Embedded newform 125.2.h.a.109.2

• Label: 125.2.h.a.109.2
• 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			109.2
Character	\(\chi\)	\(=\)	125.109
Dual form			125.2.h.a.39.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.285704 + 2.26158i) q^{2} +(1.43951 - 1.19086i) q^{3} +(-3.09594 - 0.794901i) q^{4} +(2.23368 + 0.103315i) q^{5} +(2.28196 + 3.59579i) q^{6} +(-1.01659 + 0.330310i) q^{7} +(1.00393 - 2.53564i) q^{8} +(0.0918805 - 0.481654i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.285704	+	2.26158i	−0.202023	+	1.59918i	0.486313	+	0.873784i	\(0.338341\pi\)
							−0.688336	+	0.725392i	\(0.741659\pi\)
\(3\)	1.43951	−	1.19086i	0.831100	−	0.687546i	−0.121227	−	0.992625i	\(-0.538683\pi\)
							0.952327	+	0.305079i	\(0.0986829\pi\)
\(5\)	2.23368	+	0.103315i	0.998932	+	0.0462040i
							\(7\)	−1.01659	+	0.330310i	−0.384235	+	0.124846i	−0.494764	−	0.869027i	\(-0.664746\pi\)
0.110529	+	0.993873i	\(0.464746\pi\)
\(11\)	1.56186	+	0.197308i	0.470917	+	0.0594907i	0.357211	−	0.934024i	\(-0.383728\pi\)
							0.113706	+	0.993514i	\(0.463728\pi\)
\(13\)	−3.41636	−	0.651706i	−0.947529	−	0.180751i	−0.309618	−	0.950861i	\(-0.600201\pi\)
							−0.637911	+	0.770110i	\(0.720201\pi\)
\(17\)	−1.57081	−	6.11789i	−0.380977	−	1.48381i	−0.813170	−	0.582026i	\(-0.802260\pi\)
							0.432193	−	0.901781i	\(-0.357740\pi\)
\(19\)	0.707051	−	0.854677i	0.162209	−	0.196076i	−0.683146	−	0.730281i	\(-0.739389\pi\)
							0.845355	+	0.534205i	\(0.179389\pi\)
\(23\)	−5.21894	+	5.55761i	−1.08822	+	1.15884i	−0.101770	+	0.994808i	\(0.532451\pi\)
							−0.986455	+	0.164034i	\(0.947549\pi\)
\(29\)	−0.566426	−	9.00309i	−0.105183	−	1.67183i	−0.597846	−	0.801611i	\(-0.703977\pi\)
							0.492663	−	0.870220i	\(-0.336023\pi\)
\(31\)	3.33058	−	0.855147i	0.598190	−	0.153589i	0.0625588	−	0.998041i	\(-0.480074\pi\)
							0.535631	+	0.844452i	\(0.320074\pi\)
\(37\)	−3.04110	+	5.53175i	−0.499954	+	0.909414i	0.499126	+	0.866530i	\(0.333655\pi\)
							−0.999080	+	0.0428841i	\(0.986345\pi\)
\(41\)	2.67790	−	2.51471i	0.418217	−	0.392732i	−0.446670	−	0.894699i	\(-0.647390\pi\)
							0.864887	+	0.501967i	\(0.167390\pi\)
\(43\)	2.82164	+	3.88365i	0.430296	+	0.592251i	0.968021	−	0.250869i	\(-0.0807162\pi\)
							−0.537725	+	0.843120i	\(0.680716\pi\)
\(47\)	−2.28738	−	5.77726i	−0.333648	−	0.842700i	−0.995648	−	0.0931954i	\(-0.970292\pi\)
							0.661999	−	0.749504i	\(-0.269708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.h.a.109.2	yes	240
5.2	odd	4		625.2.g.b.451.4		480
5.3	odd	4		625.2.g.b.451.21		480
5.4	even	2		625.2.h.a.174.11		240
125.27	odd	100		625.2.g.b.176.4		480
125.39	even	50	inner	125.2.h.a.39.2	✓	240
125.86	even	25		625.2.h.a.449.11		240
125.98	odd	100		625.2.g.b.176.21		480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.h.a.39.2	✓	240	125.39	even	50	inner
125.2.h.a.109.2	yes	240	1.1	even	1	trivial
625.2.g.b.176.4		480	125.27	odd	100
625.2.g.b.176.21		480	125.98	odd	100
625.2.g.b.451.4		480	5.2	odd	4
625.2.g.b.451.21		480	5.3	odd	4
625.2.h.a.174.11		240	5.4	even	2
625.2.h.a.449.11		240	125.86	even	25