Embedded newform 5.27.c.a.3.6

• Label: 5.27.c.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	5.3
Dual form			5.27.c.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1256.27 + 1256.27i) q^{2} +(745542. + 745542. i) q^{3} +6.39524e7i q^{4} +(1.73270e8 - 1.20834e9i) q^{5} -1.87321e9 q^{6} +(-1.19717e11 + 1.19717e11i) q^{7} +(-1.64649e11 - 1.64649e11i) q^{8} -1.43020e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8190 q^{2} + 1867680 q^{3} - 140947920 q^{5} - 3382472712 q^{6} - 99234643200 q^{7} - 2004366809340 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1256.27	+	1256.27i	−0.153354	+	0.153354i	−0.779614	−	0.626260i	\(-0.784585\pi\)
							0.626260	+	0.779614i	\(0.284585\pi\)
\(3\)	745542.	+	745542.i	0.467623	+	0.467623i	0.901144	−	0.433521i	\(-0.142729\pi\)
							−0.433521	+	0.901144i	\(0.642729\pi\)
\(5\)	1.73270e8	−	1.20834e9i	0.141943	−	0.989875i
							\(7\)	−1.19717e11	+	1.19717e11i	−1.23561	+	1.23561i	−0.273826	+	0.961779i	\(0.588289\pi\)
−0.961779	+	0.273826i	\(0.911711\pi\)
\(11\)	−3.70721e11			−0.0107385			−0.00536923	−	0.999986i	\(-0.501709\pi\)
							−0.00536923	+	0.999986i	\(0.501709\pi\)
\(13\)	1.62469e14	+	1.62469e14i	0.536422	+	0.536422i	0.922476	−	0.386054i	\(-0.126162\pi\)
							−0.386054	+	0.922476i	\(0.626162\pi\)
\(17\)	−5.42722e15	+	5.42722e15i	−0.547951	+	0.547951i	−0.925848	−	0.377897i	\(-0.876647\pi\)
							0.377897	+	0.925848i	\(0.376647\pi\)
\(19\)		−	6.71524e16i		−	1.59685i	−0.602093	−	0.798426i	\(-0.705666\pi\)
							0.602093	−	0.798426i	\(-0.294334\pi\)
\(23\)	−5.23969e17	−	5.23969e17i	−1.03955	−	1.03955i	−0.999185	−	0.0403601i	\(-0.987150\pi\)
							−0.0403601	−	0.999185i	\(-0.512850\pi\)
\(29\)		−	6.03827e17i		−	0.0588489i	−0.999567	−	0.0294244i	\(-0.990633\pi\)
							0.999567	−	0.0294244i	\(-0.00936744\pi\)
\(31\)	−2.68787e19			−1.10079			−0.550397	−	0.834903i	\(-0.685524\pi\)
							−0.550397	+	0.834903i	\(0.685524\pi\)
\(37\)	−7.04282e19	+	7.04282e19i	−0.289151	+	0.289151i	−0.836744	−	0.547594i	\(-0.815544\pi\)
							0.547594	+	0.836744i	\(0.315544\pi\)
\(41\)	1.01291e21			1.09492			0.547460	−	0.836832i	\(-0.315595\pi\)
							0.547460	+	0.836832i	\(0.315595\pi\)
\(43\)	−5.41974e20	−	5.41974e20i	−0.315420	−	0.315420i	0.531585	−	0.847005i	\(-0.321597\pi\)
							−0.847005	+	0.531585i	\(0.821597\pi\)
\(47\)	−3.69475e21	+	3.69475e21i	−0.676571	+	0.676571i	−0.959223	−	0.282652i	\(-0.908786\pi\)
							0.282652	+	0.959223i	\(0.408786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.27.c.a.3.6	yes	24
5.2	odd	4	inner	5.27.c.a.2.6	✓	24
5.3	odd	4		25.27.c.b.7.7		24
5.4	even	2		25.27.c.b.18.7		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.27.c.a.2.6	✓	24	5.2	odd	4	inner
5.27.c.a.3.6	yes	24	1.1	even	1	trivial
25.27.c.b.7.7		24	5.3	odd	4
25.27.c.b.18.7		24	5.4	even	2