) q^{2} +(4.32414 + 14.9767i) q^{3} +(15.1287 + 10.9916i) q^{4} +(-54.0225 + 17.5530i) q^{5} +(56.8185 + 1.88017i) q^{6} +(-123.528 + 170.022i) q^{7} +(149.586 - 108.681i) q^{8} +(-205.604 + 129.523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.12695	−	3.46841i	0.199219	−	0.613134i	−0.800682	−	0.599090i	\(-0.795529\pi\)
							0.999901	−	0.0140446i	\(-0.00447068\pi\)
\(3\)	4.32414	+	14.9767i	0.277394	+	0.960756i
							\(5\)	−54.0225	+	17.5530i	−0.966383	+	0.313997i	−0.749355	−	0.662168i	\(-0.769636\pi\)
−0.217028	+	0.976165i	\(0.569636\pi\)
\(7\)	−123.528	+	170.022i	−0.952842	+	1.31147i	−0.00258819	+	0.999997i	\(0.500824\pi\)
							−0.950254	+	0.311477i	\(0.899176\pi\)
\(11\)	371.428	−	151.962i	0.925534	−	0.378664i
							\(13\)	787.456	+	255.860i	1.29231	+	0.419898i	0.872901	−	0.487898i	\(-0.162236\pi\)
0.419412	+	0.907796i	\(0.362236\pi\)
\(17\)	−271.716	−	836.254i	−0.228030	−	0.701804i	−0.997970	−	0.0636879i	\(-0.979714\pi\)
							0.769940	−	0.638117i	\(-0.220286\pi\)
\(19\)	68.7247	+	94.5915i	0.0436746	+	0.0601130i	0.830296	−	0.557323i	\(-0.188171\pi\)
							−0.786621	+	0.617436i	\(0.788171\pi\)
\(23\)			181.300i			0.0714627i	0.999361	+	0.0357313i	\(0.0113761\pi\)
							−0.999361	+	0.0357313i	\(0.988624\pi\)
\(29\)	2737.59	+	1988.98i	0.604468	+	0.439172i	0.847462	−	0.530856i	\(-0.178129\pi\)
							−0.242994	+	0.970028i	\(0.578129\pi\)
\(31\)	3130.16	−	9633.65i	0.585009	−	1.80047i	−0.0142248	−	0.999899i	\(-0.504528\pi\)
							0.599234	−	0.800574i	\(-0.295472\pi\)
\(37\)	7064.64	+	5132.76i	0.848371	+	0.616378i	0.924696	−	0.380705i	\(-0.124319\pi\)
							−0.0763253	+	0.997083i	\(0.524319\pi\)
\(41\)	−9188.63	+	6675.93i	−0.853672	+	0.620229i	−0.926156	−	0.377140i	\(-0.876907\pi\)
							0.0724837	+	0.997370i	\(0.476907\pi\)
\(43\)			13220.1i			1.09034i	0.838325	+	0.545171i	\(0.183535\pi\)
							−0.838325	+	0.545171i	\(0.816465\pi\)
\(47\)	−5657.77	−	7787.25i	−0.373594	−	0.514209i	0.580279	−	0.814418i	\(-0.302944\pi\)
							−0.953873	+	0.300209i	\(0.902944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.12	yes	72
3.2	odd	2	inner	33.6.f.a.2.7	✓	72
11.6	odd	10	inner	33.6.f.a.17.7	yes	72
33.17	even	10	inner	33.6.f.a.17.12	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.7	✓	72	3.2	odd	2	inner
33.6.f.a.2.12	yes	72	1.1	even	1	trivial
33.6.f.a.17.7	yes	72	11.6	odd	10	inner
33.6.f.a.17.12	yes	72	33.17	even	10	inner