Embedded newform 25.6.e.a.9.12

• Label: 25.6.e.a.9.12
• Level: $25$
• Weight: $6$
• Character: 25.9
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $48$
• 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			9.12
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.80261 - 7.98661i) q^{2} +(18.8061 - 6.11048i) q^{3} +(-20.2271 - 62.2525i) q^{4} +(-7.83500 + 55.3499i) q^{5} +(60.3226 - 185.654i) q^{6} +206.656i q^{7} +(-314.114 - 102.062i) q^{8} +(119.741 - 86.9970i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.80261	−	7.98661i	1.02577	−	1.41185i	0.117686	−	0.993051i	\(-0.462452\pi\)
							0.908081	−	0.418795i	\(-0.137548\pi\)
\(3\)	18.8061	−	6.11048i	1.20641	−	0.391987i	0.364297	−	0.931283i	\(-0.381309\pi\)
							0.842117	+	0.539296i	\(0.181309\pi\)
\(5\)	−7.83500	+	55.3499i	−0.140157	+	0.990129i
							\(7\)			206.656i			1.59405i	0.603946	+	0.797025i	\(0.293594\pi\)
−0.603946	+	0.797025i	\(0.706406\pi\)
\(11\)	−356.419	−	258.954i	−0.888135	−	0.645268i	0.0472558	−	0.998883i	\(-0.484952\pi\)
							−0.935391	+	0.353615i	\(0.884952\pi\)
\(13\)	−361.134	−	497.059i	−0.592667	−	0.815736i	0.402346	−	0.915488i	\(-0.368195\pi\)
							−0.995012	+	0.0997522i	\(0.968195\pi\)
\(17\)	979.335	+	318.205i	0.821881	+	0.267045i	0.689622	−	0.724170i	\(-0.257777\pi\)
							0.132259	+	0.991215i	\(0.457777\pi\)
\(19\)	230.329	−	708.879i	0.146374	−	0.450493i	−0.850811	−	0.525472i	\(-0.823889\pi\)
							0.997185	+	0.0749785i	\(0.0238888\pi\)
\(23\)	665.261	−	915.653i	0.262224	−	0.360920i	−0.657521	−	0.753436i	\(-0.728395\pi\)
							0.919745	+	0.392515i	\(0.128395\pi\)
\(29\)	821.483	+	2528.26i	0.181386	+	0.558248i	0.999867	−	0.0162845i	\(-0.00518374\pi\)
							−0.818481	+	0.574533i	\(0.805184\pi\)
\(31\)	2436.57	−	7499.00i	0.455381	−	1.40152i	−0.415306	−	0.909682i	\(-0.636325\pi\)
							0.870687	−	0.491837i	\(-0.163675\pi\)
\(37\)	4587.64	+	6314.35i	0.550916	+	0.758271i	0.990136	−	0.140109i	\(-0.0447452\pi\)
							−0.439220	+	0.898379i	\(0.644745\pi\)
\(41\)	−3874.64	+	2815.09i	−0.359974	+	0.261537i	−0.753041	−	0.657973i	\(-0.771414\pi\)
							0.393067	+	0.919510i	\(0.371414\pi\)
\(43\)			4730.43i			0.390148i	0.980789	+	0.195074i	\(0.0624947\pi\)
							−0.980789	+	0.195074i	\(0.937505\pi\)
\(47\)	1004.94	−	326.523i	0.0663581	−	0.0215610i	−0.275650	−	0.961258i	\(-0.588893\pi\)
							0.342008	+	0.939697i	\(0.388893\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.12	✓	48
25.8	odd	20		625.6.a.g.1.45		48
25.14	even	10	inner	25.6.e.a.14.12	yes	48
25.17	odd	20		625.6.a.g.1.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.12	✓	48	1.1	even	1	trivial
25.6.e.a.14.12	yes	48	25.14	even	10	inner
625.6.a.g.1.4		48	25.17	odd	20
625.6.a.g.1.45		48	25.8	odd	20