Embedded newform 5.33.c.a.2.10

• Label: 5.33.c.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	5.2
Dual form			5.33.c.a.3.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(25524.4 + 25524.4i) q^{2} +(3.82280e7 - 3.82280e7i) q^{3} -2.99198e9i q^{4} +(-1.26030e11 - 8.60210e10i) q^{5} +1.95149e12 q^{6} +(-1.22489e13 - 1.22489e13i) q^{7} +(1.85995e14 - 1.85995e14i) q^{8} -1.06974e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	25524.4	+	25524.4i	0.389471	+	0.389471i	0.874499	−	0.485028i	\(-0.161191\pi\)
							−0.485028	+	0.874499i	\(0.661191\pi\)
\(3\)	3.82280e7	−	3.82280e7i	0.888058	−	0.888058i	−0.106278	−	0.994336i	\(-0.533893\pi\)
							0.994336	+	0.106278i	\(0.0338934\pi\)
\(5\)	−1.26030e11	−	8.60210e10i	−0.825948	−	0.563747i
							\(7\)	−1.22489e13	−	1.22489e13i	−0.368578	−	0.368578i	0.498381	−	0.866958i	\(-0.333928\pi\)
−0.866958	+	0.498381i	\(0.833928\pi\)
\(11\)	−6.03478e16			−1.31334			−0.656672	−	0.754176i	\(-0.728036\pi\)
							−0.656672	+	0.754176i	\(0.728036\pi\)
\(13\)	1.61811e17	−	1.61811e17i	0.243173	−	0.243173i	−0.574989	−	0.818161i	\(-0.694993\pi\)
							0.818161	+	0.574989i	\(0.194993\pi\)
\(17\)	3.87016e19	+	3.87016e19i	0.795328	+	0.795328i	0.982355	−	0.187027i	\(-0.0598852\pi\)
							−0.187027	+	0.982355i	\(0.559885\pi\)
\(19\)			2.45952e20i			0.852693i	0.904560	+	0.426346i	\(0.140200\pi\)
							−0.904560	+	0.426346i	\(0.859800\pi\)
\(23\)	−9.22936e19	+	9.22936e19i	−0.0150496	+	0.0150496i	−0.714592	−	0.699542i	\(-0.753387\pi\)
							0.699542	+	0.714592i	\(0.253387\pi\)
\(29\)		−	1.82230e23i		−	0.728202i	−0.931359	−	0.364101i	\(-0.881376\pi\)
							0.931359	−	0.364101i	\(-0.118624\pi\)
\(31\)	−3.83738e23			−0.527530			−0.263765	−	0.964587i	\(-0.584964\pi\)
							−0.263765	+	0.964587i	\(0.584964\pi\)
\(37\)	−1.31159e25	−	1.31159e25i	−1.06309	−	1.06309i	−0.997871	−	0.0652199i	\(-0.979225\pi\)
							−0.0652199	−	0.997871i	\(-0.520775\pi\)
\(41\)	−1.00429e26			−1.57513			−0.787567	−	0.616230i	\(-0.788659\pi\)
							−0.787567	+	0.616230i	\(0.788659\pi\)
\(43\)	7.50109e25	−	7.50109e25i	0.549072	−	0.549072i	−0.377101	−	0.926172i	\(-0.623079\pi\)
							0.926172	+	0.377101i	\(0.123079\pi\)
\(47\)	−3.79204e24	−	3.79204e24i	−0.00668817	−	0.00668817i	0.703755	−	0.710443i	\(-0.251505\pi\)
							−0.710443	+	0.703755i	\(0.751505\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.2.10	✓	30
5.2	odd	4		25.33.c.b.18.6		30
5.3	odd	4	inner	5.33.c.a.3.10	yes	30
5.4	even	2		25.33.c.b.7.6		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.10	✓	30	1.1	even	1	trivial
5.33.c.a.3.10	yes	30	5.3	odd	4	inner
25.33.c.b.7.6		30	5.4	even	2
25.33.c.b.18.6		30	5.2	odd	4