Embedded newform 76.3.l.a.23.6

• Label: 76.3.l.a.23.6
• 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			23.6
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11612 - 1.65960i) q^{2} +(-4.90961 + 0.865697i) q^{3} +(-1.50854 + 3.70463i) q^{4} +(2.37531 - 1.99312i) q^{5} +(6.91644 + 7.18176i) q^{6} +(8.41114 + 4.85617i) q^{7} +(7.83192 - 1.63125i) q^{8} +(14.8976 - 5.42229i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11612	−	1.65960i	−0.558061	−	0.829800i
							\(3\)	−4.90961	+	0.865697i	−1.63654	+	0.288566i	−0.914892	−	0.403698i	\(-0.867725\pi\)
−0.721645	+	0.692264i	\(0.756613\pi\)
\(5\)	2.37531	−	1.99312i	0.475062	−	0.398624i	−0.373575	−	0.927600i	\(-0.621868\pi\)
							0.848637	+	0.528976i	\(0.177424\pi\)
\(7\)	8.41114	+	4.85617i	1.20159	+	0.693739i	0.960909	−	0.276865i	\(-0.0892954\pi\)
							0.240682	+	0.970604i	\(0.422629\pi\)
\(11\)	−7.34500	+	4.24064i	−0.667727	+	0.385512i	−0.795215	−	0.606328i	\(-0.792642\pi\)
							0.127488	+	0.991840i	\(0.459309\pi\)
\(13\)	−3.81970	+	21.6626i	−0.293823	+	1.66635i	0.378124	+	0.925755i	\(0.376569\pi\)
							−0.671947	+	0.740599i	\(0.734542\pi\)
\(17\)	15.0608	+	5.48169i	0.885931	+	0.322453i	0.744601	−	0.667510i	\(-0.232640\pi\)
							0.141330	+	0.989963i	\(0.454862\pi\)
\(19\)	18.9737	−	0.998863i	0.998617	−	0.0525717i
							\(23\)	4.86968	−	5.80346i	0.211725	−	0.252324i	−0.649721	−	0.760172i	\(-0.725114\pi\)
0.861447	+	0.507848i	\(0.169559\pi\)
\(29\)	−2.00734	+	0.730613i	−0.0692188	+	0.0251936i	−0.376397	−	0.926458i	\(-0.622837\pi\)
							0.307179	+	0.951652i	\(0.400615\pi\)
\(31\)	−6.63446	−	3.83041i	−0.214015	−	0.123562i	0.389161	−	0.921170i	\(-0.372765\pi\)
							−0.603176	+	0.797608i	\(0.706098\pi\)
\(37\)	−29.7012			−0.802735			−0.401367	−	0.915917i	\(-0.631465\pi\)
							−0.401367	+	0.915917i	\(0.631465\pi\)
\(41\)	−5.39602	−	30.6024i	−0.131610	−	0.746399i	−0.977160	−	0.212503i	\(-0.931838\pi\)
							0.845550	−	0.533896i	\(-0.179273\pi\)
\(43\)	10.4150	+	12.4121i	0.242208	+	0.288653i	0.873430	−	0.486950i	\(-0.161891\pi\)
							−0.631222	+	0.775603i	\(0.717446\pi\)
\(47\)	17.8508	+	49.0447i	0.379804	+	1.04350i	0.971437	+	0.237296i	\(0.0762612\pi\)
							−0.591633	+	0.806207i	\(0.701517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.6	yes	108
4.3	odd	2	inner	76.3.l.a.23.2	✓	108
19.5	even	9	inner	76.3.l.a.43.2	yes	108
76.43	odd	18	inner	76.3.l.a.43.6	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.2	✓	108	4.3	odd	2	inner
76.3.l.a.23.6	yes	108	1.1	even	1	trivial
76.3.l.a.43.2	yes	108	19.5	even	9	inner
76.3.l.a.43.6	yes	108	76.43	odd	18	inner