Embedded newform 5.33.c.a.3.6

• Label: 5.33.c.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	5.3
Dual form			5.33.c.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-37971.9 + 37971.9i) q^{2} +(3.04306e7 + 3.04306e7i) q^{3} +1.41124e9i q^{4} +(7.42414e10 + 1.33309e11i) q^{5} -2.31101e12 q^{6} +(-9.26812e12 + 9.26812e12i) q^{7} +(-2.16675e14 - 2.16675e14i) q^{8} -9.78102e11i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−37971.9	+	37971.9i	−0.579405	+	0.579405i	−0.934739	−	0.355334i	\(-0.884367\pi\)
							0.355334	+	0.934739i	\(0.384367\pi\)
\(3\)	3.04306e7	+	3.04306e7i	0.706920	+	0.706920i	0.965886	−	0.258966i	\(-0.0833818\pi\)
							−0.258966	+	0.965886i	\(0.583382\pi\)
\(5\)	7.42414e10	+	1.33309e11i	0.486549	+	0.873654i
							\(7\)	−9.26812e12	+	9.26812e12i	−0.278884	+	0.278884i	−0.832663	−	0.553780i	\(-0.813185\pi\)
0.553780	+	0.832663i	\(0.313185\pi\)
\(11\)	1.98923e16			0.432914			0.216457	−	0.976292i	\(-0.430550\pi\)
							0.216457	+	0.976292i	\(0.430550\pi\)
\(13\)	−8.93581e17	−	8.93581e17i	−1.34289	−	1.34289i	−0.893170	−	0.449719i	\(-0.851524\pi\)
							−0.449719	−	0.893170i	\(-0.648476\pi\)
\(17\)	−3.66719e19	+	3.66719e19i	−0.753617	+	0.753617i	−0.975152	−	0.221536i	\(-0.928893\pi\)
							0.221536	+	0.975152i	\(0.428893\pi\)
\(19\)			2.80535e20i			0.972589i	0.873795	+	0.486295i	\(0.161652\pi\)
							−0.873795	+	0.486295i	\(0.838348\pi\)
\(23\)	−2.06972e21	−	2.06972e21i	−0.337495	−	0.337495i	0.517929	−	0.855424i	\(-0.326703\pi\)
							−0.855424	+	0.517929i	\(0.826703\pi\)
\(29\)		−	1.02359e23i		−	0.409031i	−0.978863	−	0.204516i	\(-0.934438\pi\)
							0.978863	−	0.204516i	\(-0.0655620\pi\)
\(31\)	−9.64778e23			−1.32630			−0.663148	−	0.748488i	\(-0.730780\pi\)
							−0.663148	+	0.748488i	\(0.730780\pi\)
\(37\)	7.27232e24	−	7.27232e24i	0.589448	−	0.589448i	−0.348034	−	0.937482i	\(-0.613151\pi\)
							0.937482	+	0.348034i	\(0.113151\pi\)
\(41\)	1.58648e25			0.248825			0.124412	−	0.992231i	\(-0.460295\pi\)
							0.124412	+	0.992231i	\(0.460295\pi\)
\(43\)	−9.21200e25	−	9.21200e25i	−0.674309	−	0.674309i	0.284398	−	0.958706i	\(-0.408206\pi\)
							−0.958706	+	0.284398i	\(0.908206\pi\)
\(47\)	−3.74128e26	+	3.74128e26i	−0.659863	+	0.659863i	−0.955348	−	0.295484i	\(-0.904519\pi\)
							0.295484	+	0.955348i	\(0.404519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.3.6	yes	30
5.2	odd	4	inner	5.33.c.a.2.6	✓	30
5.3	odd	4		25.33.c.b.7.10		30
5.4	even	2		25.33.c.b.18.10		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.6	✓	30	5.2	odd	4	inner
5.33.c.a.3.6	yes	30	1.1	even	1	trivial
25.33.c.b.7.10		30	5.3	odd	4
25.33.c.b.18.10		30	5.4	even	2