Embedded newform 5.27.c.a.3.8

• Label: 5.27.c.a.3.8
• Level: $5$
• Weight: $27$
• Character: 5.3
• 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			3.8
Character	\(\chi\)	\(=\)	5.3
Dual form			5.27.c.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4190.94 - 4190.94i) q^{2} +(1.61385e6 + 1.61385e6i) q^{3} +3.19809e7i q^{4} +(6.93244e8 + 1.00475e9i) q^{5} +1.35270e10 q^{6} +(-1.14147e10 + 1.14147e10i) q^{7} +(4.15279e11 + 4.15279e11i) q^{8} +2.66713e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8190 q^{2} + 1867680 q^{3} - 140947920 q^{5} - 3382472712 q^{6} - 99234643200 q^{7} - 2004366809340 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4190.94	−	4190.94i	0.511589	−	0.511589i	−0.403424	−	0.915013i	\(-0.632180\pi\)
							0.915013	+	0.403424i	\(0.132180\pi\)
\(3\)	1.61385e6	+	1.61385e6i	1.01224	+	1.01224i	0.999924	+	0.0123206i	\(0.00392188\pi\)
							0.0123206	+	0.999924i	\(0.496078\pi\)
\(5\)	6.93244e8	+	1.00475e9i	0.567905	+	0.823094i
							\(7\)	−1.14147e10	+	1.14147e10i	−0.117813	+	0.117813i	−0.763555	−	0.645743i	\(-0.776548\pi\)
0.645743	+	0.763555i	\(0.276548\pi\)
\(11\)	−3.05887e13			−0.886047			−0.443023	−	0.896510i	\(-0.646094\pi\)
							−0.443023	+	0.896510i	\(0.646094\pi\)
\(13\)	−2.50133e14	−	2.50133e14i	−0.825863	−	0.825863i	0.161079	−	0.986942i	\(-0.448503\pi\)
							−0.986942	+	0.161079i	\(0.948503\pi\)
\(17\)	2.67504e15	−	2.67504e15i	0.270081	−	0.270081i	−0.559052	−	0.829133i	\(-0.688835\pi\)
							0.829133	+	0.559052i	\(0.188835\pi\)
\(19\)		−	3.05865e16i		−	0.727332i	−0.931529	−	0.363666i	\(-0.881525\pi\)
							0.931529	−	0.363666i	\(-0.118475\pi\)
\(23\)	5.67140e17	+	5.67140e17i	1.12520	+	1.12520i	0.990948	+	0.134250i	\(0.0428624\pi\)
							0.134250	+	0.990948i	\(0.457138\pi\)
\(29\)			6.66815e18i			0.649877i	0.945735	+	0.324939i	\(0.105344\pi\)
							−0.945735	+	0.324939i	\(0.894656\pi\)
\(31\)	3.39031e19			1.38847			0.694237	−	0.719746i	\(-0.255742\pi\)
							0.694237	+	0.719746i	\(0.255742\pi\)
\(37\)	2.03546e20	−	2.03546e20i	0.835681	−	0.835681i	−0.152606	−	0.988287i	\(-0.548767\pi\)
							0.988287	+	0.152606i	\(0.0487665\pi\)
\(41\)	1.61349e21			1.74412			0.872059	−	0.489401i	\(-0.162784\pi\)
							0.872059	+	0.489401i	\(0.162784\pi\)
\(43\)	−1.40825e21	−	1.40825e21i	−0.819578	−	0.819578i	0.166469	−	0.986047i	\(-0.446764\pi\)
							−0.986047	+	0.166469i	\(0.946764\pi\)
\(47\)	−7.35127e21	+	7.35127e21i	−1.34614	+	1.34614i	−0.456328	+	0.889812i	\(0.650836\pi\)
							−0.889812	+	0.456328i	\(0.849164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.27.c.a.3.8	yes	24
5.2	odd	4	inner	5.27.c.a.2.8	✓	24
5.3	odd	4		25.27.c.b.7.5		24
5.4	even	2		25.27.c.b.18.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.27.c.a.2.8	✓	24	5.2	odd	4	inner
5.27.c.a.3.8	yes	24	1.1	even	1	trivial
25.27.c.b.7.5		24	5.3	odd	4
25.27.c.b.18.5		24	5.4	even	2