Embedded newform 25.9.f.a.2.8

• Label: 25.9.f.a.2.8
• Level: $25$
• Weight: $9$
• Character: 25.2
• Analytic conductor: $10.184$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1844652515\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(19\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			2.8
Character	\(\chi\)	\(=\)	25.2
Dual form			25.9.f.a.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.69439 - 1.37706i) q^{2} +(-50.1228 - 98.3716i) q^{3} +(-169.774 - 55.1630i) q^{4} +(-621.648 - 64.6469i) q^{5} +(300.324 + 924.303i) q^{6} +(-2315.10 - 2315.10i) q^{7} +(3408.01 + 1736.47i) q^{8} +(-3308.21 + 4553.36i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} + 62 q^{3} - 10 q^{4} - 230 q^{5} - 6 q^{6} + 2342 q^{7} + 8210 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.69439	−	1.37706i	−0.543399	−	0.0860660i	−0.121301	−	0.992616i	\(-0.538706\pi\)
							−0.422099	+	0.906550i	\(0.638706\pi\)
\(3\)	−50.1228	−	98.3716i	−0.618800	−	1.21446i	−0.961445	−	0.274999i	\(-0.911323\pi\)
							0.342644	−	0.939465i	\(-0.388677\pi\)
\(5\)	−621.648	−	64.6469i	−0.994636	−	0.103435i
							\(7\)	−2315.10	−	2315.10i	−0.964222	−	0.964222i	0.0351598	−	0.999382i	\(-0.488806\pi\)
−0.999382	+	0.0351598i	\(0.988806\pi\)
\(11\)	21888.5	−	15902.9i	1.49501	−	1.08619i	0.522697	−	0.852519i	\(-0.324926\pi\)
							0.972316	−	0.233671i	\(-0.0750740\pi\)
\(13\)	−34855.7	+	5520.59i	−1.22039	+	0.193291i	−0.733206	−	0.680006i	\(-0.761977\pi\)
							−0.487187	+	0.873298i	\(0.661977\pi\)
\(17\)	−32046.6	+	62895.0i	−0.383695	+	0.753044i	−0.999389	−	0.0349413i	\(-0.988876\pi\)
							0.615694	+	0.787985i	\(0.288876\pi\)
\(19\)	36461.3	−	11847.0i	0.279781	−	0.0909063i	−0.165765	−	0.986165i	\(-0.553009\pi\)
							0.445546	+	0.895259i	\(0.353009\pi\)
\(23\)	63645.0	−	401839.i	0.227433	−	1.43595i	−0.564544	−	0.825403i	\(-0.690948\pi\)
							0.791977	−	0.610551i	\(-0.209052\pi\)
\(29\)	−692890.	−	225134.i	−0.979653	−	0.318309i	−0.224947	−	0.974371i	\(-0.572221\pi\)
							−0.754707	+	0.656062i	\(0.772221\pi\)
\(31\)	−11513.5	−	35434.9i	−0.0124669	−	0.0383693i	0.944630	−	0.328139i	\(-0.106421\pi\)
							−0.957096	+	0.289769i	\(0.906421\pi\)
\(37\)	−43674.0	−	275747.i	−0.0233032	−	0.147131i	0.973293	−	0.229565i	\(-0.0737305\pi\)
							−0.996596	+	0.0824345i	\(0.973730\pi\)
\(41\)	2.06240e6	+	1.49842e6i	0.729857	+	0.530272i	0.889518	−	0.456900i	\(-0.151040\pi\)
							−0.159661	+	0.987172i	\(0.551040\pi\)
\(43\)	2.42753e6	−	2.42753e6i	0.710054	−	0.710054i	−0.256492	−	0.966546i	\(-0.582567\pi\)
							0.966546	+	0.256492i	\(0.0825669\pi\)
\(47\)	−4.42123e6	+	2.25273e6i	−0.906048	+	0.461655i	−0.843954	−	0.536416i	\(-0.819778\pi\)
							−0.0620945	+	0.998070i	\(0.519778\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.9.f.a.2.8	✓	152
25.13	odd	20	inner	25.9.f.a.13.8	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.9.f.a.2.8	✓	152	1.1	even	1	trivial
25.9.f.a.13.8	yes	152	25.13	odd	20	inner