Embedded newform 5.33.c.a.3.1

• Label: 5.33.c.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	5.3
Dual form			5.33.c.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-86676.0 + 86676.0i) q^{2} +(3.32299e7 + 3.32299e7i) q^{3} -1.07305e10i q^{4} +(1.18818e11 - 9.57363e10i) q^{5} -5.76047e12 q^{6} +(-3.06435e13 + 3.06435e13i) q^{7} +(5.57804e14 + 5.57804e14i) q^{8} +3.55432e14i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−86676.0	+	86676.0i	−1.32257	+	1.32257i	−0.410881	+	0.911689i	\(0.634779\pi\)
							−0.911689	+	0.410881i	\(0.865221\pi\)
\(3\)	3.32299e7	+	3.32299e7i	0.771950	+	0.771950i	0.978447	−	0.206498i	\(-0.0662066\pi\)
							−0.206498	+	0.978447i	\(0.566207\pi\)
\(5\)	1.18818e11	−	9.57363e10i	0.778683	−	0.627418i
							\(7\)	−3.06435e13	+	3.06435e13i	−0.922083	+	0.922083i	−0.997176	−	0.0750935i	\(-0.976074\pi\)
0.0750935	+	0.997176i	\(0.476074\pi\)
\(11\)	−5.53776e16			−1.20518			−0.602589	−	0.798052i	\(-0.705864\pi\)
							−0.602589	+	0.798052i	\(0.705864\pi\)
\(13\)	1.61898e17	+	1.61898e17i	0.243304	+	0.243304i	0.818215	−	0.574912i	\(-0.194964\pi\)
							−0.574912	+	0.818215i	\(0.694964\pi\)
\(17\)	−2.56755e19	+	2.56755e19i	−0.527638	+	0.527638i	−0.919868	−	0.392229i	\(-0.871704\pi\)
							0.392229	+	0.919868i	\(0.371704\pi\)
\(19\)		−	4.51346e20i		−	1.56478i	−0.622792	−	0.782388i	\(-0.714002\pi\)
							0.622792	−	0.782388i	\(-0.285998\pi\)
\(23\)	2.16132e21	+	2.16132e21i	0.352431	+	0.352431i	0.861013	−	0.508582i	\(-0.169830\pi\)
							−0.508582	+	0.861013i	\(0.669830\pi\)
\(29\)		−	1.69547e23i		−	0.677520i	−0.940873	−	0.338760i	\(-0.889992\pi\)
							0.940873	−	0.338760i	\(-0.110008\pi\)
\(31\)	6.36232e23			0.874638			0.437319	−	0.899306i	\(-0.355928\pi\)
							0.437319	+	0.899306i	\(0.355928\pi\)
\(37\)	1.46477e24	−	1.46477e24i	0.118725	−	0.118725i	−0.645248	−	0.763973i	\(-0.723246\pi\)
							0.763973	+	0.645248i	\(0.223246\pi\)
\(41\)	−4.24771e25			−0.666214			−0.333107	−	0.942889i	\(-0.608097\pi\)
							−0.333107	+	0.942889i	\(0.608097\pi\)
\(43\)	−1.43757e26	−	1.43757e26i	−1.05229	−	1.05229i	−0.998555	−	0.0537314i	\(-0.982889\pi\)
							−0.0537314	−	0.998555i	\(-0.517111\pi\)
\(47\)	4.39170e26	−	4.39170e26i	0.774580	−	0.774580i	−0.204323	−	0.978903i	\(-0.565499\pi\)
							0.978903	+	0.204323i	\(0.0654993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.3.1	yes	30
5.2	odd	4	inner	5.33.c.a.2.1	✓	30
5.3	odd	4		25.33.c.b.7.15		30
5.4	even	2		25.33.c.b.18.15		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.1	✓	30	5.2	odd	4	inner
5.33.c.a.3.1	yes	30	1.1	even	1	trivial
25.33.c.b.7.15		30	5.3	odd	4
25.33.c.b.18.15		30	5.4	even	2