Embedded newform 5.33.c.a.3.2

• Label: 5.33.c.a.3.2
• Level: $5$
• Weight: $33$
• Character: 5.3
• Analytic conductor: $32.433$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 33 \)
Character orbit:	\([\chi]\)	\(=\)	5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(32.4333275711\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.2
Character	\(\chi\)	\(=\)	5.3
Dual form			5.33.c.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-73528.8 + 73528.8i) q^{2} +(-4.29377e6 - 4.29377e6i) q^{3} -6.51799e9i q^{4} +(-1.18376e11 + 9.62823e10i) q^{5} +6.31431e11 q^{6} +(1.36489e13 - 1.36489e13i) q^{7} +(1.63456e14 + 1.63456e14i) q^{8} -1.81615e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−73528.8	+	73528.8i	−1.12196	+	1.12196i	−0.130513	+	0.991447i	\(0.541663\pi\)
							−0.991447	+	0.130513i	\(0.958337\pi\)
\(3\)	−4.29377e6	−	4.29377e6i	−0.0997466	−	0.0997466i	0.655472	−	0.755219i	\(-0.272470\pi\)
							−0.755219	+	0.655472i	\(0.772470\pi\)
\(5\)	−1.18376e11	+	9.62823e10i	−0.775786	+	0.630996i
							\(7\)	1.36489e13	−	1.36489e13i	0.410703	−	0.410703i	−0.471280	−	0.881983i	\(-0.656208\pi\)
0.881983	+	0.471280i	\(0.156208\pi\)
\(11\)	−1.07665e16			−0.234310			−0.117155	−	0.993114i	\(-0.537377\pi\)
							−0.117155	+	0.993114i	\(0.537377\pi\)
\(13\)	5.33423e17	+	5.33423e17i	0.801637	+	0.801637i	0.983351	−	0.181714i	\(-0.0581646\pi\)
							−0.181714	+	0.983351i	\(0.558165\pi\)
\(17\)	1.80495e19	−	1.80495e19i	0.370922	−	0.370922i	−0.496891	−	0.867813i	\(-0.665525\pi\)
							0.867813	+	0.496891i	\(0.165525\pi\)
\(19\)			4.66461e20i			1.61718i	0.588373	+	0.808589i	\(0.299769\pi\)
							−0.588373	+	0.808589i	\(0.700231\pi\)
\(23\)	7.85489e20	+	7.85489e20i	0.128084	+	0.128084i	0.768243	−	0.640159i	\(-0.221131\pi\)
							−0.640159	+	0.768243i	\(0.721131\pi\)
\(29\)		−	2.31649e23i		−	0.925685i	−0.886441	−	0.462842i	\(-0.846830\pi\)
							0.886441	−	0.462842i	\(-0.153170\pi\)
\(31\)	3.24657e23			0.446312			0.223156	−	0.974783i	\(-0.428364\pi\)
							0.223156	+	0.974783i	\(0.428364\pi\)
\(37\)	4.94737e24	−	4.94737e24i	0.401002	−	0.401002i	−0.477584	−	0.878586i	\(-0.658487\pi\)
							0.878586	+	0.477584i	\(0.158487\pi\)
\(41\)	−1.24479e26			−1.95234			−0.976168	−	0.217015i	\(-0.930368\pi\)
							−0.976168	+	0.217015i	\(0.930368\pi\)
\(43\)	−1.83776e25	−	1.83776e25i	−0.134522	−	0.134522i	0.636639	−	0.771162i	\(-0.280324\pi\)
							−0.771162	+	0.636639i	\(0.780324\pi\)
\(47\)	3.45135e26	−	3.45135e26i	0.608728	−	0.608728i	−0.333886	−	0.942613i	\(-0.608360\pi\)
							0.942613	+	0.333886i	\(0.108360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.3.2	yes	30
5.2	odd	4	inner	5.33.c.a.2.2	✓	30
5.3	odd	4		25.33.c.b.7.14		30
5.4	even	2		25.33.c.b.18.14		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.2	✓	30	5.2	odd	4	inner
5.33.c.a.3.2	yes	30	1.1	even	1	trivial
25.33.c.b.7.14		30	5.3	odd	4
25.33.c.b.18.14		30	5.4	even	2