Embedded newform 125.2.h.a.109.12

• Label: 125.2.h.a.109.12
• 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			109.12
Character	\(\chi\)	\(=\)	125.109
Dual form			125.2.h.a.39.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.339149 - 2.68464i) q^{2} +(0.134914 - 0.111610i) q^{3} +(-5.15511 - 1.32361i) q^{4} +(1.77465 - 1.36037i) q^{5} +(-0.253878 - 0.400048i) q^{6} +(1.24169 - 0.403450i) q^{7} +(-3.30949 + 8.35882i) q^{8} +(-0.556399 + 2.91675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.339149	−	2.68464i	0.239815	−	1.89833i	−0.168847	−	0.985642i	\(-0.554004\pi\)
							0.408661	−	0.912686i	\(-0.365996\pi\)
\(3\)	0.134914	−	0.111610i	0.0778925	−	0.0644383i	−0.597663	−	0.801747i	\(-0.703904\pi\)
							0.675555	+	0.737309i	\(0.263904\pi\)
\(5\)	1.77465	−	1.36037i	0.793649	−	0.608375i
							\(7\)	1.24169	−	0.403450i	0.469315	−	0.152490i	−0.0648059	−	0.997898i	\(-0.520643\pi\)
0.534121	+	0.845408i	\(0.320643\pi\)
\(11\)	−3.36787	−	0.425461i	−1.01545	−	0.128281i	−0.400049	−	0.916494i	\(-0.631007\pi\)
							−0.615403	+	0.788212i	\(0.711007\pi\)
\(13\)	3.08693	+	0.588864i	0.856161	+	0.163321i	0.596732	−	0.802441i	\(-0.296466\pi\)
							0.259429	+	0.965762i	\(0.416466\pi\)
\(17\)	0.342104	+	1.33241i	0.0829723	+	0.323156i	0.996497	−	0.0836272i	\(-0.0266505\pi\)
							−0.913525	+	0.406783i	\(0.866650\pi\)
\(19\)	5.41093	−	6.54070i	1.24135	−	1.50054i	0.445215	−	0.895424i	\(-0.353127\pi\)
							0.796138	−	0.605115i	\(-0.206873\pi\)
\(23\)	−2.35719	+	2.51015i	−0.491507	+	0.523402i	−0.926127	−	0.377212i	\(-0.876883\pi\)
							0.434620	+	0.900614i	\(0.356883\pi\)
\(29\)	0.0855315	+	1.35948i	0.0158828	+	0.252450i	0.997954	+	0.0639387i	\(0.0203662\pi\)
							−0.982071	+	0.188511i	\(0.939634\pi\)
\(31\)	−1.53934	+	0.395236i	−0.276474	+	0.0709864i	−0.384381	−	0.923175i	\(-0.625585\pi\)
							0.107907	+	0.994161i	\(0.465585\pi\)
\(37\)	−3.73645	+	6.79659i	−0.614269	+	1.11735i	0.367991	+	0.929829i	\(0.380046\pi\)
							−0.982260	+	0.187522i	\(0.939954\pi\)
\(41\)	−3.96670	+	3.72498i	−0.619495	+	0.581745i	−0.929423	−	0.369015i	\(-0.879695\pi\)
							0.309928	+	0.950760i	\(0.399695\pi\)
\(43\)	2.80434	+	3.85984i	0.427658	+	0.588620i	0.967414	−	0.253201i	\(-0.0814835\pi\)
							−0.539756	+	0.841822i	\(0.681483\pi\)
\(47\)	−1.16162	−	2.93392i	−0.169440	−	0.427956i	0.820099	−	0.572221i	\(-0.193918\pi\)
							−0.989539	+	0.144265i	\(0.953918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.h.a.109.12	yes	240
5.2	odd	4		625.2.g.b.451.24		480
5.3	odd	4		625.2.g.b.451.1		480
5.4	even	2		625.2.h.a.174.1		240
125.27	odd	100		625.2.g.b.176.24		480
125.39	even	50	inner	125.2.h.a.39.12	✓	240
125.86	even	25		625.2.h.a.449.1		240
125.98	odd	100		625.2.g.b.176.1		480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.h.a.39.12	✓	240	125.39	even	50	inner
125.2.h.a.109.12	yes	240	1.1	even	1	trivial
625.2.g.b.176.1		480	125.98	odd	100
625.2.g.b.176.24		480	125.27	odd	100
625.2.g.b.451.1		480	5.3	odd	4
625.2.g.b.451.24		480	5.2	odd	4
625.2.h.a.174.1		240	5.4	even	2
625.2.h.a.449.1		240	125.86	even	25