Embedded newform 25.6.e.a.19.6

• Label: 25.6.e.a.19.6
• Level: $25$
• Weight: $6$
• Character: 25.19
• 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			19.6
Character	\(\chi\)	\(=\)	25.19
Dual form			25.6.e.a.4.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.580095 + 0.188484i) q^{2} +(15.1807 + 20.8945i) q^{3} +(-25.5876 + 18.5905i) q^{4} +(-52.0512 - 20.3881i) q^{5} +(-12.7445 - 9.25945i) q^{6} -10.2303i q^{7} +(22.8118 - 31.3978i) q^{8} +(-131.033 + 403.279i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.580095	+	0.188484i	−0.102547	+	0.0333196i	−0.359841	−	0.933014i	\(-0.617169\pi\)
							0.257294	+	0.966333i	\(0.417169\pi\)
\(3\)	15.1807	+	20.8945i	0.973843	+	1.34038i	0.940081	+	0.340950i	\(0.110749\pi\)
							0.0337618	+	0.999430i	\(0.489251\pi\)
\(5\)	−52.0512	−	20.3881i	−0.931120	−	0.364713i
							\(7\)		−	10.2303i		−	0.0789121i	−0.999221	−	0.0394560i	\(-0.987437\pi\)
0.999221	−	0.0394560i	\(-0.0125625\pi\)
\(11\)	150.440	+	463.006i	0.374870	+	1.15373i	0.943566	+	0.331184i	\(0.107448\pi\)
							−0.568696	+	0.822548i	\(0.692552\pi\)
\(13\)	868.080	+	282.056i	1.42463	+	0.462890i	0.917069	−	0.398728i	\(-0.130548\pi\)
							0.507559	+	0.861617i	\(0.330548\pi\)
\(17\)	111.270	−	153.150i	0.0933804	−	0.128527i	−0.759770	−	0.650191i	\(-0.774689\pi\)
							0.853151	+	0.521664i	\(0.174689\pi\)
\(19\)	433.060	+	314.636i	0.275210	+	0.199952i	0.716825	−	0.697253i	\(-0.245594\pi\)
							−0.441616	+	0.897204i	\(0.645594\pi\)
\(23\)	−2826.19	+	918.286i	−1.11399	+	0.361958i	−0.807472	−	0.589906i	\(-0.799165\pi\)
							−0.306520	+	0.951864i	\(0.599165\pi\)
\(29\)	5587.39	−	4059.48i	1.23371	−	0.896345i	0.236550	−	0.971619i	\(-0.423983\pi\)
							0.997163	+	0.0752744i	\(0.0239833\pi\)
\(31\)	−4536.49	−	3295.95i	−0.847843	−	0.615994i	0.0767074	−	0.997054i	\(-0.475559\pi\)
							−0.924550	+	0.381060i	\(0.875559\pi\)
\(37\)	5692.75	+	1849.69i	0.683625	+	0.222123i	0.630182	−	0.776448i	\(-0.282980\pi\)
							0.0534431	+	0.998571i	\(0.482980\pi\)
\(41\)	927.964	−	2855.98i	0.0862128	−	0.265336i	−0.898651	−	0.438663i	\(-0.855452\pi\)
							0.984864	+	0.173328i	\(0.0554520\pi\)
\(43\)			13831.0i			1.14073i	0.821392	+	0.570364i	\(0.193198\pi\)
							−0.821392	+	0.570364i	\(0.806802\pi\)
\(47\)	−16043.3	−	22081.8i	−1.05938	−	1.45811i	−0.880387	−	0.474256i	\(-0.842717\pi\)
							−0.178990	−	0.983851i	\(-0.557283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.19.6	yes	48
25.2	odd	20		625.6.a.g.1.23		48
25.4	even	10	inner	25.6.e.a.4.6	✓	48
25.23	odd	20		625.6.a.g.1.26		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.6	✓	48	25.4	even	10	inner
25.6.e.a.19.6	yes	48	1.1	even	1	trivial
625.6.a.g.1.23		48	25.2	odd	20
625.6.a.g.1.26		48	25.23	odd	20