Embedded newform 25.7.f.a.2.8

• Label: 25.7.f.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	25.2
Dual form			25.7.f.a.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78809 + 0.283205i) q^{2} +(6.80972 + 13.3648i) q^{3} +(-57.7506 - 18.7643i) q^{4} +(-112.961 - 53.5248i) q^{5} +(8.39138 + 25.8260i) q^{6} +(-120.010 - 120.010i) q^{7} +(-201.185 - 102.509i) q^{8} +(296.249 - 407.752i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.78809	+	0.283205i	0.223511	+	0.0354007i	0.267185	−	0.963645i	\(-0.413906\pi\)
							−0.0436744	+	0.999046i	\(0.513906\pi\)
\(3\)	6.80972	+	13.3648i	0.252212	+	0.494993i	0.982049	−	0.188628i	\(-0.0604040\pi\)
							−0.729837	+	0.683621i	\(0.760404\pi\)
\(5\)	−112.961	−	53.5248i	−0.903685	−	0.428198i
							\(7\)	−120.010	−	120.010i	−0.349882	−	0.349882i	0.510184	−	0.860066i	\(-0.329577\pi\)
−0.860066	+	0.510184i	\(0.829577\pi\)
\(11\)	−782.056	+	568.197i	−0.587570	+	0.426895i	−0.841445	−	0.540342i	\(-0.818295\pi\)
							0.253875	+	0.967237i	\(0.418295\pi\)
\(13\)	−3129.44	+	495.655i	−1.42442	+	0.225605i	−0.820593	−	0.571513i	\(-0.806357\pi\)
							−0.603824	+	0.797118i	\(0.706357\pi\)
\(17\)	−11.6538	+	22.8720i	−0.00237204	+	0.00465540i	−0.892190	−	0.451661i	\(-0.850832\pi\)
							0.889817	+	0.456317i	\(0.150832\pi\)
\(19\)	3054.42	−	992.441i	0.445315	−	0.144692i	−0.0777713	−	0.996971i	\(-0.524780\pi\)
							0.523087	+	0.852279i	\(0.324780\pi\)
\(23\)	−528.556	+	3337.17i	−0.0434418	+	0.274280i	−0.999842	−	0.0177789i	\(-0.994340\pi\)
							0.956400	+	0.292059i	\(0.0943405\pi\)
\(29\)	−15577.1	−	5061.31i	−0.638694	−	0.207524i	−0.0282714	−	0.999600i	\(-0.509000\pi\)
							−0.610422	+	0.792076i	\(0.709000\pi\)
\(31\)	−8469.63	−	26066.8i	−0.284301	−	0.874990i	−0.986607	−	0.163115i	\(-0.947846\pi\)
							0.702306	−	0.711876i	\(-0.252154\pi\)
\(37\)	−10693.3	−	67515.0i	−0.211109	−	1.33289i	−0.834511	−	0.550991i	\(-0.814250\pi\)
							0.623402	−	0.781901i	\(-0.285750\pi\)
\(41\)	−79223.7	−	57559.4i	−1.14949	−	0.835150i	−0.161073	−	0.986942i	\(-0.551496\pi\)
							−0.988412	+	0.151792i	\(0.951496\pi\)
\(43\)	−93228.6	+	93228.6i	−1.17258	+	1.17258i	−0.190992	+	0.981592i	\(0.561170\pi\)
							−0.981592	+	0.190992i	\(0.938830\pi\)
\(47\)	87003.2	−	44330.4i	0.837996	−	0.426980i	0.0183368	−	0.999832i	\(-0.494163\pi\)
							0.819659	+	0.572852i	\(0.194163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.2.8	✓	112
25.13	odd	20	inner	25.7.f.a.13.8	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.2.8	✓	112	1.1	even	1	trivial
25.7.f.a.13.8	yes	112	25.13	odd	20	inner