Embedded newform 76.5.l.a.23.7

• Label: 76.5.l.a.23.7
• Level: $76$
• Weight: $5$
• Character: 76.23
• 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			23.7
Character	\(\chi\)	\(=\)	76.23
Dual form			76.5.l.a.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.53287 + 1.87585i) q^{2} +(3.32006 - 0.585416i) q^{3} +(8.96237 - 13.2543i) q^{4} +(31.7438 - 26.6362i) q^{5} +(-10.6312 + 8.29614i) q^{6} +(-51.6354 - 29.8117i) q^{7} +(-6.79985 + 63.6377i) q^{8} +(-65.4350 + 23.8164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.53287	+	1.87585i	−0.883218	+	0.468963i
							\(3\)	3.32006	−	0.585416i	0.368896	−	0.0650463i	0.0138726	−	0.999904i	\(-0.495584\pi\)
0.355023	+	0.934857i	\(0.384473\pi\)
\(5\)	31.7438	−	26.6362i	1.26975	−	1.06545i	0.275181	−	0.961392i	\(-0.411262\pi\)
							0.994571	−	0.104056i	\(-0.0331822\pi\)
\(7\)	−51.6354	−	29.8117i	−1.05378	−	0.608402i	−0.130078	−	0.991504i	\(-0.541523\pi\)
							−0.923706	+	0.383101i	\(0.874856\pi\)
\(11\)	−66.2459	+	38.2471i	−0.547487	+	0.316092i	−0.748108	−	0.663577i	\(-0.769037\pi\)
							0.200621	+	0.979669i	\(0.435704\pi\)
\(13\)	47.6602	−	270.294i	0.282013	−	1.59937i	−0.433752	−	0.901032i	\(-0.642811\pi\)
							0.715765	−	0.698341i	\(-0.246078\pi\)
\(17\)	−219.584	−	79.9221i	−0.759807	−	0.276547i	−0.0670806	−	0.997748i	\(-0.521368\pi\)
							−0.692727	+	0.721200i	\(0.743591\pi\)
\(19\)	−65.3930	+	355.028i	−0.181144	+	0.983457i
							\(23\)	424.356	−	505.727i	0.802184	−	0.956006i	−0.197521	−	0.980299i	\(-0.563289\pi\)
0.999705	+	0.0242926i	\(0.00773335\pi\)
\(29\)	770.112	−	280.298i	0.915709	−	0.333291i	0.159179	−	0.987250i	\(-0.449115\pi\)
							0.756530	+	0.653959i	\(0.226893\pi\)
\(31\)	−956.138	−	552.027i	−0.994941	−	0.574430i	−0.0881936	−	0.996103i	\(-0.528109\pi\)
							−0.906748	+	0.421674i	\(0.861443\pi\)
\(37\)	1009.86			0.737662			0.368831	−	0.929497i	\(-0.379758\pi\)
							0.368831	+	0.929497i	\(0.379758\pi\)
\(41\)	−198.738	−	1127.10i	−0.118226	−	0.670494i	−0.985102	−	0.171970i	\(-0.944987\pi\)
							0.866876	−	0.498524i	\(-0.166124\pi\)
\(43\)	1941.94	+	2314.31i	1.05026	+	1.25166i	0.966905	+	0.255135i	\(0.0821201\pi\)
							0.0833580	+	0.996520i	\(0.473436\pi\)
\(47\)	−580.185	−	1594.05i	−0.262646	−	0.721615i	−0.998987	−	0.0450035i	\(-0.985670\pi\)
							0.736341	−	0.676611i	\(-0.236552\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.23.7	✓	228
4.3	odd	2	inner	76.5.l.a.23.14	yes	228
19.5	even	9	inner	76.5.l.a.43.14	yes	228
76.43	odd	18	inner	76.5.l.a.43.7	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.7	✓	228	1.1	even	1	trivial
76.5.l.a.23.14	yes	228	4.3	odd	2	inner
76.5.l.a.43.7	yes	228	76.43	odd	18	inner
76.5.l.a.43.14	yes	228	19.5	even	9	inner