Embedded newform 25.3.f.a.3.1

• Label: 25.3.f.a.3.1
• Level: $25$
• Weight: $3$
• Character: 25.3
• Analytic conductor: $0.681$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.681200660901\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	25.3
Dual form			25.3.f.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69523 - 3.32707i) q^{2} +(-0.0858318 - 0.541921i) q^{3} +(-5.84445 + 8.04419i) q^{4} +(2.26962 - 4.45520i) q^{5} +(-1.65750 + 1.20425i) q^{6} +(1.68463 - 1.68463i) q^{7} +(21.9189 + 3.47161i) q^{8} +(8.27320 - 2.68812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 10 q^{2} - 10 q^{3} - 10 q^{4} - 10 q^{5} - 6 q^{6} - 10 q^{7} - 10 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69523	−	3.32707i	−0.847613	−	1.66353i	−0.743269	−	0.668993i	\(-0.766726\pi\)
							−0.104344	−	0.994541i	\(-0.533274\pi\)
\(3\)	−0.0858318	−	0.541921i	−0.0286106	−	0.180640i	0.969245	−	0.246099i	\(-0.0791489\pi\)
							−0.997855	+	0.0654589i	\(0.979149\pi\)
\(5\)	2.26962	−	4.45520i	0.453925	−	0.891040i
							\(7\)	1.68463	−	1.68463i	0.240661	−	0.240661i	−0.576462	−	0.817124i	\(-0.695567\pi\)
0.817124	+	0.576462i	\(0.195567\pi\)
\(11\)	−2.56403	+	7.89126i	−0.233093	+	0.717388i	0.764275	+	0.644890i	\(0.223097\pi\)
							−0.997369	+	0.0724975i	\(0.976903\pi\)
\(13\)	−5.04839	+	9.90802i	−0.388338	+	0.762156i	−0.999571	−	0.0292950i	\(-0.990674\pi\)
							0.611233	+	0.791451i	\(0.290674\pi\)
\(17\)	0.715822	−	4.51952i	0.0421072	−	0.265854i	−0.957649	−	0.287938i	\(-0.907030\pi\)
							0.999756	+	0.0220840i	\(0.00703014\pi\)
\(19\)	12.8236	+	17.6502i	0.674926	+	0.928956i	0.999859	−	0.0167812i	\(-0.00534187\pi\)
							−0.324933	+	0.945737i	\(0.605342\pi\)
\(23\)	−20.0785	+	10.2305i	−0.872979	+	0.444805i	−0.832273	−	0.554366i	\(-0.812961\pi\)
							−0.0407058	+	0.999171i	\(0.512961\pi\)
\(29\)	17.4721	−	24.0483i	0.602488	−	0.829253i	−0.393446	−	0.919348i	\(-0.628717\pi\)
							0.995933	+	0.0900948i	\(0.0287170\pi\)
\(31\)	−17.6079	+	12.7929i	−0.567997	+	0.412674i	−0.834377	−	0.551194i	\(-0.814172\pi\)
							0.266380	+	0.963868i	\(0.414172\pi\)
\(37\)	9.49023	+	4.83551i	0.256493	+	0.130690i	0.577510	−	0.816384i	\(-0.304025\pi\)
							−0.321017	+	0.947073i	\(0.604025\pi\)
\(41\)	−7.33421	−	22.5724i	−0.178883	−	0.550546i	0.820906	−	0.571063i	\(-0.193469\pi\)
							−0.999790	+	0.0205172i	\(0.993469\pi\)
\(43\)	14.6230	+	14.6230i	0.340070	+	0.340070i	0.856394	−	0.516323i	\(-0.172700\pi\)
							−0.516323	+	0.856394i	\(0.672700\pi\)
\(47\)	−55.8932	+	8.85261i	−1.18922	+	0.188353i	−0.719518	−	0.694474i	\(-0.755637\pi\)
							−0.469698	+	0.882827i	\(0.655637\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.3.f.a.3.1	✓	32
3.2	odd	2		225.3.r.a.28.4		32
4.3	odd	2		400.3.bg.c.353.2		32
5.2	odd	4		125.3.f.a.7.4		32
5.3	odd	4		125.3.f.b.7.1		32
5.4	even	2		125.3.f.c.118.4		32
25.6	even	5		125.3.f.a.18.4		32
25.8	odd	20		125.3.f.c.107.4		32
25.17	odd	20	inner	25.3.f.a.17.1	yes	32
25.19	even	10		125.3.f.b.18.1		32
75.17	even	20		225.3.r.a.217.4		32
100.67	even	20		400.3.bg.c.17.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.3.f.a.3.1	✓	32	1.1	even	1	trivial
25.3.f.a.17.1	yes	32	25.17	odd	20	inner
125.3.f.a.7.4		32	5.2	odd	4
125.3.f.a.18.4		32	25.6	even	5
125.3.f.b.7.1		32	5.3	odd	4
125.3.f.b.18.1		32	25.19	even	10
125.3.f.c.107.4		32	25.8	odd	20
125.3.f.c.118.4		32	5.4	even	2
225.3.r.a.28.4		32	3.2	odd	2
225.3.r.a.217.4		32	75.17	even	20
400.3.bg.c.17.2		32	100.67	even	20
400.3.bg.c.353.2		32	4.3	odd	2