Embedded newform 25.6.e.a.4.12

• Label: 25.6.e.a.4.12
• Level: $25$
• Weight: $6$
• Character: 25.4
• 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			4.12
Character	\(\chi\)	\(=\)	25.4
Dual form			25.6.e.a.19.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.3591 + 3.36586i) q^{2} +(-9.28713 + 12.7826i) q^{3} +(70.0926 + 50.9253i) q^{4} +(-25.7064 - 49.6405i) q^{5} +(-139.231 + 101.157i) q^{6} -71.6578i q^{7} +(349.815 + 481.478i) q^{8} +(-2.05400 - 6.32156i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.3591	+	3.36586i	1.83124	+	0.595006i	0.999187	+	0.0403119i	\(0.0128352\pi\)
							0.832054	+	0.554694i	\(0.187165\pi\)
\(3\)	−9.28713	+	12.7826i	−0.595770	+	0.820007i	−0.995313	−	0.0967075i	\(-0.969169\pi\)
							0.399543	+	0.916715i	\(0.369169\pi\)
\(5\)	−25.7064	−	49.6405i	−0.459850	−	0.887996i
							\(7\)		−	71.6578i		−	0.552737i	−0.961052	−	0.276368i	\(-0.910869\pi\)
0.961052	−	0.276368i	\(-0.0891310\pi\)
\(11\)	157.909	−	485.994i	0.393483	−	1.21102i	−0.536654	−	0.843802i	\(-0.680312\pi\)
							0.930137	−	0.367213i	\(-0.119688\pi\)
\(13\)	218.240	−	70.9106i	0.358160	−	0.116373i	−0.124409	−	0.992231i	\(-0.539704\pi\)
							0.482569	+	0.875858i	\(0.339704\pi\)
\(17\)	−574.007	−	790.053i	−0.481720	−	0.663031i	0.497114	−	0.867685i	\(-0.334393\pi\)
							−0.978834	+	0.204654i	\(0.934393\pi\)
\(19\)	−1471.07	+	1068.79i	−0.934865	+	0.679219i	−0.947179	−	0.320705i	\(-0.896080\pi\)
							0.0123138	+	0.999924i	\(0.496080\pi\)
\(23\)	−2306.06	−	749.283i	−0.908972	−	0.295343i	−0.183037	−	0.983106i	\(-0.558593\pi\)
							−0.725935	+	0.687763i	\(0.758593\pi\)
\(29\)	4647.35	+	3376.50i	1.02615	+	0.745541i	0.967534	−	0.252739i	\(-0.0813316\pi\)
							0.0586150	+	0.998281i	\(0.481332\pi\)
\(31\)	1965.19	−	1427.79i	0.367282	−	0.266846i	−0.388801	−	0.921322i	\(-0.627111\pi\)
							0.756083	+	0.654476i	\(0.227111\pi\)
\(37\)	−1975.95	+	642.024i	−0.237285	+	0.0770987i	−0.425246	−	0.905078i	\(-0.639812\pi\)
							0.187960	+	0.982177i	\(0.439812\pi\)
\(41\)	1350.31	+	4155.83i	0.125451	+	0.386099i	0.993983	−	0.109534i	\(-0.0349360\pi\)
							−0.868532	+	0.495633i	\(0.834936\pi\)
\(43\)			8152.90i			0.672420i	0.941787	+	0.336210i	\(0.109145\pi\)
							−0.941787	+	0.336210i	\(0.890855\pi\)
\(47\)	−8130.15	+	11190.2i	−0.536851	+	0.738912i	−0.988155	−	0.153458i	\(-0.950959\pi\)
							0.451304	+	0.892370i	\(0.350959\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.4.12	✓	48
25.12	odd	20		625.6.a.g.1.2		48
25.13	odd	20		625.6.a.g.1.47		48
25.19	even	10	inner	25.6.e.a.19.12	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.12	✓	48	1.1	even	1	trivial
25.6.e.a.19.12	yes	48	25.19	even	10	inner
625.6.a.g.1.2		48	25.12	odd	20
625.6.a.g.1.47		48	25.13	odd	20