Embedded newform 25.9.f.a.2.16

• Label: 25.9.f.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	25.2
Dual form			25.9.f.a.13.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(23.0472 + 3.65032i) q^{2} +(-22.7623 - 44.6735i) q^{3} +(274.378 + 89.1509i) q^{4} +(502.411 + 371.763i) q^{5} +(-361.534 - 1112.69i) q^{6} +(2000.67 + 2000.67i) q^{7} +(675.676 + 344.274i) q^{8} +(2378.86 - 3274.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} + 62 q^{3} - 10 q^{4} - 230 q^{5} - 6 q^{6} + 2342 q^{7} + 8210 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	23.0472	+	3.65032i	1.44045	+	0.228145i	0.827273	−	0.561801i	\(-0.189891\pi\)
							0.613177	+	0.789945i	\(0.289891\pi\)
\(3\)	−22.7623	−	44.6735i	−0.281016	−	0.551524i	0.706751	−	0.707462i	\(-0.250160\pi\)
							−0.987767	+	0.155938i	\(0.950160\pi\)
\(5\)	502.411	+	371.763i	0.803858	+	0.594821i
							\(7\)	2000.67	+	2000.67i	0.833266	+	0.833266i	0.987962	−	0.154696i	\(-0.0494398\pi\)
−0.154696	+	0.987962i	\(0.549440\pi\)
\(11\)	21563.0	−	15666.4i	1.47278	−	1.07004i	0.492984	−	0.870039i	\(-0.335906\pi\)
							0.979796	−	0.199999i	\(-0.0640937\pi\)
\(13\)	−581.795	+	92.1473i	−0.0203703	+	0.00322633i	−0.166611	−	0.986023i	\(-0.553282\pi\)
							0.146241	+	0.989249i	\(0.453282\pi\)
\(17\)	−68641.5	+	134716.i	−0.821847	+	1.61296i	−0.0321377	+	0.999483i	\(0.510231\pi\)
							−0.789709	+	0.613482i	\(0.789769\pi\)
\(19\)	−181546.	+	58987.8i	−1.39307	+	0.452635i	−0.906942	−	0.421255i	\(-0.861590\pi\)
							−0.486124	+	0.873890i	\(0.661590\pi\)
\(23\)	19768.7	−	124814.i	0.0706425	−	0.446019i	−0.926861	−	0.375404i	\(-0.877504\pi\)
							0.997504	−	0.0706148i	\(-0.0224961\pi\)
\(29\)	126362.	+	41057.4i	0.178658	+	0.0580496i	0.396980	−	0.917827i	\(-0.370058\pi\)
							−0.218322	+	0.975877i	\(0.570058\pi\)
\(31\)	−228092.	−	701995.i	−0.246981	−	0.760128i	−0.995304	−	0.0967947i	\(-0.969141\pi\)
							0.748324	−	0.663334i	\(-0.230859\pi\)
\(37\)	−309050.	−	1.95127e6i	−0.164901	−	1.04114i	−0.921816	−	0.387629i	\(-0.873294\pi\)
							0.756915	−	0.653513i	\(-0.226706\pi\)
\(41\)	478396.	+	347575.i	0.169298	+	0.123002i	0.669208	−	0.743075i	\(-0.266634\pi\)
							−0.499910	+	0.866077i	\(0.666634\pi\)
\(43\)	−2.36506e6	+	2.36506e6i	−0.691780	+	0.691780i	−0.962623	−	0.270843i	\(-0.912697\pi\)
							0.270843	+	0.962623i	\(0.412697\pi\)
\(47\)	2.71305e6	−	1.38237e6i	0.555989	−	0.283291i	−0.153333	−	0.988175i	\(-0.549001\pi\)
							0.709323	+	0.704884i	\(0.249001\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.9.f.a.2.16	✓	152
25.13	odd	20	inner	25.9.f.a.13.16	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.9.f.a.2.16	✓	152	1.1	even	1	trivial
25.9.f.a.13.16	yes	152	25.13	odd	20	inner