Embedded newform 76.3.l.a.23.18

• Label: 76.3.l.a.23.18
• Level: $76$
• Weight: $3$
• Character: 76.23
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.18
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99869 + 0.0724033i) q^{2} +(2.79584 - 0.492981i) q^{3} +(3.98952 + 0.289423i) q^{4} +(-3.69543 + 3.10084i) q^{5} +(5.62370 - 0.782889i) q^{6} +(-7.95020 - 4.59005i) q^{7} +(7.95285 + 0.867321i) q^{8} +(-0.883563 + 0.321590i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99869	+	0.0724033i	0.999345	+	0.0362016i
							\(3\)	2.79584	−	0.492981i	0.931945	−	0.164327i	0.312993	−	0.949755i	\(-0.398668\pi\)
0.618953	+	0.785428i	\(0.287557\pi\)
\(5\)	−3.69543	+	3.10084i	−0.739086	+	0.620167i	−0.932592	−	0.360932i	\(-0.882459\pi\)
							0.193506	+	0.981099i	\(0.438014\pi\)
\(7\)	−7.95020	−	4.59005i	−1.13574	−	0.655721i	−0.190370	−	0.981712i	\(-0.560969\pi\)
							−0.945373	+	0.325991i	\(0.894302\pi\)
\(11\)	−5.25230	+	3.03242i	−0.477482	+	0.275674i	−0.719367	−	0.694631i	\(-0.755568\pi\)
							0.241884	+	0.970305i	\(0.422235\pi\)
\(13\)	3.07682	−	17.4495i	0.236679	−	1.34227i	−0.602371	−	0.798216i	\(-0.705777\pi\)
							0.839049	−	0.544055i	\(-0.183112\pi\)
\(17\)	22.2652	+	8.10387i	1.30972	+	0.476698i	0.900149	−	0.435581i	\(-0.143457\pi\)
							0.409568	+	0.912279i	\(0.365679\pi\)
\(19\)	−9.89870	+	16.2178i	−0.520984	+	0.853566i
							\(23\)	15.3744	−	18.3224i	0.668450	−	0.796628i	−0.320122	−	0.947376i	\(-0.603724\pi\)
0.988572	+	0.150748i	\(0.0481684\pi\)
\(29\)	−8.08370	+	2.94223i	−0.278748	+	0.101456i	−0.477611	−	0.878571i	\(-0.658497\pi\)
							0.198863	+	0.980027i	\(0.436275\pi\)
\(31\)	−21.1510	−	12.2115i	−0.682290	−	0.393921i	0.118427	−	0.992963i	\(-0.462215\pi\)
							−0.800717	+	0.599042i	\(0.795548\pi\)
\(37\)	12.2500			0.331081			0.165541	−	0.986203i	\(-0.447063\pi\)
							0.165541	+	0.986203i	\(0.447063\pi\)
\(41\)	−3.20704	−	18.1880i	−0.0782204	−	0.443610i	−0.998615	−	0.0526186i	\(-0.983243\pi\)
							0.920394	−	0.390992i	\(-0.127868\pi\)
\(43\)	36.5902	+	43.6065i	0.850935	+	1.01411i	0.999682	+	0.0252297i	\(0.00803172\pi\)
							−0.148746	+	0.988875i	\(0.547524\pi\)
\(47\)	9.35119	+	25.6922i	0.198961	+	0.546642i	0.998546	−	0.0539119i	\(-0.0171690\pi\)
							−0.799584	+	0.600554i	\(0.794947\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.18	yes	108
4.3	odd	2	inner	76.3.l.a.23.16	✓	108
19.5	even	9	inner	76.3.l.a.43.16	yes	108
76.43	odd	18	inner	76.3.l.a.43.18	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.16	✓	108	4.3	odd	2	inner
76.3.l.a.23.18	yes	108	1.1	even	1	trivial
76.3.l.a.43.16	yes	108	19.5	even	9	inner
76.3.l.a.43.18	yes	108	76.43	odd	18	inner