Embedded newform 5.27.c.a.2.5

• Label: 5.27.c.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	5.2
Dual form			5.27.c.a.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2621.08 - 2621.08i) q^{2} +(-823115. + 823115. i) q^{3} -5.33688e7i q^{4} +(9.07026e8 - 8.16957e8i) q^{5} +4.31489e9 q^{6} +(3.09387e10 + 3.09387e10i) q^{7} +(-3.15781e11 + 3.15781e11i) q^{8} +1.18683e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8190 q^{2} + 1867680 q^{3} - 140947920 q^{5} - 3382472712 q^{6} - 99234643200 q^{7} - 2004366809340 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2621.08	−	2621.08i	−0.319956	−	0.319956i	0.528794	−	0.848750i	\(-0.322644\pi\)
							−0.848750	+	0.528794i	\(0.822644\pi\)
\(3\)	−823115.	+	823115.i	−0.516279	+	0.516279i	−0.916443	−	0.400165i	\(-0.868953\pi\)
							0.400165	+	0.916443i	\(0.368953\pi\)
\(5\)	9.07026e8	−	8.16957e8i	0.743036	−	0.669251i
							\(7\)	3.09387e10	+	3.09387e10i	0.319321	+	0.319321i	0.848506	−	0.529185i	\(-0.177502\pi\)
−0.529185	+	0.848506i	\(0.677502\pi\)
\(11\)	1.89101e13			0.547758			0.273879	−	0.961764i	\(-0.411693\pi\)
							0.273879	+	0.961764i	\(0.411693\pi\)
\(13\)	1.89565e14	−	1.89565e14i	0.625884	−	0.625884i	−0.321146	−	0.947030i	\(-0.604068\pi\)
							0.947030	+	0.321146i	\(0.104068\pi\)
\(17\)	−2.10704e14	−	2.10704e14i	−0.0212734	−	0.0212734i	0.696390	−	0.717663i	\(-0.254788\pi\)
							−0.717663	+	0.696390i	\(0.754788\pi\)
\(19\)		−	4.30980e16i		−	1.02485i	−0.858732	−	0.512425i	\(-0.828747\pi\)
							0.858732	−	0.512425i	\(-0.171253\pi\)
\(23\)	1.72142e17	−	1.72142e17i	0.341526	−	0.341526i	−0.515415	−	0.856941i	\(-0.672362\pi\)
							0.856941	+	0.515415i	\(0.172362\pi\)
\(29\)		−	1.50665e19i		−	1.46838i	−0.678945	−	0.734189i	\(-0.737562\pi\)
							0.678945	−	0.734189i	\(-0.262438\pi\)
\(31\)	−3.13975e19			−1.28586			−0.642930	−	0.765925i	\(-0.722281\pi\)
							−0.642930	+	0.765925i	\(0.722281\pi\)
\(37\)	1.99372e20	+	1.99372e20i	0.818545	+	0.818545i	0.985897	−	0.167352i	\(-0.0535218\pi\)
							−0.167352	+	0.985897i	\(0.553522\pi\)
\(41\)	−1.03660e21			−1.12053			−0.560263	−	0.828315i	\(-0.689300\pi\)
							−0.560263	+	0.828315i	\(0.689300\pi\)
\(43\)	1.56413e21	−	1.56413e21i	0.910298	−	0.910298i	−0.0859977	−	0.996295i	\(-0.527408\pi\)
							0.996295	+	0.0859977i	\(0.0274078\pi\)
\(47\)	−3.40116e21	−	3.40116e21i	−0.622810	−	0.622810i	0.323439	−	0.946249i	\(-0.395161\pi\)
							−0.946249	+	0.323439i	\(0.895161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.27.c.a.2.5	✓	24
5.2	odd	4		25.27.c.b.18.8		24
5.3	odd	4	inner	5.27.c.a.3.5	yes	24
5.4	even	2		25.27.c.b.7.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.27.c.a.2.5	✓	24	1.1	even	1	trivial
5.27.c.a.3.5	yes	24	5.3	odd	4	inner
25.27.c.b.7.8		24	5.4	even	2
25.27.c.b.18.8		24	5.2	odd	4