Embedded newform 5.33.c.a.3.3

• Label: 5.33.c.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	5.3
Dual form			5.33.c.a.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-70960.9 + 70960.9i) q^{2} +(-4.83700e7 - 4.83700e7i) q^{3} -5.77593e9i q^{4} +(1.51118e11 - 2.11291e10i) q^{5} +6.86476e12 q^{6} +(2.09740e13 - 2.09740e13i) q^{7} +(1.05090e14 + 1.05090e14i) q^{8} +2.82629e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−70960.9	+	70960.9i	−1.08278	+	1.08278i	−0.0865276	+	0.996249i	\(0.527577\pi\)
							−0.996249	+	0.0865276i	\(0.972423\pi\)
\(3\)	−4.83700e7	−	4.83700e7i	−1.12366	−	1.12366i	−0.991186	−	0.132477i	\(-0.957707\pi\)
							−0.132477	−	0.991186i	\(-0.542293\pi\)
\(5\)	1.51118e11	−	2.11291e10i	0.990366	−	0.138472i
							\(7\)	2.09740e13	−	2.09740e13i	0.631120	−	0.631120i	−0.317229	−	0.948349i	\(-0.602752\pi\)
0.948349	+	0.317229i	\(0.102752\pi\)
\(11\)	6.47623e16			1.40942			0.704708	−	0.709498i	\(-0.251078\pi\)
							0.704708	+	0.709498i	\(0.251078\pi\)
\(13\)	−4.20416e17	−	4.20416e17i	−0.631809	−	0.631809i	0.316713	−	0.948522i	\(-0.397421\pi\)
							−0.948522	+	0.316713i	\(0.897421\pi\)
\(17\)	−1.82331e19	+	1.82331e19i	−0.374695	+	0.374695i	−0.869184	−	0.494489i	\(-0.835355\pi\)
							0.494489	+	0.869184i	\(0.335355\pi\)
\(19\)		−	1.23600e20i		−	0.428508i	−0.976778	−	0.214254i	\(-0.931268\pi\)
							0.976778	−	0.214254i	\(-0.0687321\pi\)
\(23\)	6.48274e21	+	6.48274e21i	1.05709	+	1.05709i	0.998268	+	0.0588254i	\(0.0187355\pi\)
							0.0588254	+	0.998268i	\(0.481264\pi\)
\(29\)			4.68141e23i			1.87072i	0.353699	+	0.935359i	\(0.384924\pi\)
							−0.353699	+	0.935359i	\(0.615076\pi\)
\(31\)	2.94207e22			0.0404451			0.0202226	−	0.999796i	\(-0.493563\pi\)
							0.0202226	+	0.999796i	\(0.493563\pi\)
\(37\)	1.50424e25	−	1.50424e25i	1.21924	−	1.21924i	0.251340	−	0.967899i	\(-0.419129\pi\)
							0.967899	−	0.251340i	\(-0.0808711\pi\)
\(41\)	4.08942e25			0.641387			0.320693	−	0.947183i	\(-0.396084\pi\)
							0.320693	+	0.947183i	\(0.396084\pi\)
\(43\)	5.01997e25	+	5.01997e25i	0.367456	+	0.367456i	0.866549	−	0.499093i	\(-0.166333\pi\)
							−0.499093	+	0.866549i	\(0.666333\pi\)
\(47\)	4.23386e26	−	4.23386e26i	0.746742	−	0.746742i	−0.227124	−	0.973866i	\(-0.572932\pi\)
							0.973866	+	0.227124i	\(0.0729322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.3.3	yes	30
5.2	odd	4	inner	5.33.c.a.2.3	✓	30
5.3	odd	4		25.33.c.b.7.13		30
5.4	even	2		25.33.c.b.18.13		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.3	✓	30	5.2	odd	4	inner
5.33.c.a.3.3	yes	30	1.1	even	1	trivial
25.33.c.b.7.13		30	5.3	odd	4
25.33.c.b.18.13		30	5.4	even	2