Embedded newform 76.4.f.a.27.20

• Label: 76.4.f.a.27.20
• Level: $76$
• Weight: $4$
• Character: 76.27
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.20
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73738 - 2.23193i) q^{2} +(-1.31663 + 2.28047i) q^{3} +(-1.96303 - 7.75542i) q^{4} +(1.30962 - 2.26832i) q^{5} +(2.80237 + 6.90067i) q^{6} -35.2043i q^{7} +(-20.7201 - 9.09277i) q^{8} +(10.0330 + 17.3776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.73738	−	2.23193i	0.614256	−	0.789107i
							\(3\)	−1.31663	+	2.28047i	−0.253386	+	0.438877i	−0.964456	−	0.264244i	\(-0.914878\pi\)
0.711070	+	0.703121i	\(0.248211\pi\)
\(5\)	1.30962	−	2.26832i	0.117136	−	0.202885i	−0.801496	−	0.598000i	\(-0.795962\pi\)
							0.918631	+	0.395116i	\(0.129295\pi\)
\(7\)		−	35.2043i		−	1.90085i	−0.310946	−	0.950427i	\(-0.600646\pi\)
							0.310946	−	0.950427i	\(-0.399354\pi\)
\(11\)		−	32.2909i		−	0.885098i	−0.896744	−	0.442549i	\(-0.854074\pi\)
							0.896744	−	0.442549i	\(-0.145926\pi\)
\(13\)	26.0944	−	15.0656i	0.556715	−	0.321419i	−0.195111	−	0.980781i	\(-0.562507\pi\)
							0.751826	+	0.659362i	\(0.229173\pi\)
\(17\)	−20.9363	+	36.2627i	−0.298694	+	0.517353i	−0.975837	−	0.218498i	\(-0.929884\pi\)
							0.677143	+	0.735851i	\(0.263218\pi\)
\(19\)	30.4503	+	77.0180i	0.367672	+	0.929955i
							\(23\)	−8.26291	+	4.77060i	−0.0749103	+	0.0432495i	−0.536987	−	0.843590i	\(-0.680438\pi\)
0.462077	+	0.886840i	\(0.347104\pi\)
\(29\)	86.5617	−	49.9764i	0.554280	−	0.320014i	−0.196567	−	0.980490i	\(-0.562979\pi\)
							0.750846	+	0.660477i	\(0.229646\pi\)
\(31\)	206.646			1.19725			0.598624	−	0.801030i	\(-0.295714\pi\)
							0.598624	+	0.801030i	\(0.295714\pi\)
\(37\)		−	331.881i		−	1.47462i	−0.675554	−	0.737311i	\(-0.736095\pi\)
							0.675554	−	0.737311i	\(-0.263905\pi\)
\(41\)	381.777	+	220.419i	1.45423	+	0.839602i	0.998718	−	0.0506255i	\(-0.0161215\pi\)
							0.455516	+	0.890228i	\(0.349455\pi\)
\(43\)	171.539	+	99.0381i	0.608359	+	0.351237i	0.772323	−	0.635230i	\(-0.219095\pi\)
							−0.163964	+	0.986466i	\(0.552428\pi\)
\(47\)	−307.009	+	177.252i	−0.952806	+	0.550103i	−0.893951	−	0.448164i	\(-0.852078\pi\)
							−0.0588545	+	0.998267i	\(0.518745\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.20	yes	56
4.3	odd	2	inner	76.4.f.a.27.11	✓	56
19.12	odd	6	inner	76.4.f.a.31.11	yes	56
76.31	even	6	inner	76.4.f.a.31.20	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.11	✓	56	4.3	odd	2	inner
76.4.f.a.27.20	yes	56	1.1	even	1	trivial
76.4.f.a.31.11	yes	56	19.12	odd	6	inner
76.4.f.a.31.20	yes	56	76.31	even	6	inner