Embedded newform 25.6.e.a.9.2

• Label: 25.6.e.a.9.2
• 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			9.2
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.77566 + 7.94951i) q^{2} +(-24.0141 + 7.80267i) q^{3} +(-19.9480 - 61.3935i) q^{4} +(11.5397 + 54.6977i) q^{5} +(76.6701 - 235.966i) q^{6} +104.735i q^{7} +(304.214 + 98.8452i) q^{8} +(319.207 - 231.917i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.77566	+	7.94951i	−1.02100	+	1.40529i	−0.109502	+	0.993987i	\(0.534926\pi\)
							−0.911499	+	0.411302i	\(0.865074\pi\)
\(3\)	−24.0141	+	7.80267i	−1.54051	+	0.500541i	−0.951517	−	0.307598i	\(-0.900475\pi\)
							−0.588992	+	0.808139i	\(0.700475\pi\)
\(5\)	11.5397	+	54.6977i	0.206428	+	0.978462i
							\(7\)			104.735i			0.807882i	0.914785	+	0.403941i	\(0.132360\pi\)
−0.914785	+	0.403941i	\(0.867640\pi\)
\(11\)	37.6235	+	27.3351i	0.0937514	+	0.0681144i	0.633673	−	0.773601i	\(-0.281546\pi\)
							−0.539922	+	0.841715i	\(0.681546\pi\)
\(13\)	−609.619	−	839.068i	−1.00046	−	1.37702i	−0.925042	−	0.379864i	\(-0.875971\pi\)
							−0.0754184	−	0.997152i	\(-0.524029\pi\)
\(17\)	640.879	+	208.234i	0.537841	+	0.174755i	0.565327	−	0.824867i	\(-0.308750\pi\)
							−0.0274857	+	0.999622i	\(0.508750\pi\)
\(19\)	−750.043	+	2308.39i	−0.476653	+	1.46699i	0.367063	+	0.930196i	\(0.380363\pi\)
							−0.843716	+	0.536790i	\(0.819637\pi\)
\(23\)	474.086	−	652.523i	0.186869	−	0.257203i	−0.705296	−	0.708913i	\(-0.749186\pi\)
							0.892165	+	0.451710i	\(0.149186\pi\)
\(29\)	−1486.73	−	4575.69i	−0.328275	−	1.01033i	−0.969941	−	0.243342i	\(-0.921756\pi\)
							0.641666	−	0.766984i	\(-0.278244\pi\)
\(31\)	724.274	−	2229.09i	0.135363	−	0.416603i	−0.860284	−	0.509816i	\(-0.829714\pi\)
							0.995646	+	0.0932126i	\(0.0297136\pi\)
\(37\)	−4995.92	−	6876.30i	−0.599945	−	0.825753i	0.395758	−	0.918355i	\(-0.370482\pi\)
							−0.995703	+	0.0926015i	\(0.970482\pi\)
\(41\)	−2937.36	+	2134.11i	−0.272896	+	0.198270i	−0.715813	−	0.698292i	\(-0.753944\pi\)
							0.442917	+	0.896563i	\(0.353944\pi\)
\(43\)			3648.90i			0.300947i	0.988614	+	0.150474i	\(0.0480799\pi\)
							−0.988614	+	0.150474i	\(0.951920\pi\)
\(47\)	−16361.8	+	5316.27i	−1.08040	+	0.351045i	−0.794532	−	0.607223i	\(-0.792284\pi\)
							−0.285873	+	0.958267i	\(0.592284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.2	✓	48
25.8	odd	20		625.6.a.g.1.5		48
25.14	even	10	inner	25.6.e.a.14.2	yes	48
25.17	odd	20		625.6.a.g.1.44		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.2	✓	48	1.1	even	1	trivial
25.6.e.a.14.2	yes	48	25.14	even	10	inner
625.6.a.g.1.5		48	25.8	odd	20
625.6.a.g.1.44		48	25.17	odd	20