Embedded newform 5.27.c.a.2.2

• Label: 5.27.c.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	5.2
Dual form			5.27.c.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8967.06 - 8967.06i) q^{2} +(-1.25875e6 + 1.25875e6i) q^{3} +9.37076e7i q^{4} +(2.09822e8 + 1.20254e9i) q^{5} +2.25746e10 q^{6} +(5.31741e10 + 5.31741e10i) q^{7} +(2.38512e11 - 2.38512e11i) q^{8} -6.27038e11i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8190 q^{2} + 1867680 q^{3} - 140947920 q^{5} - 3382472712 q^{6} - 99234643200 q^{7} - 2004366809340 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8967.06	−	8967.06i	−1.09461	−	1.09461i	−0.995029	−	0.0995829i	\(-0.968249\pi\)
							−0.0995829	−	0.995029i	\(-0.531751\pi\)
\(3\)	−1.25875e6	+	1.25875e6i	−0.789520	+	0.789520i	−0.981415	−	0.191895i	\(-0.938537\pi\)
							0.191895	+	0.981415i	\(0.438537\pi\)
\(5\)	2.09822e8	+	1.20254e9i	0.171886	+	0.985117i
							\(7\)	5.31741e10	+	5.31741e10i	0.548815	+	0.548815i	0.926098	−	0.377283i	\(-0.123142\pi\)
−0.377283	+	0.926098i	\(0.623142\pi\)
\(11\)	−1.06320e13			−0.307970			−0.153985	−	0.988073i	\(-0.549211\pi\)
							−0.153985	+	0.988073i	\(0.549211\pi\)
\(13\)	1.39176e14	−	1.39176e14i	0.459517	−	0.459517i	−0.438980	−	0.898497i	\(-0.644660\pi\)
							0.898497	+	0.438980i	\(0.144660\pi\)
\(17\)	9.12057e15	+	9.12057e15i	0.920844	+	0.920844i	0.997089	−	0.0762448i	\(-0.0242930\pi\)
							−0.0762448	+	0.997089i	\(0.524293\pi\)
\(19\)			7.63630e16i			1.81588i	0.419104	+	0.907938i	\(0.362344\pi\)
							−0.419104	+	0.907938i	\(0.637656\pi\)
\(23\)	3.29168e17	−	3.29168e17i	0.653064	−	0.653064i	−0.300665	−	0.953730i	\(-0.597209\pi\)
							0.953730	+	0.300665i	\(0.0972087\pi\)
\(29\)			1.91977e19i			1.87100i	0.353324	+	0.935501i	\(0.385051\pi\)
							−0.353324	+	0.935501i	\(0.614949\pi\)
\(31\)	2.39548e19			0.981047			0.490524	−	0.871428i	\(-0.336806\pi\)
							0.490524	+	0.871428i	\(0.336806\pi\)
\(37\)	−5.76061e19	−	5.76061e19i	−0.236508	−	0.236508i	0.578894	−	0.815403i	\(-0.303484\pi\)
							−0.815403	+	0.578894i	\(0.803484\pi\)
\(41\)	−1.23032e21			−1.32993			−0.664965	−	0.746875i	\(-0.731553\pi\)
							−0.664965	+	0.746875i	\(0.731553\pi\)
\(43\)	−5.39742e19	+	5.39742e19i	−0.0314120	+	0.0314120i	−0.722638	−	0.691226i	\(-0.757071\pi\)
							0.691226	+	0.722638i	\(0.257071\pi\)
\(47\)	−1.83075e21	−	1.83075e21i	−0.335242	−	0.335242i	0.519331	−	0.854573i	\(-0.326181\pi\)
							−0.854573	+	0.519331i	\(0.826181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.27.c.a.2.2	✓	24
5.2	odd	4		25.27.c.b.18.11		24
5.3	odd	4	inner	5.27.c.a.3.2	yes	24
5.4	even	2		25.27.c.b.7.11		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.27.c.a.2.2	✓	24	1.1	even	1	trivial
5.27.c.a.3.2	yes	24	5.3	odd	4	inner
25.27.c.b.7.11		24	5.4	even	2
25.27.c.b.18.11		24	5.2	odd	4