Embedded newform 25.6.e.a.9.4

• Label: 25.6.e.a.9.4
• 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			9.4
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.32714 + 4.57941i) q^{2} +(3.38178 - 1.09881i) q^{3} +(-0.0126365 - 0.0388913i) q^{4} +(-43.0053 + 35.7147i) q^{5} +(-6.21976 + 19.1425i) q^{6} -56.5747i q^{7} +(-172.049 - 55.9023i) q^{8} +(-186.362 + 135.400i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.32714	+	4.57941i	−0.588161	+	0.809534i	−0.994560	−	0.104162i	\(-0.966784\pi\)
							0.406400	+	0.913695i	\(0.366784\pi\)
\(3\)	3.38178	−	1.09881i	0.216941	−	0.0704885i	−0.198530	−	0.980095i	\(-0.563617\pi\)
							0.415471	+	0.909606i	\(0.363617\pi\)
\(5\)	−43.0053	+	35.7147i	−0.769303	+	0.638885i
							\(7\)		−	56.5747i		−	0.436392i	−0.975905	−	0.218196i	\(-0.929983\pi\)
0.975905	−	0.218196i	\(-0.0700172\pi\)
\(11\)	−447.025	−	324.782i	−1.11391	−	0.809302i	−0.130634	−	0.991431i	\(-0.541701\pi\)
							−0.983275	+	0.182128i	\(0.941701\pi\)
\(13\)	616.983	+	849.204i	1.01255	+	1.39365i	0.917300	+	0.398196i	\(0.130364\pi\)
							0.0952452	+	0.995454i	\(0.469636\pi\)
\(17\)	1276.08	+	414.623i	1.07091	+	0.347961i	0.790844	−	0.612018i	\(-0.209642\pi\)
							0.280071	+	0.959979i	\(0.409642\pi\)
\(19\)	−712.985	+	2194.34i	−0.453103	+	1.39451i	0.420246	+	0.907410i	\(0.361944\pi\)
							−0.873348	+	0.487096i	\(0.838056\pi\)
\(23\)	194.036	−	267.068i	0.0764826	−	0.105269i	−0.769062	−	0.639175i	\(-0.779276\pi\)
							0.845544	+	0.533905i	\(0.179276\pi\)
\(29\)	541.998	+	1668.10i	0.119675	+	0.368321i	0.992893	−	0.119008i	\(-0.0379713\pi\)
							−0.873219	+	0.487329i	\(0.837971\pi\)
\(31\)	259.056	−	797.292i	0.0484160	−	0.149009i	−0.923926	−	0.382572i	\(-0.875038\pi\)
							0.972342	+	0.233563i	\(0.0750384\pi\)
\(37\)	6401.35	+	8810.70i	0.768718	+	1.05805i	0.996438	+	0.0843230i	\(0.0268727\pi\)
							−0.227721	+	0.973727i	\(0.573127\pi\)
\(41\)	−13126.2	+	9536.73i	−1.21949	+	0.886012i	−0.996058	−	0.0887068i	\(-0.971727\pi\)
							−0.223434	+	0.974719i	\(0.571727\pi\)
\(43\)		−	12629.3i		−	1.04162i	−0.853673	−	0.520810i	\(-0.825630\pi\)
							0.853673	−	0.520810i	\(-0.174370\pi\)
\(47\)	−2382.59	+	774.151i	−0.157328	+	0.0511188i	−0.386622	−	0.922238i	\(-0.626358\pi\)
							0.229294	+	0.973357i	\(0.426358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.4	✓	48
25.8	odd	20		625.6.a.g.1.15		48
25.14	even	10	inner	25.6.e.a.14.4	yes	48
25.17	odd	20		625.6.a.g.1.34		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.4	✓	48	1.1	even	1	trivial
25.6.e.a.14.4	yes	48	25.14	even	10	inner
625.6.a.g.1.15		48	25.8	odd	20
625.6.a.g.1.34		48	25.17	odd	20