Embedded newform 25.6.e.a.9.8

• Label: 25.6.e.a.9.8
• Level: $25$
• Weight: $6$
• Character: 25.9
• 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			9.8
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61555 - 2.22361i) q^{2} +(12.9949 - 4.22230i) q^{3} +(7.55409 + 23.2491i) q^{4} +(30.9389 + 46.5595i) q^{5} +(11.6052 - 35.7170i) q^{6} -221.003i q^{7} +(147.549 + 47.9417i) q^{8} +(-45.5512 + 33.0949i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61555	−	2.22361i	0.285591	−	0.393083i	−0.641984	−	0.766718i	\(-0.721889\pi\)
							0.927576	+	0.373635i	\(0.121889\pi\)
\(3\)	12.9949	−	4.22230i	0.833624	−	0.270861i	0.139053	−	0.990285i	\(-0.455594\pi\)
							0.694571	+	0.719424i	\(0.255594\pi\)
\(5\)	30.9389	+	46.5595i	0.553451	+	0.832882i
							\(7\)		−	221.003i		−	1.70472i	−0.522955	−	0.852360i	\(-0.675171\pi\)
0.522955	−	0.852360i	\(-0.324829\pi\)
\(11\)	196.603	+	142.840i	0.489900	+	0.355933i	0.805146	−	0.593077i	\(-0.202087\pi\)
							−0.315246	+	0.949010i	\(0.602087\pi\)
\(13\)	−354.807	−	488.350i	−0.582283	−	0.801444i	0.411660	−	0.911337i	\(-0.364949\pi\)
							−0.993943	+	0.109893i	\(0.964949\pi\)
\(17\)	−1262.91	−	410.344i	−1.05986	−	0.344370i	−0.273331	−	0.961920i	\(-0.588126\pi\)
							−0.786532	+	0.617550i	\(0.788126\pi\)
\(19\)	−651.780	+	2005.97i	−0.414207	+	1.27480i	0.498751	+	0.866745i	\(0.333792\pi\)
							−0.912958	+	0.408053i	\(0.866208\pi\)
\(23\)	1164.99	−	1603.47i	0.459199	−	0.632033i	−0.515143	−	0.857104i	\(-0.672261\pi\)
							0.974342	+	0.225071i	\(0.0722613\pi\)
\(29\)	−2042.35	−	6285.72i	−0.450958	−	1.38791i	−0.875816	−	0.482645i	\(-0.839676\pi\)
							0.424858	−	0.905260i	\(-0.360324\pi\)
\(31\)	942.570	−	2900.93i	0.176161	−	0.542167i	−0.823524	−	0.567282i	\(-0.807995\pi\)
							0.999685	+	0.0251145i	\(0.00799504\pi\)
\(37\)	4276.99	+	5886.77i	0.513611	+	0.706924i	0.984523	−	0.175256i	\(-0.0560752\pi\)
							−0.470912	+	0.882180i	\(0.656075\pi\)
\(41\)	5087.03	−	3695.94i	0.472612	−	0.343373i	−0.325846	−	0.945423i	\(-0.605649\pi\)
							0.798458	+	0.602050i	\(0.205649\pi\)
\(43\)		−	10091.2i		−	0.832284i	−0.909300	−	0.416142i	\(-0.863382\pi\)
							0.909300	−	0.416142i	\(-0.136618\pi\)
\(47\)	−8983.39	+	2918.88i	−0.593192	+	0.192740i	−0.590202	−	0.807256i	\(-0.700952\pi\)
							−0.00299047	+	0.999996i	\(0.500952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.8	✓	48
25.8	odd	20		625.6.a.g.1.30		48
25.14	even	10	inner	25.6.e.a.14.8	yes	48
25.17	odd	20		625.6.a.g.1.19		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.8	✓	48	1.1	even	1	trivial
25.6.e.a.14.8	yes	48	25.14	even	10	inner
625.6.a.g.1.19		48	25.17	odd	20
625.6.a.g.1.30		48	25.8	odd	20