Embedded newform 25.6.e.a.9.3

• Label: 25.6.e.a.9.3
• Level: $25$
• Weight: $6$
• Character: 25.9
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $48$
• 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			9.3
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.62228 + 4.98565i) q^{2} +(10.7412 - 3.49003i) q^{3} +(-1.84718 - 5.68504i) q^{4} +(55.8518 + 2.36063i) q^{5} +(-21.5076 + 66.1937i) q^{6} +133.004i q^{7} +(-152.517 - 49.5557i) q^{8} +(-93.3979 + 67.8576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.62228	+	4.98565i	−0.640335	+	0.881346i	−0.998633	−	0.0522612i	\(-0.983357\pi\)
							0.358298	+	0.933607i	\(0.383357\pi\)
\(3\)	10.7412	−	3.49003i	0.689049	−	0.223885i	0.0564960	−	0.998403i	\(-0.482007\pi\)
							0.632553	+	0.774517i	\(0.282007\pi\)
\(5\)	55.8518	+	2.36063i	0.999108	+	0.0422283i
							\(7\)			133.004i			1.02594i	0.858407	+	0.512969i	\(0.171454\pi\)
−0.858407	+	0.512969i	\(0.828546\pi\)
\(11\)	386.022	+	280.462i	0.961902	+	0.698863i	0.953592	−	0.301103i	\(-0.0973547\pi\)
							0.00831038	+	0.999965i	\(0.497355\pi\)
\(13\)	−3.32098	−	4.57094i	−0.00545015	−	0.00750148i	0.806283	−	0.591530i	\(-0.201476\pi\)
							−0.811733	+	0.584028i	\(0.801476\pi\)
\(17\)	72.0953	+	23.4252i	0.0605041	+	0.0196590i	0.339113	−	0.940746i	\(-0.389873\pi\)
							−0.278609	+	0.960405i	\(0.589873\pi\)
\(19\)	971.806	−	2990.91i	0.617583	−	1.90073i	0.272479	−	0.962162i	\(-0.412156\pi\)
							0.345104	−	0.938564i	\(-0.387844\pi\)
\(23\)	−1122.34	+	1544.77i	−0.442390	+	0.608897i	−0.970741	−	0.240129i	\(-0.922810\pi\)
							0.528351	+	0.849026i	\(0.322810\pi\)
\(29\)	−563.509	−	1734.30i	−0.124424	−	0.382939i	0.869371	−	0.494159i	\(-0.164524\pi\)
							−0.993796	+	0.111220i	\(0.964524\pi\)
\(31\)	2442.53	−	7517.35i	0.456495	−	1.40495i	−0.412875	−	0.910788i	\(-0.635476\pi\)
							0.869371	−	0.494160i	\(-0.164524\pi\)
\(37\)	−4495.65	−	6187.74i	−0.539869	−	0.743066i	0.448725	−	0.893670i	\(-0.351878\pi\)
							−0.988594	+	0.150604i	\(0.951878\pi\)
\(41\)	3575.16	−	2597.51i	0.332151	−	0.241322i	−0.409192	−	0.912448i	\(-0.634189\pi\)
							0.741343	+	0.671126i	\(0.234189\pi\)
\(43\)		−	6359.94i		−	0.524544i	−0.964994	−	0.262272i	\(-0.915528\pi\)
							0.964994	−	0.262272i	\(-0.0844718\pi\)
\(47\)	2722.41	−	884.564i	0.179766	−	0.0584097i	−0.217751	−	0.976004i	\(-0.569872\pi\)
							0.397517	+	0.917595i	\(0.369872\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.3	✓	48
25.8	odd	20		625.6.a.g.1.12		48
25.14	even	10	inner	25.6.e.a.14.3	yes	48
25.17	odd	20		625.6.a.g.1.37		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.3	✓	48	1.1	even	1	trivial
25.6.e.a.14.3	yes	48	25.14	even	10	inner
625.6.a.g.1.12		48	25.8	odd	20
625.6.a.g.1.37		48	25.17	odd	20