Embedded newform 33.5.h.a.5.2

• Label: 33.5.h.a.5.2
• Level: $33$
• Weight: $5$
• Character: 33.5
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	33.5
Dual form			33.5.h.a.20.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.11716 + 1.98759i) q^{2} +(5.92004 + 6.77888i) q^{3} +(20.5249 - 14.9122i) q^{4} +(26.2596 + 8.53227i) q^{5} +(-49.6874 - 29.7009i) q^{6} +(-49.1578 + 35.7152i) q^{7} +(-35.4248 + 48.7581i) q^{8} +(-10.9063 + 80.2624i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 90 q^{4} - 83 q^{6} - 86 q^{7} + 232 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.11716	+	1.98759i	−1.52929	+	0.496896i	−0.948397	−	0.317084i	\(-0.897296\pi\)
							−0.580892	+	0.813980i	\(0.697296\pi\)
\(3\)	5.92004	+	6.77888i	0.657782	+	0.753208i
							\(5\)	26.2596	+	8.53227i	1.05039	+	0.341291i	0.782820	−	0.622249i	\(-0.213781\pi\)
0.267566	+	0.963540i	\(0.413781\pi\)
\(7\)	−49.1578	+	35.7152i	−1.00322	+	0.728882i	−0.962776	−	0.270300i	\(-0.912877\pi\)
							−0.0404437	+	0.999182i	\(0.512877\pi\)
\(11\)	−63.8969	+	102.753i	−0.528074	+	0.849198i
							\(13\)	−47.0040	−	144.663i	−0.278130	−	0.855997i	−0.988374	−	0.152041i	\(-0.951415\pi\)
0.710244	−	0.703956i	\(-0.248585\pi\)
\(17\)	−38.7120	−	12.5783i	−0.133952	−	0.0435235i	0.241274	−	0.970457i	\(-0.422435\pi\)
							−0.375226	+	0.926934i	\(0.622435\pi\)
\(19\)	377.439	+	274.225i	1.04554	+	0.759627i	0.971359	−	0.237618i	\(-0.0763667\pi\)
							0.0741779	+	0.997245i	\(0.476367\pi\)
\(23\)		−	179.292i		−	0.338926i	−0.985537	−	0.169463i	\(-0.945797\pi\)
							0.985537	−	0.169463i	\(-0.0542033\pi\)
\(29\)	800.506	+	1101.80i	0.951850	+	1.31011i	0.950701	+	0.310110i	\(0.100366\pi\)
							0.00114925	+	0.999999i	\(0.499634\pi\)
\(31\)	−9.57512	−	29.4692i	−0.00996370	−	0.0306651i	0.945951	−	0.324309i	\(-0.105132\pi\)
							−0.955915	+	0.293644i	\(0.905132\pi\)
\(37\)	2141.57	−	1555.94i	1.56433	−	1.13655i	0.631985	−	0.774981i	\(-0.282240\pi\)
							0.932344	−	0.361571i	\(-0.117760\pi\)
\(41\)	1213.82	−	1670.68i	0.722083	−	0.993863i	−0.277369	−	0.960764i	\(-0.589462\pi\)
							0.999452	−	0.0330990i	\(-0.0105377\pi\)
\(43\)	720.538			0.389691			0.194845	−	0.980834i	\(-0.437580\pi\)
							0.194845	+	0.980834i	\(0.437580\pi\)
\(47\)	328.274	−	451.831i	0.148608	−	0.204541i	−0.728223	−	0.685340i	\(-0.759653\pi\)
							0.876831	+	0.480799i	\(0.159653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.h.a.5.2	✓	56
3.2	odd	2	inner	33.5.h.a.5.13	yes	56
11.9	even	5	inner	33.5.h.a.20.13	yes	56
33.20	odd	10	inner	33.5.h.a.20.2	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.h.a.5.2	✓	56	1.1	even	1	trivial
33.5.h.a.5.13	yes	56	3.2	odd	2	inner
33.5.h.a.20.2	yes	56	33.20	odd	10	inner
33.5.h.a.20.13	yes	56	11.9	even	5	inner