Embedded newform 5.27.c.a.2.12

• Label: 5.27.c.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	5.2
Dual form			5.27.c.a.3.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11119.4 + 11119.4i) q^{2} +(1.76277e6 - 1.76277e6i) q^{3} +1.80172e8i q^{4} +(-1.23038e8 + 1.21449e9i) q^{5} +3.92017e10 q^{6} +(1.33787e10 + 1.33787e10i) q^{7} +(-1.25719e12 + 1.25719e12i) q^{8} -3.67282e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8190 q^{2} + 1867680 q^{3} - 140947920 q^{5} - 3382472712 q^{6} - 99234643200 q^{7} - 2004366809340 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11119.4	+	11119.4i	1.35734	+	1.35734i	0.877177	+	0.480168i	\(0.159424\pi\)
							0.480168	+	0.877177i	\(0.340576\pi\)
\(3\)	1.76277e6	−	1.76277e6i	1.10565	−	1.10565i	0.111936	−	0.993715i	\(-0.464295\pi\)
							0.993715	−	0.111936i	\(-0.0357053\pi\)
\(5\)	−1.23038e8	+	1.21449e9i	−0.100793	+	0.994907i
							\(7\)	1.33787e10	+	1.33787e10i	0.138082	+	0.138082i	0.772769	−	0.634687i	\(-0.218871\pi\)
−0.634687	+	0.772769i	\(0.718871\pi\)
\(11\)	9.56116e12			0.276953			0.138476	−	0.990366i	\(-0.455779\pi\)
							0.138476	+	0.990366i	\(0.455779\pi\)
\(13\)	4.32354e13	−	4.32354e13i	0.142750	−	0.142750i	−0.632120	−	0.774870i	\(-0.717815\pi\)
							0.774870	+	0.632120i	\(0.217815\pi\)
\(17\)	7.63413e15	+	7.63413e15i	0.770767	+	0.770767i	0.978241	−	0.207473i	\(-0.0665240\pi\)
							−0.207473	+	0.978241i	\(0.566524\pi\)
\(19\)		−	3.69294e15i		−	0.0878163i	−0.999036	−	0.0439082i	\(-0.986019\pi\)
							0.999036	−	0.0439082i	\(-0.0139809\pi\)
\(23\)	6.37022e16	−	6.37022e16i	0.126384	−	0.126384i	−0.641085	−	0.767470i	\(-0.721516\pi\)
							0.767470	+	0.641085i	\(0.221516\pi\)
\(29\)		−	1.61562e19i		−	1.57458i	−0.616580	−	0.787292i	\(-0.711482\pi\)
							0.616580	−	0.787292i	\(-0.288518\pi\)
\(31\)	2.82183e19			1.15566			0.577828	−	0.816158i	\(-0.303900\pi\)
							0.577828	+	0.816158i	\(0.303900\pi\)
\(37\)	−1.75371e20	−	1.75371e20i	−0.720004	−	0.720004i	0.248602	−	0.968606i	\(-0.420029\pi\)
							−0.968606	+	0.248602i	\(0.920029\pi\)
\(41\)	−5.21645e20			−0.563878			−0.281939	−	0.959432i	\(-0.590978\pi\)
							−0.281939	+	0.959432i	\(0.590978\pi\)
\(43\)	1.20857e21	−	1.20857e21i	0.703366	−	0.703366i	−0.261765	−	0.965132i	\(-0.584305\pi\)
							0.965132	+	0.261765i	\(0.0843046\pi\)
\(47\)	5.45828e21	+	5.45828e21i	0.999502	+	0.999502i	1.00000		0.000498336i	\(-0.000158625\pi\)
							−0.000498336		1.00000i	\(0.500159\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.27.c.a.2.12	✓	24
5.2	odd	4		25.27.c.b.18.1		24
5.3	odd	4	inner	5.27.c.a.3.12	yes	24
5.4	even	2		25.27.c.b.7.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.27.c.a.2.12	✓	24	1.1	even	1	trivial
5.27.c.a.3.12	yes	24	5.3	odd	4	inner
25.27.c.b.7.1		24	5.4	even	2
25.27.c.b.18.1		24	5.2	odd	4