Embedded newform 76.3.l.a.23.4

• Label: 76.3.l.a.23.4
• 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			23.4
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75309 - 0.962633i) q^{2} +(1.94869 - 0.343607i) q^{3} +(2.14668 + 3.37517i) q^{4} +(1.01207 - 0.849231i) q^{5} +(-3.74701 - 1.27350i) q^{6} +(7.34996 + 4.24350i) q^{7} +(-0.514277 - 7.98345i) q^{8} +(-4.77790 + 1.73901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75309	−	0.962633i	−0.876547	−	0.481316i
							\(3\)	1.94869	−	0.343607i	0.649564	−	0.114536i	0.160849	−	0.986979i	\(-0.448577\pi\)
0.488715	+	0.872443i	\(0.337466\pi\)
\(5\)	1.01207	−	0.849231i	0.202415	−	0.169846i	−0.535946	−	0.844253i	\(-0.680045\pi\)
							0.738360	+	0.674406i	\(0.235600\pi\)
\(7\)	7.34996	+	4.24350i	1.04999	+	0.606214i	0.922648	−	0.385644i	\(-0.126021\pi\)
							0.127346	+	0.991858i	\(0.459354\pi\)
\(11\)	16.8475	−	9.72692i	1.53159	−	0.884265i	0.532303	−	0.846554i	\(-0.321327\pi\)
							0.999289	−	0.0377113i	\(-0.0120067\pi\)
\(13\)	2.11130	−	11.9738i	0.162408	−	0.921059i	−0.789290	−	0.614021i	\(-0.789551\pi\)
							0.951697	−	0.307038i	\(-0.0993380\pi\)
\(17\)	−7.35003	−	2.67519i	−0.432355	−	0.157364i	0.116669	−	0.993171i	\(-0.462778\pi\)
							−0.549024	+	0.835806i	\(0.685001\pi\)
\(19\)	−3.35820	+	18.7009i	−0.176747	+	0.984256i
							\(23\)	−0.634665	+	0.756365i	−0.0275941	+	0.0328854i	−0.779665	−	0.626197i	\(-0.784611\pi\)
0.752071	+	0.659082i	\(0.229055\pi\)
\(29\)	−28.3454	+	10.3169i	−0.977428	+	0.355755i	−0.780840	−	0.624732i	\(-0.785208\pi\)
							−0.196588	+	0.980486i	\(0.562986\pi\)
\(31\)	−29.7201	−	17.1589i	−0.958714	−	0.553514i	−0.0629369	−	0.998018i	\(-0.520047\pi\)
							−0.895777	+	0.444504i	\(0.853380\pi\)
\(37\)	17.7443			0.479575			0.239788	−	0.970825i	\(-0.422922\pi\)
							0.239788	+	0.970825i	\(0.422922\pi\)
\(41\)	−1.31779	−	7.47355i	−0.0321412	−	0.182282i	0.964511	−	0.264044i	\(-0.0850563\pi\)
							−0.996652	+	0.0817621i	\(0.973945\pi\)
\(43\)	3.53968	+	4.21842i	0.0823180	+	0.0981028i	0.805631	−	0.592417i	\(-0.201826\pi\)
							−0.723313	+	0.690520i	\(0.757382\pi\)
\(47\)	0.802954	+	2.20610i	0.0170841	+	0.0469382i	0.947942	−	0.318444i	\(-0.103160\pi\)
							−0.930857	+	0.365383i	\(0.880938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.4	yes	108
4.3	odd	2	inner	76.3.l.a.23.1	✓	108
19.5	even	9	inner	76.3.l.a.43.1	yes	108
76.43	odd	18	inner	76.3.l.a.43.4	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.1	✓	108	4.3	odd	2	inner
76.3.l.a.23.4	yes	108	1.1	even	1	trivial
76.3.l.a.43.1	yes	108	19.5	even	9	inner
76.3.l.a.43.4	yes	108	76.43	odd	18	inner