Embedded newform 76.5.l.a.43.9

• Label: 76.5.l.a.43.9
• 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			43.9
Character	\(\chi\)	\(=\)	76.43
Dual form			76.5.l.a.23.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.08499 + 2.54614i) q^{2} +(1.55169 + 0.273604i) q^{3} +(3.03431 - 15.7096i) q^{4} +(3.23088 + 2.71103i) q^{5} +(-5.48357 + 3.10675i) q^{6} +(-4.97171 + 2.87042i) q^{7} +(30.6382 + 56.1899i) q^{8} +(-73.7822 - 26.8545i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.08499	+	2.54614i	−0.771247	+	0.636536i
							\(3\)	1.55169	+	0.273604i	0.172410	+	0.0304005i	0.259186	−	0.965827i	\(-0.416546\pi\)
−0.0867768	+	0.996228i	\(0.527657\pi\)
\(5\)	3.23088	+	2.71103i	0.129235	+	0.108441i	0.705114	−	0.709094i	\(-0.250896\pi\)
							−0.575879	+	0.817535i	\(0.695340\pi\)
\(7\)	−4.97171	+	2.87042i	−0.101463	+	0.0585800i	−0.549873	−	0.835248i	\(-0.685324\pi\)
							0.448410	+	0.893828i	\(0.351991\pi\)
\(11\)	−133.376	−	77.0048i	−1.10228	−	0.636404i	−0.165464	−	0.986216i	\(-0.552912\pi\)
							−0.936820	+	0.349812i	\(0.886245\pi\)
\(13\)	−4.87717	−	27.6598i	−0.0288590	−	0.163667i	0.966972	−	0.254881i	\(-0.0820364\pi\)
							−0.995831	+	0.0912139i	\(0.970925\pi\)
\(17\)	485.628	−	176.754i	1.68037	−	0.611606i	0.687013	−	0.726645i	\(-0.258922\pi\)
							0.993361	+	0.115039i	\(0.0366993\pi\)
\(19\)	−250.586	−	259.861i	−0.694145	−	0.719836i
							\(23\)	−509.778	−	607.530i	−0.963663	−	1.14845i	−0.988872	−	0.148767i	\(-0.952470\pi\)
0.0252089	−	0.999682i	\(-0.491975\pi\)
\(29\)	722.448	+	262.950i	0.859035	+	0.312663i	0.733718	−	0.679454i	\(-0.237783\pi\)
							0.125316	+	0.992117i	\(0.460005\pi\)
\(31\)	−881.441	+	508.900i	−0.917212	+	0.529553i	−0.882745	−	0.469853i	\(-0.844307\pi\)
							−0.0344677	+	0.999406i	\(0.510974\pi\)
\(37\)	−1275.89			−0.931987			−0.465993	−	0.884788i	\(-0.654303\pi\)
							−0.465993	+	0.884788i	\(0.654303\pi\)
\(41\)	−91.5083	+	518.969i	−0.0544368	+	0.308727i	−0.999853	−	0.0171384i	\(-0.994544\pi\)
							0.945416	+	0.325865i	\(0.105656\pi\)
\(43\)	658.218	−	784.433i	0.355986	−	0.424247i	−0.558097	−	0.829776i	\(-0.688468\pi\)
							0.914082	+	0.405529i	\(0.132913\pi\)
\(47\)	−855.180	+	2349.59i	−0.387135	+	1.06364i	0.581151	+	0.813796i	\(0.302603\pi\)
							−0.968285	+	0.249848i	\(0.919619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.43.9	yes	228
4.3	odd	2	inner	76.5.l.a.43.1	yes	228
19.4	even	9	inner	76.5.l.a.23.1	✓	228
76.23	odd	18	inner	76.5.l.a.23.9	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.1	✓	228	19.4	even	9	inner
76.5.l.a.23.9	yes	228	76.23	odd	18	inner
76.5.l.a.43.1	yes	228	4.3	odd	2	inner
76.5.l.a.43.9	yes	228	1.1	even	1	trivial