Embedded newform 5.33.c.a.3.5

• Label: 5.33.c.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	5.3
Dual form			5.33.c.a.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-39535.2 + 39535.2i) q^{2} +(4.69567e7 + 4.69567e7i) q^{3} +1.16890e9i q^{4} +(-7.61848e10 - 1.32208e11i) q^{5} -3.71289e12 q^{6} +(2.85172e13 - 2.85172e13i) q^{7} +(-2.16015e14 - 2.16015e14i) q^{8} +2.55684e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−39535.2	+	39535.2i	−0.603260	+	0.603260i	−0.941176	−	0.337916i	\(-0.890278\pi\)
							0.337916	+	0.941176i	\(0.390278\pi\)
\(3\)	4.69567e7	+	4.69567e7i	1.09083	+	1.09083i	0.995440	+	0.0953909i	\(0.0304101\pi\)
							0.0953909	+	0.995440i	\(0.469590\pi\)
\(5\)	−7.61848e10	−	1.32208e11i	−0.499284	−	0.866438i
							\(7\)	2.85172e13	−	2.85172e13i	0.858100	−	0.858100i	−0.133014	−	0.991114i	\(-0.542466\pi\)
0.991114	+	0.133014i	\(0.0424655\pi\)
\(11\)	6.29088e16			1.36908			0.684540	−	0.728975i	\(-0.260003\pi\)
							0.684540	+	0.728975i	\(0.260003\pi\)
\(13\)	6.68945e17	+	6.68945e17i	1.00530	+	1.00530i	0.999986	+	0.00531621i	\(0.00169221\pi\)
							0.00531621	+	0.999986i	\(0.498308\pi\)
\(17\)	9.86415e17	−	9.86415e17i	0.0202711	−	0.0202711i	−0.696899	−	0.717170i	\(-0.745437\pi\)
							0.717170	+	0.696899i	\(0.245437\pi\)
\(19\)		−	3.77787e20i		−	1.30975i	−0.755736	−	0.654876i	\(-0.772721\pi\)
							0.755736	−	0.654876i	\(-0.227279\pi\)
\(23\)	5.92343e21	+	5.92343e21i	0.965890	+	0.965890i	0.999437	−	0.0335474i	\(-0.0106805\pi\)
							−0.0335474	+	0.999437i	\(0.510680\pi\)
\(29\)			2.77779e19i			0.000111002i	1.00000		5.55011e-5i	\(1.76665e-5\pi\)
							−1.00000		5.55011e-5i	\(0.999982\pi\)
\(31\)	−3.93433e23			−0.540858			−0.270429	−	0.962740i	\(-0.587166\pi\)
							−0.270429	+	0.962740i	\(0.587166\pi\)
\(37\)	−4.19934e24	+	4.19934e24i	−0.340371	+	0.340371i	−0.856507	−	0.516136i	\(-0.827370\pi\)
							0.516136	+	0.856507i	\(0.327370\pi\)
\(41\)	8.53049e25			1.33793			0.668964	−	0.743295i	\(-0.266738\pi\)
							0.668964	+	0.743295i	\(0.266738\pi\)
\(43\)	−8.89872e25	−	8.89872e25i	−0.651377	−	0.651377i	0.301948	−	0.953324i	\(-0.402363\pi\)
							−0.953324	+	0.301948i	\(0.902363\pi\)
\(47\)	3.07295e26	−	3.07295e26i	0.541988	−	0.541988i	−0.382124	−	0.924111i	\(-0.624807\pi\)
							0.924111	+	0.382124i	\(0.124807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.3.5	yes	30
5.2	odd	4	inner	5.33.c.a.2.5	✓	30
5.3	odd	4		25.33.c.b.7.11		30
5.4	even	2		25.33.c.b.18.11		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.5	✓	30	5.2	odd	4	inner
5.33.c.a.3.5	yes	30	1.1	even	1	trivial
25.33.c.b.7.11		30	5.3	odd	4
25.33.c.b.18.11		30	5.4	even	2