Embedded newform 76.3.l.a.23.10

• Label: 76.3.l.a.23.10
• 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			23.10
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0616288 + 1.99905i) q^{2} +(-2.53724 + 0.447383i) q^{3} +(-3.99240 + 0.246398i) q^{4} +(-1.56377 + 1.31216i) q^{5} +(-1.05071 - 5.04449i) q^{6} +(-3.64994 - 2.10730i) q^{7} +(-0.738609 - 7.96583i) q^{8} +(-2.21981 + 0.807946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0616288	+	1.99905i	0.0308144	+	0.999525i
							\(3\)	−2.53724	+	0.447383i	−0.845746	+	0.149128i	−0.579697	−	0.814832i	\(-0.696829\pi\)
−0.266048	+	0.963960i	\(0.585718\pi\)
\(5\)	−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	0.216095	−	0.976372i	\(-0.430668\pi\)
							−0.737516	+	0.675330i	\(0.764001\pi\)
\(11\)	−2.48083	+	1.43231i	−0.225530	+	0.130210i	−0.608508	−	0.793548i	\(-0.708232\pi\)
							0.382978	+	0.923757i	\(0.374898\pi\)
\(13\)	−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	−0.792202	−	0.610259i	\(-0.791066\pi\)
0.738552	+	0.674197i	\(0.235510\pi\)
\(29\)	−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	0.211564	−	0.977364i	\(-0.432144\pi\)
							−0.740640	+	0.671902i	\(0.765478\pi\)
\(37\)	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	0.956923	−	0.290341i	\(-0.0937688\pi\)
							−0.452098	+	0.891968i	\(0.649324\pi\)
\(47\)	4.85462	+	13.3380i	0.103290	+	0.283786i	0.980563	−	0.196205i	\(-0.0628619\pi\)
							−0.877273	+	0.479992i	\(0.840640\pi\)

(See \(a_n\) instead)

Twists

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

    

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