Embedded newform 25.6.e.a.4.11

• Label: 25.6.e.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	25.4
Dual form			25.6.e.a.19.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.05151 + 2.61610i) q^{2} +(8.11728 - 11.1725i) q^{3} +(32.0944 + 23.3179i) q^{4} +(-0.713405 + 55.8971i) q^{5} +(94.5847 - 68.7198i) q^{6} -150.001i q^{7} +(38.1710 + 52.5378i) q^{8} +(16.1571 + 49.7264i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.05151	+	2.61610i	1.42332	+	0.462465i	0.916655	−	0.399678i	\(-0.130878\pi\)
							0.506665	+	0.862143i	\(0.330878\pi\)
\(3\)	8.11728	−	11.1725i	0.520724	−	0.716715i	−0.464958	−	0.885333i	\(-0.653930\pi\)
							0.985681	+	0.168618i	\(0.0539304\pi\)
\(5\)	−0.713405	+	55.8971i	−0.0127618	+	0.999919i
							\(7\)		−	150.001i		−	1.15704i	−0.815667	−	0.578522i	\(-0.803630\pi\)
0.815667	−	0.578522i	\(-0.196370\pi\)
\(11\)	−39.8155	+	122.540i	−0.0992135	+	0.305348i	−0.988329	−	0.152335i	\(-0.951321\pi\)
							0.889115	+	0.457683i	\(0.151321\pi\)
\(13\)	−979.729	+	318.333i	−1.60786	+	0.522424i	−0.969034	−	0.246929i	\(-0.920579\pi\)
							−0.638824	+	0.769353i	\(0.720579\pi\)
\(17\)	−193.464	−	266.281i	−0.162360	−	0.223469i	0.720084	−	0.693887i	\(-0.244103\pi\)
							−0.882444	+	0.470418i	\(0.844103\pi\)
\(19\)	991.194	−	720.145i	0.629904	−	0.457652i	−0.226463	−	0.974020i	\(-0.572716\pi\)
							0.856367	+	0.516367i	\(0.172716\pi\)
\(23\)	−243.597	−	79.1494i	−0.0960179	−	0.0311981i	0.260614	−	0.965443i	\(-0.416075\pi\)
							−0.356632	+	0.934245i	\(0.616075\pi\)
\(29\)	6700.77	+	4868.39i	1.47955	+	1.07496i	0.977701	+	0.210000i	\(0.0673463\pi\)
							0.501848	+	0.864956i	\(0.332654\pi\)
\(31\)	2640.10	−	1918.14i	0.493419	−	0.358490i	−0.313079	−	0.949727i	\(-0.601360\pi\)
							0.806498	+	0.591237i	\(0.201360\pi\)
\(37\)	−2108.38	+	685.054i	−0.253189	+	0.0822661i	−0.432862	−	0.901460i	\(-0.642496\pi\)
							0.179673	+	0.983726i	\(0.442496\pi\)
\(41\)	4382.77	+	13488.8i	0.407182	+	1.25318i	0.919059	+	0.394119i	\(0.128950\pi\)
							−0.511877	+	0.859059i	\(0.671050\pi\)
\(43\)		−	12815.6i		−	1.05698i	−0.848939	−	0.528490i	\(-0.822758\pi\)
							0.848939	−	0.528490i	\(-0.177242\pi\)
\(47\)	−4582.48	+	6307.24i	−0.302591	+	0.416480i	−0.933053	−	0.359740i	\(-0.882865\pi\)
							0.630462	+	0.776220i	\(0.282865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.4.11	✓	48
25.12	odd	20		625.6.a.g.1.8		48
25.13	odd	20		625.6.a.g.1.41		48
25.19	even	10	inner	25.6.e.a.19.11	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.11	✓	48	1.1	even	1	trivial
25.6.e.a.19.11	yes	48	25.19	even	10	inner
625.6.a.g.1.8		48	25.12	odd	20
625.6.a.g.1.41		48	25.13	odd	20