Embedded newform 5.33.c.a.2.3

• Label: 5.33.c.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	5.2
Dual form			5.33.c.a.3.3

$q$-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 \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−70960.9	−	70960.9i	−1.08278	−	1.08278i	−0.996249	−	0.0865276i	\(-0.972423\pi\)
							−0.0865276	−	0.996249i	\(-0.527577\pi\)
\(3\)	−4.83700e7	+	4.83700e7i	−1.12366	+	1.12366i	−0.132477	+	0.991186i	\(0.542293\pi\)
							−0.991186	+	0.132477i	\(0.957707\pi\)
\(5\)	1.51118e11	+	2.11291e10i	0.990366	+	0.138472i
							\(7\)	2.09740e13	+	2.09740e13i	0.631120	+	0.631120i	0.948349	−	0.317229i	\(-0.102752\pi\)
−0.317229	+	0.948349i	\(0.602752\pi\)
\(11\)	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	−0.948522	−	0.316713i	\(-0.897421\pi\)
							0.316713	+	0.948522i	\(0.397421\pi\)
\(17\)	−1.82331e19	−	1.82331e19i	−0.374695	−	0.374695i	0.494489	−	0.869184i	\(-0.335355\pi\)
							−0.869184	+	0.494489i	\(0.835355\pi\)
\(19\)			1.23600e20i			0.428508i	0.976778	+	0.214254i	\(0.0687321\pi\)
							−0.976778	+	0.214254i	\(0.931268\pi\)
\(23\)	6.48274e21	−	6.48274e21i	1.05709	−	1.05709i	0.0588254	−	0.998268i	\(-0.481264\pi\)
							0.998268	−	0.0588254i	\(-0.0187355\pi\)
\(29\)		−	4.68141e23i		−	1.87072i	−0.353699	−	0.935359i	\(-0.615076\pi\)
							0.353699	−	0.935359i	\(-0.384924\pi\)
\(31\)	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	0.967899	+	0.251340i	\(0.0808711\pi\)
							0.251340	+	0.967899i	\(0.419129\pi\)
\(41\)	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	−0.499093	−	0.866549i	\(-0.666333\pi\)
							0.866549	+	0.499093i	\(0.166333\pi\)
\(47\)	4.23386e26	+	4.23386e26i	0.746742	+	0.746742i	0.973866	−	0.227124i	\(-0.0729322\pi\)
							−0.227124	+	0.973866i	\(0.572932\pi\)

(See \(a_n\) instead)

Twists

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

    

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