Embedded newform 25.9.f.a.2.3

• Label: 25.9.f.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	25.2
Dual form			25.9.f.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-23.9462 - 3.79271i) q^{2} +(-30.2881 - 59.4438i) q^{3} +(315.567 + 102.534i) q^{4} +(109.848 - 615.271i) q^{5} +(499.834 + 1538.33i) q^{6} +(2716.43 + 2716.43i) q^{7} +(-1637.61 - 834.402i) q^{8} +(1240.27 - 1707.08i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} + 62 q^{3} - 10 q^{4} - 230 q^{5} - 6 q^{6} + 2342 q^{7} + 8210 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−23.9462	−	3.79271i	−1.49664	−	0.237045i	−0.646221	−	0.763150i	\(-0.723652\pi\)
							−0.850419	+	0.526106i	\(0.823652\pi\)
\(3\)	−30.2881	−	59.4438i	−0.373927	−	0.733874i	0.624978	−	0.780642i	\(-0.285108\pi\)
							−0.998906	+	0.0467682i	\(0.985108\pi\)
\(5\)	109.848	−	615.271i	0.175757	−	0.984433i
							\(7\)	2716.43	+	2716.43i	1.13138	+	1.13138i	0.989949	+	0.141427i	\(0.0451692\pi\)
0.141427	+	0.989949i	\(0.454831\pi\)
\(11\)	16287.3	−	11833.4i	1.11245	−	0.808239i	0.129399	−	0.991593i	\(-0.458695\pi\)
							0.983047	+	0.183353i	\(0.0586952\pi\)
\(13\)	−25515.1	+	4041.20i	−0.893355	+	0.141494i	−0.586201	−	0.810166i	\(-0.699377\pi\)
							−0.307155	+	0.951660i	\(0.599377\pi\)
\(17\)	33123.8	−	65009.1i	0.396592	−	0.778356i	−0.603223	−	0.797573i	\(-0.706117\pi\)
							0.999815	+	0.0192164i	\(0.00611713\pi\)
\(19\)	136726.	−	44425.0i	1.04915	−	0.340889i	0.266814	−	0.963748i	\(-0.414029\pi\)
							0.782334	+	0.622859i	\(0.214029\pi\)
\(23\)	−48037.5	+	303297.i	−0.171660	+	1.08382i	0.739920	+	0.672695i	\(0.234863\pi\)
							−0.911580	+	0.411123i	\(0.865137\pi\)
\(29\)	574888.	+	186793.i	0.812815	+	0.264099i	0.685789	−	0.727800i	\(-0.259457\pi\)
							0.127025	+	0.991899i	\(0.459457\pi\)
\(31\)	−46990.1	−	144621.i	−0.0508815	−	0.156597i	0.922387	−	0.386267i	\(-0.126235\pi\)
							−0.973269	+	0.229669i	\(0.926235\pi\)
\(37\)	−309111.	−	1.95165e6i	−0.164933	−	1.04135i	−0.921768	−	0.387741i	\(-0.873255\pi\)
							0.756835	−	0.653606i	\(-0.226745\pi\)
\(41\)	−4.30676e6	−	3.12904e6i	−1.52411	−	1.10733i	−0.959405	−	0.282032i	\(-0.908991\pi\)
							−0.564701	−	0.825296i	\(-0.691009\pi\)
\(43\)	1.27511e6	−	1.27511e6i	0.372971	−	0.372971i	−0.495587	−	0.868558i	\(-0.665047\pi\)
							0.868558	+	0.495587i	\(0.165047\pi\)
\(47\)	2.43709e6	−	1.24176e6i	0.499437	−	0.254476i	−0.186077	−	0.982535i	\(-0.559577\pi\)
							0.685514	+	0.728059i	\(0.259577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.9.f.a.2.3	✓	152
25.13	odd	20	inner	25.9.f.a.13.3	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.9.f.a.2.3	✓	152	1.1	even	1	trivial
25.9.f.a.13.3	yes	152	25.13	odd	20	inner