Embedded newform 25.4.e.a.14.2

• Label: 25.4.e.a.14.2
• Level: $25$
• Weight: $4$
• Character: 25.14
• Analytic conductor: $1.475$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47504775014\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			14.2
Character	\(\chi\)	\(=\)	25.14
Dual form			25.4.e.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.23725 - 3.07931i) q^{2} +(8.89950 + 2.89162i) q^{3} +(-2.00472 + 6.16988i) q^{4} +(-3.48187 - 10.6243i) q^{5} +(-11.0062 - 33.8735i) q^{6} +4.54748i q^{7} +(-5.47554 + 1.77911i) q^{8} +(48.9961 + 35.5978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{2} - 5 q^{3} + 13 q^{4} + 15 q^{5} - 7 q^{6} - 110 q^{8} + 47 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.23725	−	3.07931i	−0.790986	−	1.08870i	−0.993985	−	0.109520i	\(-0.965069\pi\)
							0.202999	−	0.979179i	\(-0.434931\pi\)
\(3\)	8.89950	+	2.89162i	1.71271	+	0.556493i	0.990781	−	0.135475i	\(-0.0432561\pi\)
							0.721928	+	0.691968i	\(0.243256\pi\)
\(5\)	−3.48187	−	10.6243i	−0.311428	−	0.950270i
							\(7\)			4.54748i			0.245541i	0.992435	+	0.122770i	\(0.0391779\pi\)
−0.992435	+	0.122770i	\(0.960822\pi\)
\(11\)	10.9297	−	7.94089i	0.299584	−	0.217661i	−0.427830	−	0.903859i	\(-0.640722\pi\)
							0.727414	+	0.686198i	\(0.240722\pi\)
\(13\)	−40.2463	+	55.3943i	−0.858639	+	1.18182i	0.123253	+	0.992375i	\(0.460667\pi\)
							−0.981892	+	0.189440i	\(0.939333\pi\)
\(17\)	−35.1667	+	11.4264i	−0.501716	+	0.163017i	−0.548932	−	0.835867i	\(-0.684965\pi\)
							0.0472154	+	0.998885i	\(0.484965\pi\)
\(19\)	0.479342	+	1.47526i	0.00578782	+	0.0178131i	0.953909	−	0.300097i	\(-0.0970190\pi\)
							−0.948121	+	0.317910i	\(0.897019\pi\)
\(23\)	0.969915	+	1.33497i	0.00879310	+	0.0121027i	0.813391	−	0.581717i	\(-0.197619\pi\)
							−0.804598	+	0.593820i	\(0.797619\pi\)
\(29\)	49.0853	−	151.069i	0.314307	−	0.967339i	−0.661731	−	0.749741i	\(-0.730178\pi\)
							0.976039	−	0.217597i	\(-0.0698220\pi\)
\(31\)	−83.6862	−	257.560i	−0.484855	−	1.49223i	−0.832191	−	0.554489i	\(-0.812914\pi\)
							0.347337	−	0.937741i	\(-0.387086\pi\)
\(37\)	81.9442	−	112.787i	0.364096	−	0.501135i	−0.587188	−	0.809450i	\(-0.699765\pi\)
							0.951284	+	0.308315i	\(0.0997652\pi\)
\(41\)	80.5761	+	58.5420i	0.306924	+	0.222993i	0.730575	−	0.682832i	\(-0.239252\pi\)
							−0.423652	+	0.905825i	\(0.639252\pi\)
\(43\)			135.690i			0.481223i	0.970622	+	0.240611i	\(0.0773479\pi\)
							−0.970622	+	0.240611i	\(0.922652\pi\)
\(47\)	−171.103	−	55.5946i	−0.531019	−	0.172539i	0.0312212	−	0.999512i	\(-0.490060\pi\)
							−0.562240	+	0.826974i	\(0.690060\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.4.e.a.14.2	yes	24
3.2	odd	2		225.4.m.a.64.5		24
5.2	odd	4		125.4.d.b.51.10		48
5.3	odd	4		125.4.d.b.51.3		48
5.4	even	2		125.4.e.a.74.5		24
25.3	odd	20		625.4.a.g.1.21		24
25.9	even	10	inner	25.4.e.a.9.2	✓	24
25.12	odd	20		125.4.d.b.76.10		48
25.13	odd	20		125.4.d.b.76.3		48
25.16	even	5		125.4.e.a.49.5		24
25.22	odd	20		625.4.a.g.1.4		24
75.59	odd	10		225.4.m.a.109.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.4.e.a.9.2	✓	24	25.9	even	10	inner
25.4.e.a.14.2	yes	24	1.1	even	1	trivial
125.4.d.b.51.3		48	5.3	odd	4
125.4.d.b.51.10		48	5.2	odd	4
125.4.d.b.76.3		48	25.13	odd	20
125.4.d.b.76.10		48	25.12	odd	20
125.4.e.a.49.5		24	25.16	even	5
125.4.e.a.74.5		24	5.4	even	2
225.4.m.a.64.5		24	3.2	odd	2
225.4.m.a.109.5		24	75.59	odd	10
625.4.a.g.1.4		24	25.22	odd	20
625.4.a.g.1.21		24	25.3	odd	20