Embedded newform 114.3.k.a.5.14

• Label: 114.3.k.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	114.5
Dual form			114.3.k.a.23.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.483690 + 1.32893i) q^{2} +(2.99116 + 0.230107i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(1.76037 - 2.09793i) q^{5} +(1.14100 + 4.08633i) q^{6} +(2.58777 + 4.48215i) q^{7} +(-2.44949 - 1.41421i) q^{8} +(8.89410 + 1.37658i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 6 q^{3} - 12 q^{6} + 6 q^{9}+O(q^{10}) \)

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
							\(3\)	2.99116	+	0.230107i	0.997054	+	0.0767024i
\(5\)	1.76037	−	2.09793i	0.352074	−	0.419585i								−0.560720	−	0.828005i	\(-0.689476\pi\)
							0.912794	+	0.408420i	\(0.133920\pi\)
\(7\)	2.58777	+	4.48215i	0.369682	+	0.640307i	0.989516	−	0.144426i	\(-0.0461334\pi\)
							−0.619834	+	0.784733i	\(0.712800\pi\)
\(11\)	−3.69159	−	2.13134i	−0.335599	−	0.193758i	0.322725	−	0.946493i	\(-0.395401\pi\)
							−0.658324	+	0.752735i	\(0.728734\pi\)
\(13\)	1.06594	+	6.04526i	0.0819955	+	0.465020i	0.997965	+	0.0637708i	\(0.0203127\pi\)
							−0.915969	+	0.401249i	\(0.868576\pi\)
\(17\)	−6.30955	−	17.3353i	−0.371150	−	1.01973i	−0.974918	−	0.222566i	\(-0.928557\pi\)
							0.603768	−	0.797160i	\(-0.293665\pi\)
\(19\)	−18.1760	+	5.53462i	−0.956633	+	0.291296i
							\(23\)	−7.44896	−	8.87733i	−0.323868	−	0.385971i	0.579403	−	0.815041i	\(-0.303286\pi\)
−0.903271	+	0.429070i	\(0.858841\pi\)
\(29\)	11.5670	−	31.7801i	0.398862	−	1.09586i	−0.563977	−	0.825790i	\(-0.690729\pi\)
							0.962839	−	0.270075i	\(-0.0870484\pi\)
\(31\)	−16.6859	−	28.9008i	−0.538254	−	0.932282i	−0.998998	−	0.0447498i	\(-0.985751\pi\)
							0.460745	−	0.887533i	\(-0.347582\pi\)
\(37\)	−7.78523			−0.210412			−0.105206	−	0.994450i	\(-0.533550\pi\)
							−0.105206	+	0.994450i	\(0.533550\pi\)
\(41\)	−21.0926	−	3.71920i	−0.514454	−	0.0907121i	−0.0896069	−	0.995977i	\(-0.528561\pi\)
							−0.424847	+	0.905265i	\(0.639672\pi\)
\(43\)	−14.1304	−	11.8568i	−0.328614	−	0.275740i	0.463521	−	0.886086i	\(-0.346586\pi\)
							−0.792135	+	0.610346i	\(0.791030\pi\)
\(47\)	8.11477	−	22.2951i	0.172655	−	0.474365i	−0.822940	−	0.568129i	\(-0.807668\pi\)
							0.995594	+	0.0937638i	\(0.0298898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.3.k.a.5.14	yes	84
3.2	odd	2	inner	114.3.k.a.5.1	✓	84
19.4	even	9	inner	114.3.k.a.23.1	yes	84
57.23	odd	18	inner	114.3.k.a.23.14	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.k.a.5.1	✓	84	3.2	odd	2	inner
114.3.k.a.5.14	yes	84	1.1	even	1	trivial
114.3.k.a.23.1	yes	84	19.4	even	9	inner
114.3.k.a.23.14	yes	84	57.23	odd	18	inner