Embedded newform 25.8.e.a.4.8

• Label: 25.8.e.a.4.8
• Level: $25$
• Weight: $8$
• Character: 25.4
• Analytic conductor: $7.810$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.80962563710\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.8
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.30461 - 0.748813i) q^{2} +(-13.2509 + 18.2384i) q^{3} +(-98.8037 - 71.7851i) q^{4} +(254.041 - 116.569i) q^{5} +(44.1954 - 32.1098i) q^{6} +591.855i q^{7} +(356.264 + 490.355i) q^{8} +(518.770 + 1596.61i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.30461	−	0.748813i	−0.203701	−	0.0661863i	0.205390	−	0.978680i	\(-0.434154\pi\)
							−0.409090	+	0.912494i	\(0.634154\pi\)
\(3\)	−13.2509	+	18.2384i	−0.283350	+	0.389997i	−0.926840	−	0.375457i	\(-0.877486\pi\)
							0.643490	+	0.765454i	\(0.277486\pi\)
\(5\)	254.041	−	116.569i	0.908884	−	0.417049i
							\(7\)			591.855i			0.652187i	0.945337	+	0.326094i	\(0.105733\pi\)
−0.945337	+	0.326094i	\(0.894267\pi\)
\(11\)	−1704.94	+	5247.27i	−0.386221	+	1.18866i	0.549370	+	0.835579i	\(0.314868\pi\)
							−0.935591	+	0.353086i	\(0.885132\pi\)
\(13\)	6040.21	−	1962.58i	0.762518	−	0.247757i	0.0981590	−	0.995171i	\(-0.468705\pi\)
							0.664359	+	0.747414i	\(0.268705\pi\)
\(17\)	15125.2	+	20818.0i	0.746671	+	1.02770i	0.998207	+	0.0598562i	\(0.0190642\pi\)
							−0.251536	+	0.967848i	\(0.580936\pi\)
\(19\)	12451.2	−	9046.32i	0.416460	−	0.302576i	−0.359752	−	0.933048i	\(-0.617139\pi\)
							0.776212	+	0.630472i	\(0.217139\pi\)
\(23\)	−53726.1	−	17456.7i	−0.920741	−	0.299167i	−0.189970	−	0.981790i	\(-0.560839\pi\)
							−0.730771	+	0.682623i	\(0.760839\pi\)
\(29\)	82871.4	+	60209.6i	0.630974	+	0.458430i	0.856738	−	0.515753i	\(-0.172488\pi\)
							−0.225763	+	0.974182i	\(0.572488\pi\)
\(31\)	−36685.0	+	26653.2i	−0.221168	+	0.160688i	−0.692852	−	0.721079i	\(-0.743646\pi\)
							0.471684	+	0.881768i	\(0.343646\pi\)
\(37\)	−312498.	+	101537.i	−1.01424	+	0.329547i	−0.768542	−	0.639799i	\(-0.779017\pi\)
							−0.245699	+	0.969346i	\(0.579017\pi\)
\(41\)	181163.	+	557563.i	0.410513	+	1.26343i	0.916203	+	0.400714i	\(0.131238\pi\)
							−0.505691	+	0.862715i	\(0.668762\pi\)
\(43\)			136911.i			0.262602i	0.991343	+	0.131301i	\(0.0419154\pi\)
							−0.991343	+	0.131301i	\(0.958085\pi\)
\(47\)	−316436.	+	435537.i	−0.444573	+	0.611903i	−0.971221	−	0.238181i	\(-0.923449\pi\)
							0.526647	+	0.850084i	\(0.323449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.8	✓	64
25.12	odd	20		625.8.a.g.1.36		64
25.13	odd	20		625.8.a.g.1.29		64
25.19	even	10	inner	25.8.e.a.19.8	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.8	✓	64	1.1	even	1	trivial
25.8.e.a.19.8	yes	64	25.19	even	10	inner
625.8.a.g.1.29		64	25.13	odd	20
625.8.a.g.1.36		64	25.12	odd	20