Embedded newform 114.3.k.a.5.8

• Label: 114.3.k.a.5.8
• Level: $114$
• Weight: $3$
• Character: 114.5
• Analytic conductor: $3.106$
• Analytic rank: $0$
• Dimension: $84$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	114.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.10627501371\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.8
Character	\(\chi\)	\(=\)	114.5
Dual form			114.3.k.a.23.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.483690 + 1.32893i) q^{2} +(-2.95157 - 0.536869i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(5.47600 - 6.52605i) q^{5} +(-0.714185 - 4.18210i) q^{6} +(6.07385 + 10.5202i) q^{7} +(-2.44949 - 1.41421i) q^{8} +(8.42354 + 3.16921i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 6 q^{3} - 12 q^{6} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.483690	+	1.32893i	0.241845	+	0.664463i
							\(3\)	−2.95157	−	0.536869i	−0.983857	−	0.178956i
\(5\)	5.47600	−	6.52605i	1.09520	−	1.30521i								0.146438	−	0.989220i	\(-0.453219\pi\)
							0.948763	−	0.315990i	\(-0.102337\pi\)
\(7\)	6.07385	+	10.5202i	0.867693	+	1.50289i	0.864348	+	0.502893i	\(0.167731\pi\)
							0.00334435	+	0.999994i	\(0.498935\pi\)
\(11\)	6.93412	+	4.00341i	0.630374	+	0.363947i	0.780897	−	0.624660i	\(-0.214762\pi\)
							−0.150523	+	0.988607i	\(0.548096\pi\)
\(13\)	−1.68659	−	9.56510i	−0.129737	−	0.735777i	−0.978381	−	0.206812i	\(-0.933691\pi\)
							0.848643	−	0.528965i	\(-0.177420\pi\)
\(17\)	4.97107	+	13.6579i	0.292416	+	0.803406i	0.995712	+	0.0925089i	\(0.0294887\pi\)
							−0.703296	+	0.710897i	\(0.748289\pi\)
\(19\)	18.9025	−	1.92197i	0.994871	−	0.101156i
							\(23\)	−14.6697	−	17.4826i	−0.637812	−	0.760114i	0.346211	−	0.938157i	\(-0.387468\pi\)
−0.984023	+	0.178042i	\(0.943024\pi\)
\(29\)	−0.608189	+	1.67099i	−0.0209720	+	0.0576202i	−0.949737	−	0.313049i	\(-0.898650\pi\)
							0.928765	+	0.370669i	\(0.120872\pi\)
\(31\)	9.59304	+	16.6156i	0.309453	+	0.535988i	0.978243	−	0.207463i	\(-0.0665207\pi\)
							−0.668790	+	0.743452i	\(0.733187\pi\)
\(37\)	−4.28922			−0.115925			−0.0579625	−	0.998319i	\(-0.518460\pi\)
							−0.0579625	+	0.998319i	\(0.518460\pi\)
\(41\)	−49.8250	−	8.78550i	−1.21524	−	0.214280i	−0.470967	−	0.882151i	\(-0.656095\pi\)
							−0.744278	+	0.667870i	\(0.767206\pi\)
\(43\)	−44.9754	−	37.7388i	−1.04594	−	0.877647i	−0.0532784	−	0.998580i	\(-0.516967\pi\)
							−0.992661	+	0.120933i	\(0.961412\pi\)
\(47\)	−20.2265	+	55.5719i	−0.430352	+	1.18238i	0.515246	+	0.857042i	\(0.327701\pi\)
							−0.945597	+	0.325339i	\(0.894522\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.3.k.a.5.8	yes	84
3.2	odd	2	inner	114.3.k.a.5.6	✓	84
19.4	even	9	inner	114.3.k.a.23.6	yes	84
57.23	odd	18	inner	114.3.k.a.23.8	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.k.a.5.6	✓	84	3.2	odd	2	inner
114.3.k.a.5.8	yes	84	1.1	even	1	trivial
114.3.k.a.23.6	yes	84	19.4	even	9	inner
114.3.k.a.23.8	yes	84	57.23	odd	18	inner