Embedded newform 25.9.f.a.2.2

• Label: 25.9.f.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	25.2
Dual form			25.9.f.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-27.0051 - 4.27718i) q^{2} +(48.0141 + 94.2330i) q^{3} +(467.510 + 151.903i) q^{4} +(614.484 + 114.167i) q^{5} +(-893.573 - 2750.14i) q^{6} +(-244.540 - 244.540i) q^{7} +(-5738.84 - 2924.09i) q^{8} +(-2718.05 + 3741.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} + 62 q^{3} - 10 q^{4} - 230 q^{5} - 6 q^{6} + 2342 q^{7} + 8210 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−27.0051	−	4.27718i	−1.68782	−	0.267324i	−0.762630	−	0.646835i	\(-0.776092\pi\)
							−0.925187	+	0.379511i	\(0.876092\pi\)
\(3\)	48.0141	+	94.2330i	0.592767	+	1.16337i	0.971317	+	0.237790i	\(0.0764229\pi\)
							−0.378550	+	0.925581i	\(0.623577\pi\)
\(5\)	614.484	+	114.167i	0.983175	+	0.182667i
							\(7\)	−244.540	−	244.540i	−0.101849	−	0.101849i	0.654346	−	0.756195i	\(-0.272944\pi\)
−0.756195	+	0.654346i	\(0.772944\pi\)
\(11\)	15282.4	−	11103.3i	1.04381	−	0.758371i	0.0727824	−	0.997348i	\(-0.476812\pi\)
							0.971025	+	0.238977i	\(0.0768121\pi\)
\(13\)	39006.2	−	6177.97i	1.36571	−	0.216308i	0.569833	−	0.821761i	\(-0.307008\pi\)
							0.795881	+	0.605453i	\(0.207008\pi\)
\(17\)	−61847.9	+	121383.i	−0.740507	+	1.45333i	0.145355	+	0.989380i	\(0.453567\pi\)
							−0.885863	+	0.463948i	\(0.846433\pi\)
\(19\)	83082.9	−	26995.3i	0.637525	−	0.207145i	0.0276196	−	0.999619i	\(-0.491207\pi\)
							0.609906	+	0.792474i	\(0.291207\pi\)
\(23\)	−8477.63	+	53525.7i	−0.0302945	+	0.191272i	−0.998195	−	0.0600635i	\(-0.980870\pi\)
							0.967900	+	0.251335i	\(0.0808697\pi\)
\(29\)	−1.13321e6	−	368202.i	−1.60221	−	0.520588i	−0.634554	−	0.772879i	\(-0.718816\pi\)
							−0.967651	+	0.252291i	\(0.918816\pi\)
\(31\)	277996.	+	855583.i	0.301017	+	0.926435i	0.981134	+	0.193331i	\(0.0619292\pi\)
							−0.680116	+	0.733104i	\(0.738071\pi\)
\(37\)	301883.	+	1.90602e6i	0.161077	+	1.01700i	0.927272	+	0.374389i	\(0.122148\pi\)
							−0.766195	+	0.642608i	\(0.777852\pi\)
\(41\)	−861553.	−	625955.i	−0.304892	−	0.221517i	0.424809	−	0.905283i	\(-0.360341\pi\)
							−0.729702	+	0.683766i	\(0.760341\pi\)
\(43\)	531726.	−	531726.i	0.155530	−	0.155530i	−0.625053	−	0.780583i	\(-0.714923\pi\)
							0.780583	+	0.625053i	\(0.214923\pi\)
\(47\)	−3.20845e6	+	1.63479e6i	−0.657513	+	0.335019i	−0.750714	−	0.660628i	\(-0.770290\pi\)
							0.0932009	+	0.995647i	\(0.470290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.9.f.a.2.2	✓	152
25.13	odd	20	inner	25.9.f.a.13.2	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.9.f.a.2.2	✓	152	1.1	even	1	trivial
25.9.f.a.13.2	yes	152	25.13	odd	20	inner