Embedded newform 76.3.l.a.23.14

• Label: 76.3.l.a.23.14
• 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			23.14
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.33218 - 1.49175i) q^{2} +(2.53724 - 0.447383i) q^{3} +(-0.450619 - 3.97454i) q^{4} +(-1.56377 + 1.31216i) q^{5} +(2.71266 - 4.38091i) q^{6} +(3.64994 + 2.10730i) q^{7} +(-6.52931 - 4.62257i) q^{8} +(-2.21981 + 0.807946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33218	−	1.49175i	0.666088	−	0.745874i
							\(3\)	2.53724	−	0.447383i	0.845746	−	0.149128i	0.266048	−	0.963960i	\(-0.414282\pi\)
0.579697	+	0.814832i	\(0.303171\pi\)
\(5\)	−1.56377	+	1.31216i	−0.312754	+	0.262431i	−0.785629	−	0.618698i	\(-0.787661\pi\)
							0.472875	+	0.881129i	\(0.343216\pi\)
\(7\)	3.64994	+	2.10730i	0.521421	+	0.301042i	0.737516	−	0.675330i	\(-0.235999\pi\)
							−0.216095	+	0.976372i	\(0.569332\pi\)
\(11\)	2.48083	−	1.43231i	0.225530	−	0.130210i	−0.382978	−	0.923757i	\(-0.625102\pi\)
							0.608508	+	0.793548i	\(0.291768\pi\)
\(13\)	−2.61046	+	14.8046i	−0.200804	+	1.13882i	0.703102	+	0.711089i	\(0.251798\pi\)
							−0.903907	+	0.427730i	\(0.859314\pi\)
\(17\)	3.09325	+	1.12585i	0.181956	+	0.0662265i	0.431391	−	0.902165i	\(-0.358023\pi\)
							−0.249435	+	0.968391i	\(0.580245\pi\)
\(19\)	5.53691	−	18.1753i	0.291416	−	0.956596i
							\(23\)	1.23396	−	1.47057i	0.0536503	−	0.0639380i	−0.738552	−	0.674197i	\(-0.764490\pi\)
0.792202	+	0.610259i	\(0.208934\pi\)
\(29\)	−12.3704	+	4.50246i	−0.426565	+	0.155257i	−0.546377	−	0.837540i	\(-0.683993\pi\)
							0.119811	+	0.992797i	\(0.461771\pi\)
\(31\)	16.4013	+	9.46932i	0.529075	+	0.305462i	0.740640	−	0.671902i	\(-0.234522\pi\)
							−0.211564	+	0.977364i	\(0.567856\pi\)
\(37\)	31.8537			0.860910			0.430455	−	0.902612i	\(-0.358353\pi\)
							0.430455	+	0.902612i	\(0.358353\pi\)
\(41\)	−13.4830	−	76.4661i	−0.328855	−	1.86503i	−0.481073	−	0.876680i	\(-0.659753\pi\)
							0.152219	−	0.988347i	\(-0.451358\pi\)
\(43\)	−21.7075	−	25.8700i	−0.504825	−	0.601627i	0.452098	−	0.891968i	\(-0.350676\pi\)
							−0.956923	+	0.290341i	\(0.906231\pi\)
\(47\)	−4.85462	−	13.3380i	−0.103290	−	0.283786i	0.877273	−	0.479992i	\(-0.159360\pi\)
							−0.980563	+	0.196205i	\(0.937138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.14	yes	108
4.3	odd	2	inner	76.3.l.a.23.10	✓	108
19.5	even	9	inner	76.3.l.a.43.10	yes	108
76.43	odd	18	inner	76.3.l.a.43.14	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.10	✓	108	4.3	odd	2	inner
76.3.l.a.23.14	yes	108	1.1	even	1	trivial
76.3.l.a.43.10	yes	108	19.5	even	9	inner
76.3.l.a.43.14	yes	108	76.43	odd	18	inner