Embedded newform 76.4.f.a.27.8

• Label: 76.4.f.a.27.8
• Level: $76$
• Weight: $4$
• Character: 76.27
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• 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			27.8
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90417 - 2.09144i) q^{2} +(2.62655 - 4.54932i) q^{3} +(-0.748247 + 7.96493i) q^{4} +(6.06129 - 10.4985i) q^{5} +(-14.5160 + 3.16942i) q^{6} -11.2918i q^{7} +(18.0830 - 13.6017i) q^{8} +(-0.297553 - 0.515377i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90417	−	2.09144i	−0.673227	−	0.739436i
							\(3\)	2.62655	−	4.54932i	0.505480	−	0.875517i	−0.494500	−	0.869178i	\(-0.664649\pi\)
0.999980	−	0.00633960i	\(-0.00201797\pi\)
\(5\)	6.06129	−	10.4985i	0.542139	−	0.939012i	−0.456642	−	0.889650i	\(-0.650948\pi\)
							0.998781	−	0.0493613i	\(-0.0157186\pi\)
\(7\)		−	11.2918i		−	0.609698i	−0.952401	−	0.304849i	\(-0.901394\pi\)
							0.952401	−	0.304849i	\(-0.0986060\pi\)
\(11\)		−	29.9549i		−	0.821067i	−0.911846	−	0.410533i	\(-0.865343\pi\)
							0.911846	−	0.410533i	\(-0.134657\pi\)
\(13\)	−53.1083	+	30.6621i	−1.13305	+	0.654164i	−0.944699	−	0.327939i	\(-0.893646\pi\)
							−0.188346	+	0.982103i	\(0.560313\pi\)
\(17\)	−20.5816	+	35.6484i	−0.293633	+	0.508588i	−0.974666	−	0.223665i	\(-0.928198\pi\)
							0.681033	+	0.732253i	\(0.261531\pi\)
\(19\)	−7.83224	−	82.4479i	−0.0945704	−	0.995518i
							\(23\)	−6.97278	+	4.02574i	−0.0632142	+	0.0364967i	−0.531274	−	0.847200i	\(-0.678287\pi\)
0.468060	+	0.883697i	\(0.344953\pi\)
\(29\)	103.570	−	59.7960i	0.663187	−	0.382891i	−0.130303	−	0.991474i	\(-0.541595\pi\)
							0.793490	+	0.608583i	\(0.208262\pi\)
\(31\)	213.949			1.23956			0.619779	−	0.784776i	\(-0.287222\pi\)
							0.619779	+	0.784776i	\(0.287222\pi\)
\(37\)		−	69.8697i		−	0.310446i	−0.987879	−	0.155223i	\(-0.950390\pi\)
							0.987879	−	0.155223i	\(-0.0496096\pi\)
\(41\)	425.468	+	245.644i	1.62066	+	0.935687i	0.986744	+	0.162282i	\(0.0518855\pi\)
							0.633913	+	0.773405i	\(0.281448\pi\)
\(43\)	−219.957	−	126.992i	−0.780074	−	0.450376i	0.0563824	−	0.998409i	\(-0.482043\pi\)
							−0.836457	+	0.548033i	\(0.815377\pi\)
\(47\)	−104.827	+	60.5221i	−0.325333	+	0.187831i	−0.653767	−	0.756696i	\(-0.726812\pi\)
							0.328434	+	0.944527i	\(0.393479\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.8	yes	56
4.3	odd	2	inner	76.4.f.a.27.2	✓	56
19.12	odd	6	inner	76.4.f.a.31.2	yes	56
76.31	even	6	inner	76.4.f.a.31.8	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.2	✓	56	4.3	odd	2	inner
76.4.f.a.27.8	yes	56	1.1	even	1	trivial
76.4.f.a.31.2	yes	56	19.12	odd	6	inner
76.4.f.a.31.8	yes	56	76.31	even	6	inner