Embedded newform 125.2.h.a.109.7

• Label: 125.2.h.a.109.7
• 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			109.7
Character	\(\chi\)	\(=\)	125.109
Dual form			125.2.h.a.39.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00606389 + 0.0480006i) q^{2} +(-0.795440 + 0.658045i) q^{3} +(1.93490 + 0.496798i) q^{4} +(-1.50197 + 1.65653i) q^{5} +(-0.0267631 - 0.0421719i) q^{6} +(1.35113 - 0.439010i) q^{7} +(-0.0712009 + 0.179833i) q^{8} +(-0.362443 + 1.89999i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00606389	+	0.0480006i	−0.00428782	+	0.0339416i	−0.994113	−	0.108344i	\(-0.965445\pi\)
							0.989826	+	0.142286i	\(0.0454452\pi\)
\(3\)	−0.795440	+	0.658045i	−0.459247	+	0.379922i	−0.838084	−	0.545541i	\(-0.816324\pi\)
							0.378837	+	0.925463i	\(0.376324\pi\)
\(5\)	−1.50197	+	1.65653i	−0.671701	+	0.740822i
							\(7\)	1.35113	−	0.439010i	0.510680	−	0.165930i	−0.0423314	−	0.999104i	\(-0.513479\pi\)
0.553012	+	0.833174i	\(0.313479\pi\)
\(11\)	1.92974	+	0.243783i	0.581838	+	0.0735032i	0.410743	−	0.911751i	\(-0.365269\pi\)
							0.171095	+	0.985255i	\(0.445269\pi\)
\(13\)	−0.711401	−	0.135707i	−0.197307	−	0.0376384i	0.0877803	−	0.996140i	\(-0.472023\pi\)
							−0.285088	+	0.958501i	\(0.592023\pi\)
\(17\)	−1.40985	−	5.49099i	−0.341938	−	1.33176i	−0.874359	−	0.485280i	\(-0.838718\pi\)
							0.532421	−	0.846480i	\(-0.321282\pi\)
\(19\)	3.35842	−	4.05963i	0.770474	−	0.931343i	−0.228636	−	0.973512i	\(-0.573427\pi\)
							0.999111	+	0.0421685i	\(0.0134266\pi\)
\(23\)	−4.26032	+	4.53678i	−0.888337	+	0.945983i	−0.998853	−	0.0478913i	\(-0.984750\pi\)
							0.110515	+	0.993874i	\(0.464750\pi\)
\(29\)	−0.436935	−	6.94489i	−0.0811368	−	1.28963i	−0.802322	−	0.596891i	\(-0.796402\pi\)
							0.721186	−	0.692742i	\(-0.243598\pi\)
\(31\)	−5.42248	+	1.39226i	−0.973906	+	0.250057i	−0.701929	−	0.712247i	\(-0.747678\pi\)
							−0.271977	+	0.962304i	\(0.587678\pi\)
\(37\)	4.28990	−	7.80330i	0.705255	−	1.28285i	−0.244436	−	0.969665i	\(-0.578603\pi\)
							0.949691	−	0.313189i	\(-0.101397\pi\)
\(41\)	2.96940	−	2.78845i	0.463742	−	0.435482i	−0.417123	−	0.908850i	\(-0.636962\pi\)
							0.880865	+	0.473367i	\(0.156962\pi\)
\(43\)	3.11056	+	4.28132i	0.474356	+	0.652895i	0.977408	−	0.211361i	\(-0.0677896\pi\)
							−0.503052	+	0.864256i	\(0.667790\pi\)
\(47\)	3.19392	+	8.06691i	0.465881	+	1.17668i	0.952899	+	0.303288i	\(0.0980845\pi\)
							−0.487018	+	0.873392i	\(0.661916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.h.a.109.7	yes	240
5.2	odd	4		625.2.g.b.451.12		480
5.3	odd	4		625.2.g.b.451.13		480
5.4	even	2		625.2.h.a.174.6		240
125.27	odd	100		625.2.g.b.176.12		480
125.39	even	50	inner	125.2.h.a.39.7	✓	240
125.86	even	25		625.2.h.a.449.6		240
125.98	odd	100		625.2.g.b.176.13		480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.h.a.39.7	✓	240	125.39	even	50	inner
125.2.h.a.109.7	yes	240	1.1	even	1	trivial
625.2.g.b.176.12		480	125.27	odd	100
625.2.g.b.176.13		480	125.98	odd	100
625.2.g.b.451.12		480	5.2	odd	4
625.2.g.b.451.13		480	5.3	odd	4
625.2.h.a.174.6		240	5.4	even	2
625.2.h.a.449.6		240	125.86	even	25