Embedded newform 76.5.l.a.43.16

• Label: 76.5.l.a.43.16
• Level: $76$
• Weight: $5$
• Character: 76.43
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			43.16
Character	\(\chi\)	\(=\)	76.43
Dual form			76.5.l.a.23.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33273 - 3.77145i) q^{2} +(-4.88717 - 0.861739i) q^{3} +(-12.4477 + 10.0526i) q^{4} +(-29.7827 - 24.9906i) q^{5} +(3.26326 + 19.5802i) q^{6} +(20.5177 - 11.8459i) q^{7} +(54.5024 + 33.5483i) q^{8} +(-52.9733 - 19.2807i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33273	−	3.77145i	−0.333182	−	0.942862i
							\(3\)	−4.88717	−	0.861739i	−0.543018	−	0.0957488i	−0.104591	−	0.994515i	\(-0.533353\pi\)
−0.438428	+	0.898767i	\(0.644464\pi\)
\(5\)	−29.7827	−	24.9906i	−1.19131	−	0.999625i	−0.999836	−	0.0181271i	\(-0.994230\pi\)
							−0.191471	−	0.981498i	\(-0.561326\pi\)
\(7\)	20.5177	−	11.8459i	0.418729	−	0.241753i	−0.275804	−	0.961214i	\(-0.588944\pi\)
							0.694533	+	0.719461i	\(0.255611\pi\)
\(11\)	101.352	+	58.5153i	0.837616	+	0.483598i	0.856453	−	0.516225i	\(-0.172663\pi\)
							−0.0188372	+	0.999823i	\(0.505996\pi\)
\(13\)	52.2772	+	296.479i	0.309332	+	1.75431i	0.602376	+	0.798212i	\(0.294221\pi\)
							−0.293044	+	0.956099i	\(0.594668\pi\)
\(17\)	28.4336	−	10.3490i	0.0983862	−	0.0358097i	−0.292358	−	0.956309i	\(-0.594440\pi\)
							0.390744	+	0.920499i	\(0.372218\pi\)
\(19\)	−259.020	−	251.455i	−0.717506	−	0.696552i
							\(23\)	296.930	+	353.867i	0.561304	+	0.668936i	0.969822	−	0.243814i	\(-0.0783988\pi\)
−0.408518	+	0.912750i	\(0.633954\pi\)
\(29\)	−279.219	−	101.627i	−0.332008	−	0.120841i	0.170637	−	0.985334i	\(-0.445418\pi\)
							−0.502645	+	0.864493i	\(0.667640\pi\)
\(31\)	−602.887	+	348.077i	−0.627354	+	0.362203i	−0.779726	−	0.626120i	\(-0.784642\pi\)
							0.152373	+	0.988323i	\(0.451309\pi\)
\(37\)	−1760.84			−1.28623			−0.643113	−	0.765771i	\(-0.722357\pi\)
							−0.643113	+	0.765771i	\(0.722357\pi\)
\(41\)	−278.144	+	1577.43i	−0.165463	+	0.938390i	0.783122	+	0.621868i	\(0.213626\pi\)
							−0.948585	+	0.316522i	\(0.897485\pi\)
\(43\)	−914.707	+	1090.10i	−0.494703	+	0.589565i	−0.954407	−	0.298508i	\(-0.903511\pi\)
							0.459704	+	0.888072i	\(0.347956\pi\)
\(47\)	1067.38	−	2932.59i	0.483194	−	1.32757i	−0.423545	−	0.905875i	\(-0.639214\pi\)
							0.906740	−	0.421691i	\(-0.138563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.43.16	yes	228
4.3	odd	2	inner	76.5.l.a.43.24	yes	228
19.4	even	9	inner	76.5.l.a.23.24	yes	228
76.23	odd	18	inner	76.5.l.a.23.16	✓	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.16	✓	228	76.23	odd	18	inner
76.5.l.a.23.24	yes	228	19.4	even	9	inner
76.5.l.a.43.16	yes	228	1.1	even	1	trivial
76.5.l.a.43.24	yes	228	4.3	odd	2	inner