Embedded newform 125.2.h.a.109.8

• Label: 125.2.h.a.109.8
• 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			109.8
Character	\(\chi\)	\(=\)	125.109
Dual form			125.2.h.a.39.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.161861 - 1.28126i) q^{2} +(-1.71483 + 1.41863i) q^{3} +(0.321728 + 0.0826056i) q^{4} +(0.720344 - 2.11686i) q^{5} +(1.54008 + 2.42678i) q^{6} +(2.27945 - 0.740638i) q^{7} +(1.10874 - 2.80036i) q^{8} +(0.365988 - 1.91858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.161861	−	1.28126i	0.114453	−	0.905991i	−0.825432	−	0.564501i	\(-0.809069\pi\)
							0.939886	−	0.341489i	\(-0.110931\pi\)
\(3\)	−1.71483	+	1.41863i	−0.990059	+	0.819048i	−0.983496	−	0.180930i	\(-0.942089\pi\)
							−0.00656345	+	0.999978i	\(0.502089\pi\)
\(5\)	0.720344	−	2.11686i	0.322147	−	0.946690i
							\(7\)	2.27945	−	0.740638i	0.861551	−	0.279935i	0.155275	−	0.987871i	\(-0.450374\pi\)
0.706276	+	0.707936i	\(0.250374\pi\)
\(11\)	4.57197	+	0.577574i	1.37850	+	0.174145i	0.779399	−	0.626528i	\(-0.215524\pi\)
							0.599102	+	0.800673i	\(0.295524\pi\)
\(13\)	−5.66906	−	1.08143i	−1.57231	−	0.299935i	−0.673811	−	0.738903i	\(-0.735344\pi\)
							−0.898502	+	0.438969i	\(0.855344\pi\)
\(17\)	1.07959	+	4.20471i	0.261838	+	1.01979i	0.954423	+	0.298456i	\(0.0964715\pi\)
							−0.692586	+	0.721336i	\(0.743528\pi\)
\(19\)	−3.02560	+	3.65732i	−0.694120	+	0.839048i	−0.993350	−	0.115134i	\(-0.963270\pi\)
							0.299230	+	0.954181i	\(0.403270\pi\)
\(23\)	−0.551231	+	0.587001i	−0.114940	+	0.122398i	−0.783865	−	0.620931i	\(-0.786755\pi\)
							0.668925	+	0.743330i	\(0.266755\pi\)
\(29\)	0.0122026	+	0.193954i	0.00226596	+	0.0360164i	0.998997	−	0.0447721i	\(-0.0142562\pi\)
							−0.996731	+	0.0807885i	\(0.974256\pi\)
\(31\)	0.935449	−	0.240183i	0.168012	−	0.0431381i	−0.163747	−	0.986502i	\(-0.552358\pi\)
							0.331759	+	0.943364i	\(0.392358\pi\)
\(37\)	−3.98946	+	7.25681i	−0.655864	+	1.19301i	0.314014	+	0.949418i	\(0.398326\pi\)
							−0.969877	+	0.243593i	\(0.921674\pi\)
\(41\)	0.442053	−	0.415115i	0.0690370	−	0.0648301i	−0.649261	−	0.760566i	\(-0.724922\pi\)
							0.718298	+	0.695736i	\(0.244922\pi\)
\(43\)	6.60429	+	9.09003i	1.00715	+	1.38622i	0.920837	+	0.389948i	\(0.127507\pi\)
							0.0863081	+	0.996268i	\(0.472493\pi\)
\(47\)	−1.47059	−	3.71428i	−0.214507	−	0.541783i	0.782186	−	0.623046i	\(-0.214105\pi\)
							−0.996693	+	0.0812622i	\(0.974105\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.h.a.109.8	yes	240
5.2	odd	4		625.2.g.b.451.17		480
5.3	odd	4		625.2.g.b.451.8		480
5.4	even	2		625.2.h.a.174.5		240
125.27	odd	100		625.2.g.b.176.17		480
125.39	even	50	inner	125.2.h.a.39.8	✓	240
125.86	even	25		625.2.h.a.449.5		240
125.98	odd	100		625.2.g.b.176.8		480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.h.a.39.8	✓	240	125.39	even	50	inner
125.2.h.a.109.8	yes	240	1.1	even	1	trivial
625.2.g.b.176.8		480	125.98	odd	100
625.2.g.b.176.17		480	125.27	odd	100
625.2.g.b.451.8		480	5.3	odd	4
625.2.g.b.451.17		480	5.2	odd	4
625.2.h.a.174.5		240	5.4	even	2
625.2.h.a.449.5		240	125.86	even	25