Embedded newform 25.6.d.a.6.3

• Label: 25.6.d.a.6.3
• 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			6.3
Character	\(\chi\)	\(=\)	25.6
Dual form			25.6.d.a.21.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94032 - 5.97168i) q^{2} +(9.69103 - 7.04095i) q^{3} +(-6.00755 + 4.36474i) q^{4} +(46.5920 - 30.8899i) q^{5} +(-60.8499 - 44.2101i) q^{6} -78.1631 q^{7} +(-124.833 - 90.6962i) q^{8} +(-30.7500 + 94.6386i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94032	−	5.97168i	−0.343002	−	1.05565i	−0.962644	−	0.270769i	\(-0.912722\pi\)
							0.619642	−	0.784885i	\(-0.287278\pi\)
\(3\)	9.69103	−	7.04095i	0.621680	−	0.451677i	−0.231828	−	0.972757i	\(-0.574471\pi\)
							0.853508	+	0.521080i	\(0.174471\pi\)
\(5\)	46.5920	−	30.8899i	0.833463	−	0.552575i
							\(7\)	−78.1631			−0.602916			−0.301458	−	0.953479i	\(-0.597473\pi\)
−0.301458	+	0.953479i	\(0.597473\pi\)
\(11\)	35.4725	+	109.173i	0.0883915	+	0.272041i	0.985475	−	0.169820i	\(-0.0543185\pi\)
							−0.897084	+	0.441861i	\(0.854319\pi\)
\(13\)	239.107	−	735.894i	0.392404	−	1.20769i	−0.538561	−	0.842586i	\(-0.681032\pi\)
							0.930965	−	0.365108i	\(-0.118968\pi\)
\(17\)	1740.48	+	1264.53i	1.46065	+	1.06122i	0.983193	+	0.182570i	\(0.0584418\pi\)
							0.477458	+	0.878655i	\(0.341558\pi\)
\(19\)	391.300	+	284.296i	0.248671	+	0.180670i	0.705138	−	0.709070i	\(-0.250885\pi\)
							−0.456466	+	0.889741i	\(0.650885\pi\)
\(23\)	−252.970	−	778.562i	−0.0997125	−	0.306883i	0.888741	−	0.458410i	\(-0.151581\pi\)
							−0.988453	+	0.151527i	\(0.951581\pi\)
\(29\)	508.051	−	369.120i	0.112179	−	0.0815029i	−0.530281	−	0.847822i	\(-0.677914\pi\)
							0.642460	+	0.766319i	\(0.277914\pi\)
\(31\)	568.911	+	413.338i	0.106326	+	0.0772505i	0.639678	−	0.768643i	\(-0.279068\pi\)
							−0.533352	+	0.845894i	\(0.679068\pi\)
\(37\)	1061.79	−	3267.85i	0.127507	−	0.392425i	−0.866843	−	0.498582i	\(-0.833854\pi\)
							0.994349	+	0.106156i	\(0.0338544\pi\)
\(41\)	−4380.37	+	13481.4i	−0.406959	+	1.25249i	0.512289	+	0.858813i	\(0.328798\pi\)
							−0.919248	+	0.393678i	\(0.871202\pi\)
\(43\)	21923.1			1.80813			0.904067	−	0.427390i	\(-0.140567\pi\)
							0.904067	+	0.427390i	\(0.140567\pi\)
\(47\)	−18769.2	+	13636.6i	−1.23937	+	0.900453i	−0.997556	−	0.0698703i	\(-0.977741\pi\)
							−0.241811	+	0.970323i	\(0.577741\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.6.3	✓	44
25.11	even	5		625.6.a.d.1.6		22
25.14	even	10		625.6.a.c.1.17		22
25.21	even	5	inner	25.6.d.a.21.3	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.3	✓	44	1.1	even	1	trivial
25.6.d.a.21.3	yes	44	25.21	even	5	inner
625.6.a.c.1.17		22	25.14	even	10
625.6.a.d.1.6		22	25.11	even	5