Embedded newform 25.6.e.a.9.7

• Label: 25.6.e.a.9.7
• 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			9.7
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.958035 - 1.31862i) q^{2} +(-8.25369 + 2.68179i) q^{3} +(9.06761 + 27.9072i) q^{4} +(-55.8091 + 3.21620i) q^{5} +(-4.37106 + 13.4527i) q^{6} +127.304i q^{7} +(95.0904 + 30.8967i) q^{8} +(-135.660 + 98.5625i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.958035	−	1.31862i	0.169358	−	0.233102i	−0.715898	−	0.698204i	\(-0.753983\pi\)
							0.885257	+	0.465103i	\(0.153983\pi\)
\(3\)	−8.25369	+	2.68179i	−0.529475	+	0.172037i	−0.561541	−	0.827449i	\(-0.689791\pi\)
							0.0320663	+	0.999486i	\(0.489791\pi\)
\(5\)	−55.8091	+	3.21620i	−0.998344	+	0.0575332i
							\(7\)			127.304i			0.981967i	0.871169	+	0.490984i	\(0.163362\pi\)
−0.871169	+	0.490984i	\(0.836638\pi\)
\(11\)	416.042	+	302.272i	1.03671	+	0.753211i	0.969640	−	0.244538i	\(-0.0786362\pi\)
							0.0670661	+	0.997749i	\(0.478636\pi\)
\(13\)	−641.996	−	883.631i	−1.05360	−	1.45015i	−0.885650	−	0.464353i	\(-0.846287\pi\)
							−0.167945	−	0.985796i	\(-0.553713\pi\)
\(17\)	1001.17	+	325.300i	0.840207	+	0.273000i	0.697338	−	0.716742i	\(-0.254368\pi\)
							0.142868	+	0.989742i	\(0.454368\pi\)
\(19\)	321.628	−	989.871i	0.204395	−	0.629063i	−0.795343	−	0.606160i	\(-0.792709\pi\)
							0.999738	−	0.0229032i	\(-0.00729097\pi\)
\(23\)	−18.0426	+	24.8335i	−0.00711181	+	0.00978856i	−0.812558	−	0.582880i	\(-0.801926\pi\)
							0.805446	+	0.592669i	\(0.201926\pi\)
\(29\)	2247.22	+	6916.22i	0.496192	+	1.52712i	0.815091	+	0.579334i	\(0.196687\pi\)
							−0.318898	+	0.947789i	\(0.603313\pi\)
\(31\)	−3012.22	+	9270.65i	−0.562965	+	1.73263i	0.110955	+	0.993825i	\(0.464609\pi\)
							−0.673920	+	0.738804i	\(0.735391\pi\)
\(37\)	2797.22	+	3850.04i	0.335909	+	0.462339i	0.943241	−	0.332109i	\(-0.107760\pi\)
							−0.607332	+	0.794448i	\(0.707760\pi\)
\(41\)	1724.06	−	1252.60i	0.160174	−	0.116373i	−0.504811	−	0.863230i	\(-0.668438\pi\)
							0.664985	+	0.746857i	\(0.268438\pi\)
\(43\)		−	11588.1i		−	0.955740i	−0.878431	−	0.477870i	\(-0.841409\pi\)
							0.878431	−	0.477870i	\(-0.158591\pi\)
\(47\)	12896.4	−	4190.31i	0.851579	−	0.276695i	0.149472	−	0.988766i	\(-0.452243\pi\)
							0.702107	+	0.712071i	\(0.252243\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.7	✓	48
25.8	odd	20		625.6.a.g.1.28		48
25.14	even	10	inner	25.6.e.a.14.7	yes	48
25.17	odd	20		625.6.a.g.1.21		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.7	✓	48	1.1	even	1	trivial
25.6.e.a.14.7	yes	48	25.14	even	10	inner
625.6.a.g.1.21		48	25.17	odd	20
625.6.a.g.1.28		48	25.8	odd	20