Embedded newform 115.3.i.a.14.15

• Label: 115.3.i.a.14.15
• 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			14.15
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.435846 + 1.48436i) q^{2} +(4.32985 - 3.75184i) q^{3} +(1.35166 - 0.868661i) q^{4} +(-4.07982 - 2.89052i) q^{5} +(7.45621 + 4.79182i) q^{6} +(-4.16584 + 9.12191i) q^{7} +(6.55516 + 5.68008i) q^{8} +(3.39049 - 23.5814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.435846	+	1.48436i	0.217923	+	0.742178i	0.993789	+	0.111280i	\(0.0354951\pi\)
							−0.775866	+	0.630898i	\(0.782687\pi\)
\(3\)	4.32985	−	3.75184i	1.44328	−	1.25061i	0.527130	−	0.849785i	\(-0.323268\pi\)
							0.916153	−	0.400828i	\(-0.131277\pi\)
\(5\)	−4.07982	−	2.89052i	−0.815963	−	0.578104i
							\(7\)	−4.16584	+	9.12191i	−0.595119	+	1.30313i	0.337179	+	0.941440i	\(0.390527\pi\)
−0.932299	+	0.361689i	\(0.882200\pi\)
\(11\)	0.740148	−	2.52071i	0.0672861	−	0.229156i	−0.918984	−	0.394294i	\(-0.870989\pi\)
							0.986270	+	0.165139i	\(0.0528072\pi\)
\(13\)	−6.09926	+	2.78544i	−0.469174	+	0.214264i	−0.635950	−	0.771730i	\(-0.719392\pi\)
							0.166776	+	0.985995i	\(0.446664\pi\)
\(17\)	17.2196	+	11.0664i	1.01292	+	0.650963i	0.938147	−	0.346237i	\(-0.112541\pi\)
							0.0747717	+	0.997201i	\(0.476177\pi\)
\(19\)	0.984014	+	1.53116i	0.0517902	+	0.0805872i	0.866183	−	0.499727i	\(-0.166566\pi\)
							−0.814393	+	0.580314i	\(0.802930\pi\)
\(23\)	−16.4646	+	16.0598i	−0.715852	+	0.698252i
							\(29\)	−16.2887	−	10.4681i	−0.561679	−	0.360969i	0.228786	−	0.973477i	\(-0.426525\pi\)
−0.790465	+	0.612508i	\(0.790161\pi\)
\(31\)	−14.0574	+	16.2231i	−0.453465	+	0.523326i	−0.935739	−	0.352694i	\(-0.885266\pi\)
							0.482274	+	0.876020i	\(0.339811\pi\)
\(37\)	−5.82972	+	40.5466i	−0.157560	+	1.09585i	0.745552	+	0.666448i	\(0.232186\pi\)
							−0.903112	+	0.429406i	\(0.858723\pi\)
\(41\)	−2.75489	−	19.1607i	−0.0671925	−	0.467334i	−0.995441	−	0.0953746i	\(-0.969595\pi\)
							0.928249	−	0.371960i	\(-0.121314\pi\)
\(43\)	−4.59546	−	5.30344i	−0.106871	−	0.123336i	0.699794	−	0.714345i	\(-0.253275\pi\)
							−0.806665	+	0.591009i	\(0.798730\pi\)
\(47\)		−	33.3321i		−	0.709195i	−0.935019	−	0.354597i	\(-0.884618\pi\)
							0.935019	−	0.354597i	\(-0.115382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.i.a.14.15	yes	220
5.4	even	2	inner	115.3.i.a.14.8	✓	220
23.5	odd	22	inner	115.3.i.a.74.8	yes	220
115.74	odd	22	inner	115.3.i.a.74.15	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.i.a.14.8	✓	220	5.4	even	2	inner
115.3.i.a.14.15	yes	220	1.1	even	1	trivial
115.3.i.a.74.8	yes	220	23.5	odd	22	inner
115.3.i.a.74.15	yes	220	115.74	odd	22	inner