Embedded newform 125.2.h.a.109.4

• Label: 125.2.h.a.109.4
• Level: $125$
• Weight: $2$
• Character: 125.109
• Analytic conductor: $0.998$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 125 = 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	125.h (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.998130025266\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{50})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{50}]$

Embedding invariants

Embedding label			109.4
Character	\(\chi\)	\(=\)	125.109
Dual form			125.2.h.a.39.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.158227 + 1.25250i) q^{2} +(0.573928 - 0.474794i) q^{3} +(0.393454 + 0.101022i) q^{4} +(-0.459635 - 2.18832i) q^{5} +(0.503868 + 0.793968i) q^{6} +(4.01162 - 1.30345i) q^{7} +(-1.11826 + 2.82441i) q^{8} +(-0.458180 + 2.40187i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{27}{50}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.158227	+	1.25250i	−0.111883	+	0.885649i	0.831863	+	0.554980i	\(0.187274\pi\)
							−0.943747	+	0.330669i	\(0.892726\pi\)
\(3\)	0.573928	−	0.474794i	0.331357	−	0.274123i	−0.456835	−	0.889551i	\(-0.651017\pi\)
							0.788193	+	0.615429i	\(0.211017\pi\)
\(5\)	−0.459635	−	2.18832i	−0.205555	−	0.978646i
							\(7\)	4.01162	−	1.30345i	1.51625	−	0.492659i	0.571541	−	0.820574i	\(-0.306346\pi\)
0.944707	+	0.327915i	\(0.106346\pi\)
\(11\)	−4.86804	−	0.614976i	−1.46777	−	0.185422i	−0.649545	−	0.760323i	\(-0.725041\pi\)
							−0.818223	+	0.574901i	\(0.805041\pi\)
\(13\)	−0.601337	−	0.114711i	−0.166781	−	0.0318152i	0.103314	−	0.994649i	\(-0.467055\pi\)
							−0.270095	+	0.962834i	\(0.587055\pi\)
\(17\)	0.0971403	+	0.378337i	0.0235600	+	0.0917601i	0.979276	−	0.202531i	\(-0.0649166\pi\)
							−0.955716	+	0.294291i	\(0.904917\pi\)
\(19\)	−2.39459	+	2.89456i	−0.549356	+	0.664058i	−0.969982	−	0.243176i	\(-0.921811\pi\)
							0.420626	+	0.907234i	\(0.361811\pi\)
\(23\)	4.06964	−	4.33373i	0.848579	−	0.903645i	−0.147770	−	0.989022i	\(-0.547209\pi\)
							0.996349	+	0.0853768i	\(0.0272094\pi\)
\(29\)	−0.409318	−	6.50592i	−0.0760084	−	1.20812i	−0.832466	−	0.554076i	\(-0.813072\pi\)
							0.756457	−	0.654043i	\(-0.226928\pi\)
\(31\)	−4.18444	+	1.07438i	−0.751548	+	0.192965i	−0.604985	−	0.796237i	\(-0.706821\pi\)
							−0.146563	+	0.989201i	\(0.546821\pi\)
\(37\)	1.41629	−	2.57622i	0.232837	−	0.423529i	−0.733916	−	0.679240i	\(-0.762310\pi\)
							0.966753	+	0.255711i	\(0.0823096\pi\)
\(41\)	−3.09266	+	2.90420i	−0.482992	+	0.453559i	−0.887387	−	0.461025i	\(-0.847482\pi\)
							0.404396	+	0.914584i	\(0.367482\pi\)
\(43\)	−5.23366	−	7.20351i	−0.798125	−	1.09852i	−0.993048	−	0.117706i	\(-0.962446\pi\)
							0.194923	−	0.980818i	\(-0.437554\pi\)
\(47\)	3.31214	+	8.36551i	0.483125	+	1.22023i	0.943162	+	0.332335i	\(0.107836\pi\)
							−0.460036	+	0.887900i	\(0.652164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.h.a.109.4	yes	240
5.2	odd	4		625.2.g.b.451.9		480
5.3	odd	4		625.2.g.b.451.16		480
5.4	even	2		625.2.h.a.174.9		240
125.27	odd	100		625.2.g.b.176.9		480
125.39	even	50	inner	125.2.h.a.39.4	✓	240
125.86	even	25		625.2.h.a.449.9		240
125.98	odd	100		625.2.g.b.176.16		480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.h.a.39.4	✓	240	125.39	even	50	inner
125.2.h.a.109.4	yes	240	1.1	even	1	trivial
625.2.g.b.176.9		480	125.27	odd	100
625.2.g.b.176.16		480	125.98	odd	100
625.2.g.b.451.9		480	5.2	odd	4
625.2.g.b.451.16		480	5.3	odd	4
625.2.h.a.174.9		240	5.4	even	2
625.2.h.a.449.9		240	125.86	even	25