Embedded newform 25.6.d.a.6.9

• Label: 25.6.d.a.6.9
• 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			6.9
Character	\(\chi\)	\(=\)	25.6
Dual form			25.6.d.a.21.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.36682 + 7.28432i) q^{2} +(-19.7149 + 14.3237i) q^{3} +(-21.5710 + 15.6723i) q^{4} +(54.0142 - 14.4039i) q^{5} +(-151.000 - 109.708i) q^{6} -209.197 q^{7} +(33.0689 + 24.0259i) q^{8} +(108.418 - 333.675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.36682	+	7.28432i	0.418399	+	1.28770i	0.909175	+	0.416413i	\(0.136713\pi\)
							−0.490777	+	0.871285i	\(0.663287\pi\)
\(3\)	−19.7149	+	14.3237i	−1.26471	+	0.918867i	−0.998979	−	0.0451797i	\(-0.985614\pi\)
							−0.265733	+	0.964047i	\(0.585614\pi\)
\(5\)	54.0142	−	14.4039i	0.966235	−	0.257664i
							\(7\)	−209.197			−1.61365			−0.806826	−	0.590789i	\(-0.798817\pi\)
−0.806826	+	0.590789i	\(0.798817\pi\)
\(11\)	66.6115	+	205.009i	0.165985	+	0.510848i	0.999107	−	0.0422400i	\(-0.0134494\pi\)
							−0.833123	+	0.553088i	\(0.813449\pi\)
\(13\)	−208.768	+	642.522i	−0.342615	+	1.05446i	0.620234	+	0.784417i	\(0.287038\pi\)
							−0.962848	+	0.270042i	\(0.912962\pi\)
\(17\)	1006.40	+	731.195i	0.844598	+	0.613636i	0.923651	−	0.383234i	\(-0.125190\pi\)
							−0.0790535	+	0.996870i	\(0.525190\pi\)
\(19\)	−923.285	−	670.806i	−0.586748	−	0.426297i	0.254402	−	0.967098i	\(-0.418121\pi\)
							−0.841151	+	0.540801i	\(0.818121\pi\)
\(23\)	425.364	+	1309.14i	0.167664	+	0.516018i	0.999223	−	0.0394194i	\(-0.0125508\pi\)
							−0.831558	+	0.555437i	\(0.812551\pi\)
\(29\)	4153.04	−	3017.36i	0.917004	−	0.666242i	−0.0257726	−	0.999668i	\(-0.508205\pi\)
							0.942776	+	0.333426i	\(0.108205\pi\)
\(31\)	3660.52	+	2659.53i	0.684131	+	0.497050i	0.874726	−	0.484619i	\(-0.161042\pi\)
							−0.190595	+	0.981669i	\(0.561042\pi\)
\(37\)	1020.96	−	3142.20i	0.122604	−	0.377337i	−0.870853	−	0.491544i	\(-0.836433\pi\)
							0.993457	+	0.114207i	\(0.0364326\pi\)
\(41\)	−4525.52	+	13928.1i	−0.420444	+	1.29399i	0.486845	+	0.873488i	\(0.338148\pi\)
							−0.907289	+	0.420507i	\(0.861852\pi\)
\(43\)	8486.60			0.699943			0.349971	−	0.936760i	\(-0.386191\pi\)
							0.349971	+	0.936760i	\(0.386191\pi\)
\(47\)	12327.8	−	8956.68i	0.814032	−	0.591429i	−0.100965	−	0.994890i	\(-0.532193\pi\)
							0.914997	+	0.403461i	\(0.132193\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.6.9	✓	44
25.11	even	5		625.6.a.d.1.19		22
25.14	even	10		625.6.a.c.1.4		22
25.21	even	5	inner	25.6.d.a.21.9	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.9	✓	44	1.1	even	1	trivial
25.6.d.a.21.9	yes	44	25.21	even	5	inner
625.6.a.c.1.4		22	25.14	even	10
625.6.a.d.1.19		22	25.11	even	5