Embedded newform 5.33.c.a.2.8

• Label: 5.33.c.a.2.8
• Level: $5$
• Weight: $33$
• Character: 5.2
• 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			2.8
Character	\(\chi\)	\(=\)	5.2
Dual form			5.33.c.a.3.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5100.58 - 5100.58i) q^{2} +(-7.18185e6 + 7.18185e6i) q^{3} -4.24294e9i q^{4} +(1.52141e11 + 1.16729e10i) q^{5} +7.32632e10 q^{6} +(2.17171e13 + 2.17171e13i) q^{7} +(-4.35482e13 + 4.35482e13i) q^{8} +1.74986e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5100.58	−	5100.58i	−0.0778287	−	0.0778287i	0.667121	−	0.744950i	\(-0.267526\pi\)
							−0.744950	+	0.667121i	\(0.767526\pi\)
\(3\)	−7.18185e6	+	7.18185e6i	−0.166838	+	0.166838i	−0.785588	−	0.618750i	\(-0.787640\pi\)
							0.618750	+	0.785588i	\(0.287640\pi\)
\(5\)	1.52141e11	+	1.16729e10i	0.997070	+	0.0764996i
							\(7\)	2.17171e13	+	2.17171e13i	0.653481	+	0.653481i	0.953830	−	0.300348i	\(-0.0971030\pi\)
−0.300348	+	0.953830i	\(0.597103\pi\)
\(11\)	−5.19447e16			−1.13047			−0.565234	−	0.824930i	\(-0.691214\pi\)
							−0.565234	+	0.824930i	\(0.691214\pi\)
\(13\)	6.04100e17	−	6.04100e17i	0.907852	−	0.907852i	−0.0882469	−	0.996099i	\(-0.528126\pi\)
							0.996099	+	0.0882469i	\(0.0281265\pi\)
\(17\)	2.55450e19	+	2.55450e19i	0.524957	+	0.524957i	0.919064	−	0.394107i	\(-0.128946\pi\)
							−0.394107	+	0.919064i	\(0.628946\pi\)
\(19\)			1.62776e20i			0.564329i	0.959366	+	0.282165i	\(0.0910524\pi\)
							−0.959366	+	0.282165i	\(0.908948\pi\)
\(23\)	−2.79781e21	+	2.79781e21i	−0.456218	+	0.456218i	−0.897412	−	0.441194i	\(-0.854555\pi\)
							0.441194	+	0.897412i	\(0.354555\pi\)
\(29\)			3.52661e23i			1.40926i	0.709577	+	0.704628i	\(0.248886\pi\)
							−0.709577	+	0.704628i	\(0.751114\pi\)
\(31\)	4.98494e23			0.685287			0.342643	−	0.939466i	\(-0.388678\pi\)
							0.342643	+	0.939466i	\(0.388678\pi\)
\(37\)	1.32176e25	+	1.32176e25i	1.07134	+	1.07134i	0.997252	+	0.0740830i	\(0.0236030\pi\)
							0.0740830	+	0.997252i	\(0.476397\pi\)
\(41\)	1.17803e26			1.84762			0.923811	−	0.382849i	\(-0.125057\pi\)
							0.923811	+	0.382849i	\(0.125057\pi\)
\(43\)	9.57807e25	−	9.57807e25i	0.701105	−	0.701105i	−0.263543	−	0.964648i	\(-0.584891\pi\)
							0.964648	+	0.263543i	\(0.0848911\pi\)
\(47\)	−4.63469e26	−	4.63469e26i	−0.817439	−	0.817439i	0.168297	−	0.985736i	\(-0.446173\pi\)
							−0.985736	+	0.168297i	\(0.946173\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.2.8	✓	30
5.2	odd	4		25.33.c.b.18.8		30
5.3	odd	4	inner	5.33.c.a.3.8	yes	30
5.4	even	2		25.33.c.b.7.8		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.8	✓	30	1.1	even	1	trivial
5.33.c.a.3.8	yes	30	5.3	odd	4	inner
25.33.c.b.7.8		30	5.4	even	2
25.33.c.b.18.8		30	5.2	odd	4