Embedded newform 25.8.e.a.4.12

• Label: 25.8.e.a.4.12
• Level: $25$
• Weight: $8$
• Character: 25.4
• Analytic conductor: $7.810$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.80962563710\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.15239 + 2.97379i) q^{2} +(10.6184 - 14.6149i) q^{3} +(-28.6314 - 20.8020i) q^{4} +(-161.344 - 228.239i) q^{5} +(140.645 - 102.185i) q^{6} -933.124i q^{7} +(-924.215 - 1272.07i) q^{8} +(574.974 + 1769.59i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.15239	+	2.97379i	0.808964	+	0.262848i	0.684159	−	0.729333i	\(-0.260169\pi\)
							0.124805	+	0.992181i	\(0.460169\pi\)
\(3\)	10.6184	−	14.6149i	0.227056	−	0.312516i	−0.680255	−	0.732975i	\(-0.738131\pi\)
							0.907311	+	0.420459i	\(0.138131\pi\)
\(5\)	−161.344	−	228.239i	−0.577241	−	0.816574i
							\(7\)		−	933.124i		−	1.02824i	−0.857717	−	0.514122i	\(-0.828118\pi\)
0.857717	−	0.514122i	\(-0.171882\pi\)
\(11\)	391.799	−	1205.83i	0.0887542	−	0.273157i	−0.896821	−	0.442393i	\(-0.854130\pi\)
							0.985576	+	0.169235i	\(0.0541298\pi\)
\(13\)	3418.27	−	1110.66i	0.431524	−	0.140211i	−0.0851977	−	0.996364i	\(-0.527152\pi\)
							0.516722	+	0.856153i	\(0.327152\pi\)
\(17\)	−239.809	−	330.068i	−0.0118384	−	0.0162942i	0.803057	−	0.595903i	\(-0.203206\pi\)
							−0.814895	+	0.579608i	\(0.803206\pi\)
\(19\)	23165.5	−	16830.7i	0.774824	−	0.562943i	−0.128597	−	0.991697i	\(-0.541047\pi\)
							0.903421	+	0.428754i	\(0.141047\pi\)
\(23\)	−56471.6	−	18348.7i	−0.967794	−	0.314455i	−0.217869	−	0.975978i	\(-0.569910\pi\)
							−0.749925	+	0.661523i	\(0.769910\pi\)
\(29\)	196643.	+	142870.i	1.49722	+	1.08780i	0.971472	+	0.237156i	\(0.0762153\pi\)
							0.525750	+	0.850639i	\(0.323785\pi\)
\(31\)	−48534.6	+	35262.5i	−0.292608	+	0.212592i	−0.724398	−	0.689382i	\(-0.757882\pi\)
							0.431790	+	0.901974i	\(0.357882\pi\)
\(37\)	102818.	−	33407.7i	0.333706	−	0.108428i	−0.137371	−	0.990520i	\(-0.543865\pi\)
							0.471077	+	0.882092i	\(0.343865\pi\)
\(41\)	−252748.	−	777877.i	−0.572722	−	1.76266i	−0.643809	−	0.765186i	\(-0.722647\pi\)
							0.0710873	−	0.997470i	\(-0.477353\pi\)
\(43\)		−	164592.i		−	0.315697i	−0.987463	−	0.157848i	\(-0.949544\pi\)
							0.987463	−	0.157848i	\(-0.0504558\pi\)
\(47\)	560741.	−	771794.i	0.787807	−	1.08432i	−0.206570	−	0.978432i	\(-0.566230\pi\)
							0.994378	−	0.105892i	\(-0.0337698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.12	✓	64
25.12	odd	20		625.8.a.g.1.20		64
25.13	odd	20		625.8.a.g.1.45		64
25.19	even	10	inner	25.8.e.a.19.12	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.12	✓	64	1.1	even	1	trivial
25.8.e.a.19.12	yes	64	25.19	even	10	inner
625.8.a.g.1.20		64	25.12	odd	20
625.8.a.g.1.45		64	25.13	odd	20