Embedded newform 5.27.c.a.2.1

• Label: 5.27.c.a.2.1
• Level: $5$
• Weight: $27$
• Character: 5.2
• Analytic conductor: $21.415$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 27 \)
Character orbit:	\([\chi]\)	\(=\)	5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.4146460365\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	5.2
Dual form			5.27.c.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9199.72 - 9199.72i) q^{2} +(1.92148e6 - 1.92148e6i) q^{3} +1.02161e8i q^{4} +(1.20945e9 + 1.65366e8i) q^{5} -3.53542e10 q^{6} +(2.78396e10 + 2.78396e10i) q^{7} +(3.22468e11 - 3.22468e11i) q^{8} -4.84231e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8190 q^{2} + 1867680 q^{3} - 140947920 q^{5} - 3382472712 q^{6} - 99234643200 q^{7} - 2004366809340 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9199.72	−	9199.72i	−1.12301	−	1.12301i	−0.991286	−	0.131726i	\(-0.957948\pi\)
							−0.131726	−	0.991286i	\(-0.542052\pi\)
\(3\)	1.92148e6	−	1.92148e6i	1.20520	−	1.20520i	0.232638	−	0.972563i	\(-0.425264\pi\)
							0.972563	−	0.232638i	\(-0.0747357\pi\)
\(5\)	1.20945e9	+	1.65366e8i	0.990782	+	0.135468i
							\(7\)	2.78396e10	+	2.78396e10i	0.287335	+	0.287335i	0.836026	−	0.548690i	\(-0.184873\pi\)
−0.548690	+	0.836026i	\(0.684873\pi\)
\(11\)	5.43326e13			1.57382			0.786911	−	0.617067i	\(-0.211679\pi\)
							0.786911	+	0.617067i	\(0.211679\pi\)
\(13\)	9.46117e13	−	9.46117e13i	0.312379	−	0.312379i	−0.533452	−	0.845830i	\(-0.679105\pi\)
							0.845830	+	0.533452i	\(0.179105\pi\)
\(17\)	−9.19005e15	−	9.19005e15i	−0.927859	−	0.927859i	0.0697089	−	0.997567i	\(-0.477793\pi\)
							−0.997567	+	0.0697089i	\(0.977793\pi\)
\(19\)		−	5.32221e16i		−	1.26560i	−0.774317	−	0.632798i	\(-0.781906\pi\)
							0.774317	−	0.632798i	\(-0.218094\pi\)
\(23\)	−1.98118e17	+	1.98118e17i	−0.393063	+	0.393063i	−0.875778	−	0.482715i	\(-0.839651\pi\)
							0.482715	+	0.875778i	\(0.339651\pi\)
\(29\)			1.15444e19i			1.12511i	0.826759	+	0.562557i	\(0.190182\pi\)
							−0.826759	+	0.562557i	\(0.809818\pi\)
\(31\)	7.75453e18			0.317580			0.158790	−	0.987312i	\(-0.449241\pi\)
							0.158790	+	0.987312i	\(0.449241\pi\)
\(37\)	−1.01083e19	−	1.01083e19i	−0.0415006	−	0.0415006i	0.686052	−	0.727553i	\(-0.259342\pi\)
							−0.727553	+	0.686052i	\(0.759342\pi\)
\(41\)	6.54637e20			0.707637			0.353819	−	0.935314i	\(-0.384883\pi\)
							0.353819	+	0.935314i	\(0.384883\pi\)
\(43\)	−1.72278e21	+	1.72278e21i	−1.00263	+	1.00263i	−0.00263267	+	0.999997i	\(0.500838\pi\)
							−0.999997	+	0.00263267i	\(0.999162\pi\)
\(47\)	−1.35367e21	−	1.35367e21i	−0.247879	−	0.247879i	0.572221	−	0.820100i	\(-0.306082\pi\)
							−0.820100	+	0.572221i	\(0.806082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.27.c.a.2.1	✓	24
5.2	odd	4		25.27.c.b.18.12		24
5.3	odd	4	inner	5.27.c.a.3.1	yes	24
5.4	even	2		25.27.c.b.7.12		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.27.c.a.2.1	✓	24	1.1	even	1	trivial
5.27.c.a.3.1	yes	24	5.3	odd	4	inner
25.27.c.b.7.12		24	5.4	even	2
25.27.c.b.18.12		24	5.2	odd	4