Embedded newform 76.4.f.a.27.12

• Label: 76.4.f.a.27.12
• 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			27.12
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03532 + 2.63213i) q^{2} +(3.69266 - 6.39588i) q^{3} +(-5.85623 - 5.45019i) q^{4} +(-3.68797 + 6.38776i) q^{5} +(13.0117 + 16.3413i) q^{6} -17.6395i q^{7} +(20.4087 - 9.77170i) q^{8} +(-13.7715 - 23.8529i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03532	+	2.63213i	−0.366040	+	0.930599i
							\(3\)	3.69266	−	6.39588i	0.710653	−	1.23089i	−0.253960	−	0.967215i	\(-0.581733\pi\)
0.964612	−	0.263672i	\(-0.0849336\pi\)
\(5\)	−3.68797	+	6.38776i	−0.329862	+	0.571339i	−0.982484	−	0.186345i	\(-0.940336\pi\)
							0.652622	+	0.757684i	\(0.273669\pi\)
\(7\)		−	17.6395i		−	0.952444i	−0.879325	−	0.476222i	\(-0.842006\pi\)
							0.879325	−	0.476222i	\(-0.157994\pi\)
\(11\)		−	72.2293i		−	1.97981i	−0.141721	−	0.989907i	\(-0.545264\pi\)
							0.141721	−	0.989907i	\(-0.454736\pi\)
\(13\)	−12.1703	+	7.02650i	−0.259648	+	0.149908i	−0.624174	−	0.781285i	\(-0.714564\pi\)
							0.364526	+	0.931193i	\(0.381231\pi\)
\(17\)	45.1701	−	78.2369i	0.644433	−	1.11619i	−0.340000	−	0.940426i	\(-0.610427\pi\)
							0.984432	−	0.175765i	\(-0.0562398\pi\)
\(19\)	52.6052	+	63.9663i	0.635182	+	0.772362i
							\(23\)	−100.885	+	58.2459i	−0.914607	+	0.528048i	−0.881910	−	0.471417i	\(-0.843743\pi\)
−0.0326961	+	0.999465i	\(0.510409\pi\)
\(29\)	−16.4077	+	9.47300i	−0.105063	+	0.0606583i	−0.551611	−	0.834102i	\(-0.685987\pi\)
							0.446548	+	0.894760i	\(0.352653\pi\)
\(31\)	−95.8582			−0.555375			−0.277688	−	0.960671i	\(-0.589568\pi\)
							−0.277688	+	0.960671i	\(0.589568\pi\)
\(37\)			143.359i			0.636974i	0.947927	+	0.318487i	\(0.103175\pi\)
							−0.947927	+	0.318487i	\(0.896825\pi\)
\(41\)	155.727	+	89.9089i	0.593182	+	0.342474i	0.766355	−	0.642418i	\(-0.222069\pi\)
							−0.173173	+	0.984891i	\(0.555402\pi\)
\(43\)	−139.215	−	80.3760i	−0.493724	−	0.285052i	0.232394	−	0.972622i	\(-0.425344\pi\)
							−0.726118	+	0.687570i	\(0.758677\pi\)
\(47\)	439.310	−	253.636i	1.36340	−	0.787161i	0.373327	−	0.927700i	\(-0.378217\pi\)
							0.990075	+	0.140539i	\(0.0448835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.12	✓	56
4.3	odd	2	inner	76.4.f.a.27.21	yes	56
19.12	odd	6	inner	76.4.f.a.31.21	yes	56
76.31	even	6	inner	76.4.f.a.31.12	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.12	✓	56	1.1	even	1	trivial
76.4.f.a.27.21	yes	56	4.3	odd	2	inner
76.4.f.a.31.12	yes	56	76.31	even	6	inner
76.4.f.a.31.21	yes	56	19.12	odd	6	inner