Embedded newform 76.5.l.a.43.19

• Label: 76.5.l.a.43.19
• Level: $76$
• Weight: $5$
• Character: 76.43
• 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			43.19
Character	\(\chi\)	\(=\)	76.43
Dual form			76.5.l.a.23.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.548361 + 3.96223i) q^{2} +(1.05794 + 0.186543i) q^{3} +(-15.3986 - 4.34547i) q^{4} +(1.47493 + 1.23761i) q^{5} +(-1.31926 + 4.08950i) q^{6} +(-26.6122 + 15.3646i) q^{7} +(25.6618 - 58.6300i) q^{8} +(-75.0307 - 27.3089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.548361	+	3.96223i	−0.137090	+	0.990559i
							\(3\)	1.05794	+	0.186543i	0.117548	+	0.0207270i	0.232113	−	0.972689i	\(-0.425436\pi\)
−0.114564	+	0.993416i	\(0.536547\pi\)
\(5\)	1.47493	+	1.23761i	0.0589970	+	0.0495044i	0.671809	−	0.740724i	\(-0.265517\pi\)
							−0.612812	+	0.790228i	\(0.709962\pi\)
\(7\)	−26.6122	+	15.3646i	−0.543106	+	0.313562i	−0.746337	−	0.665568i	\(-0.768189\pi\)
							0.203231	+	0.979131i	\(0.434856\pi\)
\(11\)	−29.9069	−	17.2667i	−0.247164	−	0.142700i	0.371301	−	0.928513i	\(-0.378912\pi\)
							−0.618465	+	0.785812i	\(0.712245\pi\)
\(13\)	−42.0617	−	238.544i	−0.248886	−	1.41150i	−0.811292	−	0.584641i	\(-0.801235\pi\)
							0.562406	−	0.826861i	\(-0.309876\pi\)
\(17\)	−280.382	+	102.051i	−0.970181	+	0.353117i	−0.778015	−	0.628245i	\(-0.783773\pi\)
							−0.192166	+	0.981362i	\(0.561551\pi\)
\(19\)	242.275	−	267.627i	0.671121	−	0.741348i
							\(23\)	421.407	+	502.214i	0.796611	+	0.949365i	0.999555	−	0.0298259i	\(-0.00949528\pi\)
−0.202944	+	0.979190i	\(0.565051\pi\)
\(29\)	−1218.39	−	443.456i	−1.44873	−	0.527296i	−0.506497	−	0.862242i	\(-0.669060\pi\)
							−0.942237	+	0.334946i	\(0.891282\pi\)
\(31\)	−961.830	+	555.313i	−1.00086	+	0.577849i	−0.908504	−	0.417877i	\(-0.862774\pi\)
							−0.0923598	+	0.995726i	\(0.529441\pi\)
\(37\)	198.579			0.145054			0.0725270	−	0.997366i	\(-0.476894\pi\)
							0.0725270	+	0.997366i	\(0.476894\pi\)
\(41\)	−143.868	+	815.918i	−0.0855850	+	0.485377i	0.911644	+	0.410981i	\(0.134814\pi\)
							−0.997229	+	0.0743955i	\(0.976297\pi\)
\(43\)	−2296.43	+	2736.77i	−1.24198	+	1.48014i	−0.423197	+	0.906038i	\(0.639092\pi\)
							−0.818786	+	0.574099i	\(0.805352\pi\)
\(47\)	173.023	−	475.377i	0.0783265	−	0.215200i	−0.894349	−	0.447371i	\(-0.852360\pi\)
							0.972675	+	0.232170i	\(0.0745827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.43.19	yes	228
4.3	odd	2	inner	76.5.l.a.43.11	yes	228
19.4	even	9	inner	76.5.l.a.23.11	✓	228
76.23	odd	18	inner	76.5.l.a.23.19	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.11	✓	228	19.4	even	9	inner
76.5.l.a.23.19	yes	228	76.23	odd	18	inner
76.5.l.a.43.11	yes	228	4.3	odd	2	inner
76.5.l.a.43.19	yes	228	1.1	even	1	trivial