Embedded newform 76.5.l.a.43.3

• Label: 76.5.l.a.43.3
• 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			43.3
Character	\(\chi\)	\(=\)	76.43
Dual form			76.5.l.a.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.94825 - 0.641375i) q^{2} +(-13.6721 - 2.41076i) q^{3} +(15.1773 + 5.06461i) q^{4} +(3.65652 + 3.06818i) q^{5} +(52.4346 + 18.2872i) q^{6} +(76.9302 - 44.4157i) q^{7} +(-56.6753 - 29.7306i) q^{8} +(105.000 + 38.2168i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.94825	−	0.641375i	−0.987061	−	0.160344i
							\(3\)	−13.6721	−	2.41076i	−1.51912	−	0.267862i	−0.649035	−	0.760758i	\(-0.724827\pi\)
−0.870088	+	0.492896i	\(0.835938\pi\)
\(5\)	3.65652	+	3.06818i	0.146261	+	0.122727i	0.712982	−	0.701182i	\(-0.247344\pi\)
							−0.566722	+	0.823909i	\(0.691788\pi\)
\(7\)	76.9302	−	44.4157i	1.57000	−	0.906442i	0.573837	−	0.818970i	\(-0.305454\pi\)
							0.996167	−	0.0874725i	\(-0.0278790\pi\)
\(11\)	−136.857	−	79.0144i	−1.13105	−	0.653011i	−0.186850	−	0.982388i	\(-0.559828\pi\)
							−0.944198	+	0.329377i	\(0.893161\pi\)
\(13\)	33.9535	+	192.560i	0.200908	+	1.13941i	0.903750	+	0.428061i	\(0.140803\pi\)
							−0.702841	+	0.711347i	\(0.748086\pi\)
\(17\)	−66.6579	+	24.2615i	−0.230650	+	0.0839498i	−0.454760	−	0.890614i	\(-0.650275\pi\)
							0.224110	+	0.974564i	\(0.428053\pi\)
\(19\)	−290.672	−	214.081i	−0.805186	−	0.593022i
							\(23\)	−110.879	−	132.140i	−0.209600	−	0.249792i	0.650994	−	0.759083i	\(-0.274352\pi\)
−0.860595	+	0.509291i	\(0.829908\pi\)
\(29\)	−687.056	−	250.068i	−0.816951	−	0.297346i	−0.100459	−	0.994941i	\(-0.532031\pi\)
							−0.716492	+	0.697595i	\(0.754253\pi\)
\(31\)	−701.660	+	405.104i	−0.730136	+	0.421544i	−0.818472	−	0.574547i	\(-0.805178\pi\)
							0.0883362	+	0.996091i	\(0.471845\pi\)
\(37\)	−665.757			−0.486309			−0.243155	−	0.969988i	\(-0.578182\pi\)
							−0.243155	+	0.969988i	\(0.578182\pi\)
\(41\)	26.5664	−	150.666i	0.0158039	−	0.0896286i	−0.975886	−	0.218283i	\(-0.929955\pi\)
							0.991690	+	0.128654i	\(0.0410657\pi\)
\(43\)	−1917.80	+	2285.55i	−1.03721	+	1.23610i	−0.0660157	+	0.997819i	\(0.521029\pi\)
							−0.971196	+	0.238282i	\(0.923416\pi\)
\(47\)	−617.989	+	1697.91i	−0.279760	+	0.768633i	0.717630	+	0.696425i	\(0.245227\pi\)
							−0.997390	+	0.0722086i	\(0.976995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.43.3	yes	228
4.3	odd	2	inner	76.5.l.a.43.12	yes	228
19.4	even	9	inner	76.5.l.a.23.12	yes	228
76.23	odd	18	inner	76.5.l.a.23.3	✓	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.3	✓	228	76.23	odd	18	inner
76.5.l.a.23.12	yes	228	19.4	even	9	inner
76.5.l.a.43.3	yes	228	1.1	even	1	trivial
76.5.l.a.43.12	yes	228	4.3	odd	2	inner