Embedded newform 5.33.c.a.2.11

• Label: 5.33.c.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	5.2
Dual form			5.33.c.a.3.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(32395.0 + 32395.0i) q^{2} +(1.63684e7 - 1.63684e7i) q^{3} -2.19610e9i q^{4} +(5.30856e10 + 1.43056e11i) q^{5} +1.06051e12 q^{6} +(-2.79912e13 - 2.79912e13i) q^{7} +(2.10278e14 - 2.10278e14i) q^{8} +1.31717e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	32395.0	+	32395.0i	0.494308	+	0.494308i	0.909660	−	0.415353i	\(-0.136342\pi\)
							−0.415353	+	0.909660i	\(0.636342\pi\)
\(3\)	1.63684e7	−	1.63684e7i	0.380247	−	0.380247i	−0.490944	−	0.871191i	\(-0.663348\pi\)
							0.871191	+	0.490944i	\(0.163348\pi\)
\(5\)	5.30856e10	+	1.43056e11i	0.347902	+	0.937531i
							\(7\)	−2.79912e13	−	2.79912e13i	−0.842273	−	0.842273i	0.146881	−	0.989154i	\(-0.453076\pi\)
−0.989154	+	0.146881i	\(0.953076\pi\)
\(11\)	6.58116e16			1.43225			0.716126	−	0.697971i	\(-0.245914\pi\)
							0.716126	+	0.697971i	\(0.245914\pi\)
\(13\)	−2.99215e16	+	2.99215e16i	−0.0449666	+	0.0449666i	−0.729233	−	0.684266i	\(-0.760123\pi\)
							0.684266	+	0.729233i	\(0.260123\pi\)
\(17\)	−1.92770e19	−	1.92770e19i	−0.396148	−	0.396148i	0.480724	−	0.876872i	\(-0.340374\pi\)
							−0.876872	+	0.480724i	\(0.840374\pi\)
\(19\)		−	5.51904e20i		−	1.91340i	−0.291073	−	0.956701i	\(-0.594012\pi\)
							0.291073	−	0.956701i	\(-0.405988\pi\)
\(23\)	3.28869e21	−	3.28869e21i	0.536263	−	0.536263i	−0.386166	−	0.922429i	\(-0.626201\pi\)
							0.922429	+	0.386166i	\(0.126201\pi\)
\(29\)		−	3.10854e23i		−	1.24219i	−0.783734	−	0.621096i	\(-0.786688\pi\)
							0.783734	−	0.621096i	\(-0.213312\pi\)
\(31\)	1.32479e24			1.82121			0.910606	−	0.413275i	\(-0.135615\pi\)
							0.910606	+	0.413275i	\(0.135615\pi\)
\(37\)	2.81341e24	+	2.81341e24i	0.228037	+	0.228037i	0.811872	−	0.583835i	\(-0.198449\pi\)
							−0.583835	+	0.811872i	\(0.698449\pi\)
\(41\)	3.62740e25			0.568923			0.284462	−	0.958687i	\(-0.408185\pi\)
							0.284462	+	0.958687i	\(0.408185\pi\)
\(43\)	1.14542e26	−	1.14542e26i	0.838433	−	0.838433i	−0.150219	−	0.988653i	\(-0.547998\pi\)
							0.988653	+	0.150219i	\(0.0479980\pi\)
\(47\)	−7.62585e24	−	7.62585e24i	−0.0134500	−	0.0134500i	0.700350	−	0.713800i	\(-0.253027\pi\)
							−0.713800	+	0.700350i	\(0.753027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.2.11	✓	30
5.2	odd	4		25.33.c.b.18.5		30
5.3	odd	4	inner	5.33.c.a.3.11	yes	30
5.4	even	2		25.33.c.b.7.5		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.11	✓	30	1.1	even	1	trivial
5.33.c.a.3.11	yes	30	5.3	odd	4	inner
25.33.c.b.7.5		30	5.4	even	2
25.33.c.b.18.5		30	5.2	odd	4