Embedded newform 25.7.f.a.2.13

• Label: 25.7.f.a.2.13
• Level: $25$
• Weight: $7$
• Character: 25.2
• Analytic conductor: $5.751$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.75135209050\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	25.2
Dual form			25.7.f.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.4157 + 1.96646i) q^{2} +(-5.46221 - 10.7202i) q^{3} +(89.4159 + 29.0530i) q^{4} +(84.3839 - 92.2191i) q^{5} +(-46.7365 - 143.840i) q^{6} +(324.444 + 324.444i) q^{7} +(336.207 + 171.306i) q^{8} +(343.409 - 472.662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.4157	+	1.96646i	1.55197	+	0.245807i	0.872764	−	0.488143i	\(-0.162326\pi\)
							0.679203	+	0.733950i	\(0.262326\pi\)
\(3\)	−5.46221	−	10.7202i	−0.202304	−	0.397044i	0.767457	−	0.641100i	\(-0.221522\pi\)
							−0.969761	+	0.244057i	\(0.921522\pi\)
\(5\)	84.3839	−	92.2191i	0.675071	−	0.737753i
							\(7\)	324.444	+	324.444i	0.945901	+	0.945901i	0.998610	−	0.0527093i	\(-0.0167857\pi\)
−0.0527093	+	0.998610i	\(0.516786\pi\)
\(11\)	−1691.96	+	1229.28i	−1.27119	+	0.923577i	−0.999249	−	0.0387357i	\(-0.987667\pi\)
							−0.271945	+	0.962313i	\(0.587667\pi\)
\(13\)	−2372.89	+	375.829i	−1.08006	+	0.171065i	−0.671011	−	0.741448i	\(-0.734139\pi\)
							−0.409049	+	0.912512i	\(0.634139\pi\)
\(17\)	795.899	−	1562.04i	0.161999	−	0.317940i	−0.795711	−	0.605676i	\(-0.792903\pi\)
							0.957710	+	0.287736i	\(0.0929026\pi\)
\(19\)	1685.38	−	547.613i	0.245718	−	0.0798386i	−0.183569	−	0.983007i	\(-0.558765\pi\)
							0.429287	+	0.903168i	\(0.358765\pi\)
\(23\)	−2942.06	+	18575.4i	−0.241807	+	1.52671i	0.505850	+	0.862621i	\(0.331179\pi\)
							−0.747657	+	0.664085i	\(0.768821\pi\)
\(29\)	833.143	+	270.705i	0.0341606	+	0.0110995i	0.326047	−	0.945353i	\(-0.394283\pi\)
							−0.291887	+	0.956453i	\(0.594283\pi\)
\(31\)	−520.631	−	1602.34i	−0.0174761	−	0.0537859i	0.941938	−	0.335786i	\(-0.109002\pi\)
							−0.959414	+	0.282000i	\(0.909002\pi\)
\(37\)	5189.65	+	32766.2i	0.102455	+	0.646875i	0.984456	+	0.175629i	\(0.0561958\pi\)
							−0.882002	+	0.471247i	\(0.843804\pi\)
\(41\)	44693.0	+	32471.4i	0.648467	+	0.471139i	0.862749	−	0.505633i	\(-0.168741\pi\)
							−0.214282	+	0.976772i	\(0.568741\pi\)
\(43\)	42849.1	−	42849.1i	0.538935	−	0.538935i	−0.384281	−	0.923216i	\(-0.625551\pi\)
							0.923216	+	0.384281i	\(0.125551\pi\)
\(47\)	171763.	−	87517.9i	1.65439	−	0.842953i	0.658478	−	0.752600i	\(-0.271201\pi\)
							0.995910	−	0.0903525i	\(-0.0287994\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.2.13	✓	112
25.13	odd	20	inner	25.7.f.a.13.13	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.2.13	✓	112	1.1	even	1	trivial
25.7.f.a.13.13	yes	112	25.13	odd	20	inner