Embedded newform 25.6.e.a.4.4

• Label: 25.6.e.a.4.4
• Level: $25$
• Weight: $6$
• Character: 25.4
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	25.4
Dual form			25.6.e.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.43609 - 1.76629i) q^{2} +(-17.9990 + 24.7735i) q^{3} +(0.542750 + 0.394331i) q^{4} +(37.4160 - 41.5336i) q^{5} +(141.602 - 102.880i) q^{6} -28.6321i q^{7} +(105.256 + 144.873i) q^{8} +(-214.672 - 660.693i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.43609	−	1.76629i	−0.960974	−	0.312239i	−0.213807	−	0.976876i	\(-0.568586\pi\)
							−0.747167	+	0.664636i	\(0.768586\pi\)
\(3\)	−17.9990	+	24.7735i	−1.15464	+	1.58922i	−0.425330	+	0.905038i	\(0.639842\pi\)
							−0.729309	+	0.684185i	\(0.760158\pi\)
\(5\)	37.4160	−	41.5336i	0.669318	−	0.742976i
							\(7\)		−	28.6321i		−	0.220855i	−0.993884	−	0.110428i	\(-0.964778\pi\)
0.993884	−	0.110428i	\(-0.0352220\pi\)
\(11\)	0.178988	−	0.550870i	0.000446008	−	0.00137267i	−0.950833	−	0.309703i	\(-0.899770\pi\)
							0.951279	+	0.308331i	\(0.0997703\pi\)
\(13\)	444.375	−	144.386i	0.729276	−	0.236956i	0.0792355	−	0.996856i	\(-0.474752\pi\)
							0.650040	+	0.759900i	\(0.274752\pi\)
\(17\)	−591.867	−	814.635i	−0.496709	−	0.683661i	0.484899	−	0.874570i	\(-0.338856\pi\)
							−0.981608	+	0.190909i	\(0.938856\pi\)
\(19\)	1060.80	−	770.713i	0.674136	−	0.489789i	−0.197271	−	0.980349i	\(-0.563208\pi\)
							0.871407	+	0.490560i	\(0.163208\pi\)
\(23\)	−3099.69	−	1007.15i	−1.22180	−	0.396986i	−0.374061	−	0.927404i	\(-0.622035\pi\)
							−0.847736	+	0.530418i	\(0.822035\pi\)
\(29\)	−1539.53	−	1118.53i	−0.339932	−	0.246975i	0.404701	−	0.914449i	\(-0.367376\pi\)
							−0.744633	+	0.667474i	\(0.767376\pi\)
\(31\)	3736.64	−	2714.83i	0.698357	−	0.507386i	−0.181040	−	0.983476i	\(-0.557946\pi\)
							0.879397	+	0.476090i	\(0.157946\pi\)
\(37\)	−829.390	+	269.485i	−0.0995989	+	0.0323616i	−0.358393	−	0.933571i	\(-0.616675\pi\)
							0.258794	+	0.965933i	\(0.416675\pi\)
\(41\)	−3989.02	−	12276.9i	−0.370601	−	1.14059i	−0.946399	−	0.323000i	\(-0.895309\pi\)
							0.575798	−	0.817592i	\(-0.304691\pi\)
\(43\)		−	2723.59i		−	0.224632i	−0.993673	−	0.112316i	\(-0.964173\pi\)
							0.993673	−	0.112316i	\(-0.0358268\pi\)
\(47\)	−4785.94	+	6587.28i	−0.316026	+	0.434972i	−0.937249	−	0.348662i	\(-0.886636\pi\)
							0.621223	+	0.783634i	\(0.286636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.4.4	✓	48
25.12	odd	20		625.6.a.g.1.35		48
25.13	odd	20		625.6.a.g.1.14		48
25.19	even	10	inner	25.6.e.a.19.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.4	✓	48	1.1	even	1	trivial
25.6.e.a.19.4	yes	48	25.19	even	10	inner
625.6.a.g.1.14		48	25.13	odd	20
625.6.a.g.1.35		48	25.12	odd	20