Embedded newform 25.6.d.a.6.2

• Label: 25.6.d.a.6.2
• Level: $25$
• Weight: $6$
• Character: 25.6
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			6.2
Character	\(\chi\)	\(=\)	25.6
Dual form			25.6.d.a.21.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46926 - 7.59959i) q^{2} +(-14.2493 + 10.3527i) q^{3} +(-25.7680 + 18.7216i) q^{4} +(27.5974 + 48.6147i) q^{5} +(113.862 + 82.7252i) q^{6} +40.0850 q^{7} +(-0.963073 - 0.699714i) q^{8} +(20.7724 - 63.9307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46926	−	7.59959i	−0.436507	−	1.34343i	−0.891534	−	0.452953i	\(-0.850370\pi\)
							0.455027	−	0.890478i	\(-0.349630\pi\)
\(3\)	−14.2493	+	10.3527i	−0.914092	+	0.664127i	−0.942046	−	0.335482i	\(-0.891101\pi\)
							0.0279544	+	0.999609i	\(0.491101\pi\)
\(5\)	27.5974	+	48.6147i	0.493677	+	0.869646i
							\(7\)	40.0850			0.309198			0.154599	−	0.987977i	\(-0.450591\pi\)
0.154599	+	0.987977i	\(0.450591\pi\)
\(11\)	184.215	+	566.955i	0.459032	+	1.41275i	0.866336	+	0.499462i	\(0.166469\pi\)
							−0.407304	+	0.913293i	\(0.633531\pi\)
\(13\)	−219.959	+	676.965i	−0.360980	+	1.11098i	0.591480	+	0.806320i	\(0.298544\pi\)
							−0.952460	+	0.304664i	\(0.901456\pi\)
\(17\)	−1067.70	−	775.728i	−0.896038	−	0.651009i	0.0414076	−	0.999142i	\(-0.486816\pi\)
							−0.937445	+	0.348133i	\(0.886816\pi\)
\(19\)	600.819	+	436.521i	0.381821	+	0.277409i	0.762096	−	0.647464i	\(-0.224171\pi\)
							−0.380275	+	0.924874i	\(0.624171\pi\)
\(23\)	−662.953	−	2040.36i	−0.261314	−	0.804243i	−0.992520	−	0.122085i	\(-0.961042\pi\)
							0.731205	−	0.682157i	\(-0.238958\pi\)
\(29\)	530.629	−	385.525i	0.117165	−	0.0851250i	−0.527660	−	0.849456i	\(-0.676930\pi\)
							0.644824	+	0.764331i	\(0.276930\pi\)
\(31\)	2836.74	+	2061.01i	0.530170	+	0.385191i	0.820421	−	0.571760i	\(-0.193739\pi\)
							−0.290252	+	0.956950i	\(0.593739\pi\)
\(37\)	−4254.72	+	13094.7i	−0.510936	+	1.57250i	0.279620	+	0.960111i	\(0.409791\pi\)
							−0.790556	+	0.612389i	\(0.790209\pi\)
\(41\)	3587.99	−	11042.7i	0.333343	−	1.02592i	−0.634189	−	0.773178i	\(-0.718666\pi\)
							0.967532	−	0.252747i	\(-0.0813340\pi\)
\(43\)	17718.5			1.46136			0.730678	−	0.682722i	\(-0.239204\pi\)
							0.730678	+	0.682722i	\(0.239204\pi\)
\(47\)	2629.24	−	1910.26i	0.173615	−	0.126138i	−0.497585	−	0.867415i	\(-0.665780\pi\)
							0.671199	+	0.741277i	\(0.265780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.6.2	✓	44
25.11	even	5		625.6.a.d.1.4		22
25.14	even	10		625.6.a.c.1.19		22
25.21	even	5	inner	25.6.d.a.21.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.2	✓	44	1.1	even	1	trivial
25.6.d.a.21.2	yes	44	25.21	even	5	inner
625.6.a.c.1.19		22	25.14	even	10
625.6.a.d.1.4		22	25.11	even	5