Embedded newform 25.6.e.a.4.1

• Label: 25.6.e.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	25.4
Dual form			25.6.e.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.71189 - 3.15559i) q^{2} +(-6.12321 + 8.42787i) q^{3} +(58.4746 + 42.4843i) q^{4} +(-38.3382 + 40.6839i) q^{5} +(86.0628 - 62.5283i) q^{6} -211.819i q^{7} +(-241.764 - 332.759i) q^{8} +(41.5558 + 127.895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.71189	−	3.15559i	−1.71684	−	0.557834i	−0.725389	−	0.688339i	\(-0.758340\pi\)
							−0.991448	+	0.130505i	\(0.958340\pi\)
\(3\)	−6.12321	+	8.42787i	−0.392804	+	0.540648i	−0.958920	−	0.283677i	\(-0.908446\pi\)
							0.566116	+	0.824326i	\(0.308446\pi\)
\(5\)	−38.3382	+	40.6839i	−0.685815	+	0.727776i
							\(7\)		−	211.819i		−	1.63388i	−0.576724	−	0.816939i	\(-0.695669\pi\)
0.576724	−	0.816939i	\(-0.304331\pi\)
\(11\)	137.931	−	424.509i	0.343702	−	1.05780i	−0.618574	−	0.785727i	\(-0.712289\pi\)
							0.962275	−	0.272078i	\(-0.0877108\pi\)
\(13\)	418.615	−	136.016i	0.687000	−	0.223220i	0.0553425	−	0.998467i	\(-0.482375\pi\)
							0.631657	+	0.775248i	\(0.282375\pi\)
\(17\)	142.489	+	196.120i	0.119581	+	0.164588i	0.864611	−	0.502442i	\(-0.167565\pi\)
							−0.745030	+	0.667031i	\(0.767565\pi\)
\(19\)	1824.52	−	1325.59i	1.15948	−	0.842413i	0.169770	−	0.985484i	\(-0.445697\pi\)
							0.989713	+	0.143070i	\(0.0456975\pi\)
\(23\)	89.2099	+	28.9861i	0.0351636	+	0.0114254i	0.326546	−	0.945181i	\(-0.394115\pi\)
							−0.291382	+	0.956607i	\(0.594115\pi\)
\(29\)	−1048.08	−	761.474i	−0.231419	−	0.168136i	0.466033	−	0.884767i	\(-0.345683\pi\)
							−0.697452	+	0.716632i	\(0.745683\pi\)
\(31\)	1265.73	−	919.607i	0.236558	−	0.171869i	−0.463191	−	0.886259i	\(-0.653295\pi\)
							0.699748	+	0.714389i	\(0.253295\pi\)
\(37\)	107.846	−	35.0413i	0.0129509	−	0.00420800i	−0.302534	−	0.953138i	\(-0.597833\pi\)
							0.315485	+	0.948930i	\(0.397833\pi\)
\(41\)	−3708.10	−	11412.4i	−0.344502	−	1.06027i	−0.961850	−	0.273578i	\(-0.911793\pi\)
							0.617348	−	0.786690i	\(-0.288207\pi\)
\(43\)			12177.7i			1.00437i	0.864759	+	0.502187i	\(0.167471\pi\)
							−0.864759	+	0.502187i	\(0.832529\pi\)
\(47\)	5599.75	−	7707.39i	0.369763	−	0.508936i	−0.583073	−	0.812420i	\(-0.698150\pi\)
							0.952837	+	0.303484i	\(0.0981499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.4.1	✓	48
25.12	odd	20		625.6.a.g.1.46		48
25.13	odd	20		625.6.a.g.1.3		48
25.19	even	10	inner	25.6.e.a.19.1	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.1	✓	48	1.1	even	1	trivial
25.6.e.a.19.1	yes	48	25.19	even	10	inner
625.6.a.g.1.3		48	25.13	odd	20
625.6.a.g.1.46		48	25.12	odd	20