Embedded newform 25.7.f.a.3.4

• Label: 25.7.f.a.3.4
• Level: $25$
• Weight: $7$
• Character: 25.3
• 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			3.4
Character	\(\chi\)	\(=\)	25.3
Dual form			25.7.f.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.99542 - 5.87885i) q^{2} +(-4.81859 - 30.4234i) q^{3} +(12.0299 - 16.5578i) q^{4} +(95.2241 - 80.9775i) q^{5} +(-164.421 + 119.459i) q^{6} +(-61.2445 + 61.2445i) q^{7} +(-550.448 - 87.1824i) q^{8} +(-209.045 + 67.9227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.99542	−	5.87885i	−0.374428	−	0.734856i	0.624506	−	0.781020i	\(-0.285300\pi\)
							−0.998934	+	0.0461638i	\(0.985300\pi\)
\(3\)	−4.81859	−	30.4234i	−0.178466	−	1.12679i	−0.900475	−	0.434908i	\(-0.856781\pi\)
							0.722008	−	0.691884i	\(-0.243219\pi\)
\(5\)	95.2241	−	80.9775i	0.761793	−	0.647820i
							\(7\)	−61.2445	+	61.2445i	−0.178555	+	0.178555i	−0.790726	−	0.612170i	\(-0.790297\pi\)
0.612170	+	0.790726i	\(0.290297\pi\)
\(11\)	−346.091	+	1065.16i	−0.260023	+	0.800270i	0.732775	+	0.680471i	\(0.238225\pi\)
							−0.992798	+	0.119798i	\(0.961775\pi\)
\(13\)	15.2049	−	29.8412i	0.00692074	−	0.0135827i	−0.887520	−	0.460769i	\(-0.847574\pi\)
							0.894441	+	0.447186i	\(0.147574\pi\)
\(17\)	703.355	−	4440.81i	0.143162	−	0.903889i	−0.806641	−	0.591041i	\(-0.798717\pi\)
							0.949803	−	0.312848i	\(-0.101283\pi\)
\(19\)	4618.13	+	6356.31i	0.673295	+	0.926711i	0.999829	−	0.0184746i	\(-0.00588098\pi\)
							−0.326535	+	0.945185i	\(0.605881\pi\)
\(23\)	11360.9	−	5788.68i	0.933749	−	0.475769i	0.0801984	−	0.996779i	\(-0.474445\pi\)
							0.853551	+	0.521010i	\(0.174445\pi\)
\(29\)	−252.945	+	348.149i	−0.0103713	+	0.0142748i	−0.814171	−	0.580625i	\(-0.802808\pi\)
							0.803800	+	0.594900i	\(0.202808\pi\)
\(31\)	−863.059	+	627.049i	−0.0289705	+	0.0210483i	−0.602176	−	0.798363i	\(-0.705700\pi\)
							0.573206	+	0.819411i	\(0.305700\pi\)
\(37\)	−63893.0	−	32555.1i	−1.26139	−	0.642708i	−0.310008	−	0.950734i	\(-0.600332\pi\)
							−0.951378	+	0.308026i	\(0.900332\pi\)
\(41\)	−36632.3	−	112743.i	−0.531511	−	1.63582i	−0.751069	−	0.660223i	\(-0.770462\pi\)
							0.219558	−	0.975599i	\(-0.429538\pi\)
\(43\)	52089.5	+	52089.5i	0.655156	+	0.655156i	0.954230	−	0.299074i	\(-0.0966778\pi\)
							−0.299074	+	0.954230i	\(0.596678\pi\)
\(47\)	179143.	−	28373.5i	1.72547	−	0.273287i	0.786574	−	0.617496i	\(-0.211853\pi\)
							0.938893	+	0.344209i	\(0.111853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.3.4	✓	112
25.17	odd	20	inner	25.7.f.a.17.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.3.4	✓	112	1.1	even	1	trivial
25.7.f.a.17.4	yes	112	25.17	odd	20	inner