Embedded newform 5.33.c.a.2.7

• Label: 5.33.c.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	5.2
Dual form			5.33.c.a.3.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7096.95 - 7096.95i) q^{2} +(-4.70380e7 + 4.70380e7i) q^{3} -4.19423e9i q^{4} +(-3.87281e10 - 1.47591e11i) q^{5} +6.67652e11 q^{6} +(-3.04405e13 - 3.04405e13i) q^{7} +(-6.02475e13 + 6.02475e13i) q^{8} -2.57212e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2792232 q^{3} + 229266409900 q^{5} - 645476451240 q^{6} + 21807690136848 q^{7} + 340768936037220 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7096.95	−	7096.95i	−0.108291	−	0.108291i	0.650885	−	0.759176i	\(-0.274398\pi\)
							−0.759176	+	0.650885i	\(0.774398\pi\)
\(3\)	−4.70380e7	+	4.70380e7i	−1.09272	+	1.09272i	−0.0974813	+	0.995237i	\(0.531079\pi\)
							−0.995237	+	0.0974813i	\(0.968921\pi\)
\(5\)	−3.87281e10	−	1.47591e11i	−0.253809	−	0.967254i
							\(7\)	−3.04405e13	−	3.04405e13i	−0.915973	−	0.915973i	0.0807601	−	0.996734i	\(-0.474265\pi\)
−0.996734	+	0.0807601i	\(0.974265\pi\)
\(11\)	3.11706e16			0.678362			0.339181	−	0.940721i	\(-0.389850\pi\)
							0.339181	+	0.940721i	\(0.389850\pi\)
\(13\)	8.11181e17	−	8.11181e17i	1.21906	−	1.21906i	0.251094	−	0.967963i	\(-0.419210\pi\)
							0.967963	−	0.251094i	\(-0.0807904\pi\)
\(17\)	−5.99103e19	−	5.99103e19i	−1.23117	−	1.23117i	−0.963514	−	0.267659i	\(-0.913750\pi\)
							−0.267659	−	0.963514i	\(-0.586250\pi\)
\(19\)			3.24007e19i			0.112330i	0.998421	+	0.0561652i	\(0.0178874\pi\)
							−0.998421	+	0.0561652i	\(0.982113\pi\)
\(23\)	7.07294e20	−	7.07294e20i	0.115333	−	0.115333i	−0.647085	−	0.762418i	\(-0.724012\pi\)
							0.762418	+	0.647085i	\(0.224012\pi\)
\(29\)		−	1.29019e23i		−	0.515568i	−0.966203	−	0.257784i	\(-0.917008\pi\)
							0.966203	−	0.257784i	\(-0.0829924\pi\)
\(31\)	1.94976e23			0.268037			0.134019	−	0.990979i	\(-0.457212\pi\)
							0.134019	+	0.990979i	\(0.457212\pi\)
\(37\)	−1.30388e25	−	1.30388e25i	−1.05684	−	1.05684i	−0.998284	−	0.0585575i	\(-0.981350\pi\)
							−0.0585575	−	0.998284i	\(-0.518650\pi\)
\(41\)	1.97064e25			0.309076			0.154538	−	0.987987i	\(-0.450611\pi\)
							0.154538	+	0.987987i	\(0.450611\pi\)
\(43\)	−1.49066e26	+	1.49066e26i	−1.09114	+	1.09114i	−0.0957369	+	0.995407i	\(0.530521\pi\)
							−0.995407	+	0.0957369i	\(0.969479\pi\)
\(47\)	−1.46092e25	−	1.46092e25i	−0.0257669	−	0.0257669i	0.694106	−	0.719873i	\(-0.255800\pi\)
							−0.719873	+	0.694106i	\(0.755800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.33.c.a.2.7	✓	30
5.2	odd	4		25.33.c.b.18.9		30
5.3	odd	4	inner	5.33.c.a.3.7	yes	30
5.4	even	2		25.33.c.b.7.9		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.33.c.a.2.7	✓	30	1.1	even	1	trivial
5.33.c.a.3.7	yes	30	5.3	odd	4	inner
25.33.c.b.7.9		30	5.4	even	2
25.33.c.b.18.9		30	5.2	odd	4