Embedded newform 25.6.e.a.4.8

• Label: 25.6.e.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	25.4
Dual form			25.6.e.a.19.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.22114 + 0.721693i) q^{2} +(-9.09397 + 12.5168i) q^{3} +(-21.4759 - 15.6032i) q^{4} +(-55.6985 - 4.76195i) q^{5} +(-29.2323 + 21.2385i) q^{6} -43.5518i q^{7} +(-80.3681 - 110.617i) q^{8} +(1.12167 + 3.45216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.22114	+	0.721693i	0.392646	+	0.127579i	0.498685	−	0.866784i	\(-0.333817\pi\)
							−0.106038	+	0.994362i	\(0.533817\pi\)
\(3\)	−9.09397	+	12.5168i	−0.583379	+	0.802952i	−0.994061	−	0.108827i	\(-0.965291\pi\)
							0.410682	+	0.911779i	\(0.365291\pi\)
\(5\)	−55.6985	−	4.76195i	−0.996365	−	0.0851844i
							\(7\)		−	43.5518i		−	0.335939i	−0.985792	−	0.167970i	\(-0.946279\pi\)
0.985792	−	0.167970i	\(-0.0537210\pi\)
\(11\)	−175.258	+	539.388i	−0.436713	+	1.34406i	0.454608	+	0.890691i	\(0.349779\pi\)
							−0.891321	+	0.453372i	\(0.850221\pi\)
\(13\)	−283.723	+	92.1871i	−0.465624	+	0.151291i	−0.532427	−	0.846476i	\(-0.678720\pi\)
							0.0668030	+	0.997766i	\(0.478720\pi\)
\(17\)	−404.419	−	556.635i	−0.339398	−	0.467142i	0.604867	−	0.796326i	\(-0.293226\pi\)
							−0.944266	+	0.329185i	\(0.893226\pi\)
\(19\)	912.609	−	663.049i	0.579964	−	0.421368i	−0.258747	−	0.965945i	\(-0.583310\pi\)
							0.838711	+	0.544577i	\(0.183310\pi\)
\(23\)	19.1117	+	6.20976i	0.00753319	+	0.00244768i	0.312781	−	0.949825i	\(-0.398739\pi\)
							−0.305248	+	0.952273i	\(0.598739\pi\)
\(29\)	−5688.17	−	4132.70i	−1.25597	−	0.912512i	−0.257413	−	0.966301i	\(-0.582870\pi\)
							−0.998552	+	0.0537893i	\(0.982870\pi\)
\(31\)	−997.324	+	724.598i	−0.186394	+	0.135423i	−0.677069	−	0.735919i	\(-0.736750\pi\)
							0.490675	+	0.871343i	\(0.336750\pi\)
\(37\)	−12087.9	+	3927.61i	−1.45160	+	0.471655i	−0.925494	−	0.378762i	\(-0.876350\pi\)
							−0.526110	+	0.850417i	\(0.676350\pi\)
\(41\)	5775.90	+	17776.4i	0.536612	+	1.65152i	0.740141	+	0.672452i	\(0.234759\pi\)
							−0.203530	+	0.979069i	\(0.565241\pi\)
\(43\)		−	14422.3i		−	1.18950i	−0.803910	−	0.594750i	\(-0.797251\pi\)
							0.803910	−	0.594750i	\(-0.202749\pi\)
\(47\)	3658.54	−	5035.55i	0.241581	−	0.332508i	−0.670959	−	0.741494i	\(-0.734118\pi\)
							0.912541	+	0.408986i	\(0.134118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.4.8	✓	48
25.12	odd	20		625.6.a.g.1.20		48
25.13	odd	20		625.6.a.g.1.29		48
25.19	even	10	inner	25.6.e.a.19.8	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.8	✓	48	1.1	even	1	trivial
25.6.e.a.19.8	yes	48	25.19	even	10	inner
625.6.a.g.1.20		48	25.12	odd	20
625.6.a.g.1.29		48	25.13	odd	20