Embedded newform 76.3.l.a.23.3

• Label: 76.3.l.a.23.3
• 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			23.3
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77891 + 0.914055i) q^{2} +(0.408304 - 0.0719951i) q^{3} +(2.32901 - 3.25203i) q^{4} +(-5.90028 + 4.95092i) q^{5} +(-0.660528 + 0.501285i) q^{6} +(6.78454 + 3.91705i) q^{7} +(-1.17055 + 7.91390i) q^{8} +(-8.29570 + 3.01939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77891	+	0.914055i	−0.889453	+	0.457027i
							\(3\)	0.408304	−	0.0719951i	0.136101	−	0.0239984i	−0.105182	−	0.994453i	\(-0.533543\pi\)
0.241284	+	0.970455i	\(0.422432\pi\)
\(5\)	−5.90028	+	4.95092i	−1.18006	+	0.990185i	−0.180077	+	0.983652i	\(0.557635\pi\)
							−0.999979	+	0.00653221i	\(0.997921\pi\)
\(7\)	6.78454	+	3.91705i	0.969220	+	0.559579i	0.898998	−	0.437952i	\(-0.144296\pi\)
							0.0702214	+	0.997531i	\(0.477629\pi\)
\(11\)	−8.63202	+	4.98370i	−0.784729	+	0.453063i	−0.838104	−	0.545511i	\(-0.816336\pi\)
							0.0533746	+	0.998575i	\(0.483002\pi\)
\(13\)	−1.47454	+	8.36251i	−0.113426	+	0.643270i	0.874092	+	0.485761i	\(0.161458\pi\)
							−0.987518	+	0.157509i	\(0.949654\pi\)
\(17\)	−1.47255	−	0.535965i	−0.0866206	−	0.0315273i	0.298346	−	0.954458i	\(-0.403565\pi\)
							−0.384967	+	0.922930i	\(0.625787\pi\)
\(19\)	11.4350	+	15.1737i	0.601841	+	0.798616i
							\(23\)	23.4756	−	27.9772i	1.02068	−	1.21640i	0.0445957	−	0.999005i	\(-0.485800\pi\)
0.976084	−	0.217394i	\(-0.0697555\pi\)
\(29\)	31.1272	−	11.3294i	1.07335	−	0.390668i	0.255923	−	0.966697i	\(-0.417621\pi\)
							0.817429	+	0.576029i	\(0.195398\pi\)
\(31\)	12.4518	+	7.18904i	0.401671	+	0.231905i	0.687205	−	0.726464i	\(-0.258838\pi\)
							−0.285534	+	0.958369i	\(0.592171\pi\)
\(37\)	−49.4191			−1.33565			−0.667826	−	0.744317i	\(-0.732775\pi\)
							−0.667826	+	0.744317i	\(0.732775\pi\)
\(41\)	11.5309	+	65.3949i	0.281241	+	1.59500i	0.718411	+	0.695619i	\(0.244870\pi\)
							−0.437170	+	0.899379i	\(0.644019\pi\)
\(43\)	7.57741	+	9.03041i	0.176219	+	0.210009i	0.846923	−	0.531716i	\(-0.178452\pi\)
							−0.670704	+	0.741725i	\(0.734008\pi\)
\(47\)	−24.9279	−	68.4890i	−0.530382	−	1.45721i	−0.858618	−	0.512616i	\(-0.828676\pi\)
							0.328236	−	0.944596i	\(-0.393546\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.3	✓	108
4.3	odd	2	inner	76.3.l.a.23.8	yes	108
19.5	even	9	inner	76.3.l.a.43.8	yes	108
76.43	odd	18	inner	76.3.l.a.43.3	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.3	✓	108	1.1	even	1	trivial
76.3.l.a.23.8	yes	108	4.3	odd	2	inner
76.3.l.a.43.3	yes	108	76.43	odd	18	inner
76.3.l.a.43.8	yes	108	19.5	even	9	inner