Embedded newform 115.3.i.a.14.2

• Label: 115.3.i.a.14.2
• Level: $115$
• Weight: $3$
• Character: 115.14
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.2
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01059 - 3.44174i) q^{2} +(-0.190798 + 0.165327i) q^{3} +(-7.45930 + 4.79380i) q^{4} +(-4.45644 + 2.26720i) q^{5} +(0.761832 + 0.489600i) q^{6} +(0.896990 - 1.96413i) q^{7} +(13.1937 + 11.4324i) q^{8} +(-1.27176 + 8.84530i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01059	−	3.44174i	−0.505294	−	1.72087i	−0.677237	−	0.735765i	\(-0.736823\pi\)
							0.171944	−	0.985107i	\(-0.444995\pi\)
\(3\)	−0.190798	+	0.165327i	−0.0635993	+	0.0551091i	−0.686080	−	0.727526i	\(-0.740670\pi\)
							0.622481	+	0.782635i	\(0.286125\pi\)
\(5\)	−4.45644	+	2.26720i	−0.891287	+	0.453439i
							\(7\)	0.896990	−	1.96413i	0.128141	−	0.280590i	−0.834677	−	0.550740i	\(-0.814346\pi\)
0.962818	+	0.270149i	\(0.0870730\pi\)
\(11\)	−0.960740	+	3.27198i	−0.0873400	+	0.297453i	−0.991566	−	0.129601i	\(-0.958630\pi\)
							0.904226	+	0.427054i	\(0.140449\pi\)
\(13\)	−4.05435	+	1.85156i	−0.311873	+	0.142428i	−0.565202	−	0.824952i	\(-0.691202\pi\)
							0.253329	+	0.967380i	\(0.418475\pi\)
\(17\)	−21.6269	−	13.8987i	−1.27217	−	0.817573i	−0.282268	−	0.959336i	\(-0.591087\pi\)
							−0.989901	+	0.141763i	\(0.954723\pi\)
\(19\)	8.58011	+	13.3509i	0.451585	+	0.702679i	0.990167	−	0.139894i	\(-0.0446761\pi\)
							−0.538582	+	0.842573i	\(0.681040\pi\)
\(23\)	−13.9414	+	18.2931i	−0.606147	+	0.795352i
							\(29\)	−33.7893	−	21.7151i	−1.16515	−	0.748795i	−0.192550	−	0.981287i	\(-0.561676\pi\)
−0.972598	+	0.232492i	\(0.925312\pi\)
\(31\)	−21.1617	+	24.4219i	−0.682635	+	0.787803i	−0.986297	−	0.164977i	\(-0.947245\pi\)
							0.303662	+	0.952780i	\(0.401791\pi\)
\(37\)	−6.08737	+	42.3386i	−0.164524	+	1.14429i	0.725450	+	0.688275i	\(0.241632\pi\)
							−0.889974	+	0.456012i	\(0.849277\pi\)
\(41\)	0.928015	+	6.45449i	0.0226345	+	0.157427i	0.998004	−	0.0631509i	\(-0.0201149\pi\)
							−0.975369	+	0.220577i	\(0.929206\pi\)
\(43\)	−31.9577	−	36.8811i	−0.743202	−	0.857701i	0.250689	−	0.968068i	\(-0.419343\pi\)
							−0.993891	+	0.110367i	\(0.964797\pi\)
\(47\)		−	2.62661i		−	0.0558852i	−0.999610	−	0.0279426i	\(-0.991104\pi\)
							0.999610	−	0.0279426i	\(-0.00889557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.i.a.14.2	✓	220
5.4	even	2	inner	115.3.i.a.14.21	yes	220
23.5	odd	22	inner	115.3.i.a.74.21	yes	220
115.74	odd	22	inner	115.3.i.a.74.2	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.i.a.14.2	✓	220	1.1	even	1	trivial
115.3.i.a.14.21	yes	220	5.4	even	2	inner
115.3.i.a.74.2	yes	220	115.74	odd	22	inner
115.3.i.a.74.21	yes	220	23.5	odd	22	inner