Embedded newform 76.4.f.a.27.13

• Label: 76.4.f.a.27.13
• 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			27.13
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.585298 - 2.76721i) q^{2} +(-4.88508 + 8.46120i) q^{3} +(-7.31485 + 3.23928i) q^{4} +(6.65041 - 11.5188i) q^{5} +(26.2731 + 8.56569i) q^{6} -4.05088i q^{7} +(13.2451 + 18.3457i) q^{8} +(-34.2280 - 59.2846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.585298	−	2.76721i	−0.206934	−	0.978355i
							\(3\)	−4.88508	+	8.46120i	−0.940134	+	1.62836i	−0.174920	+	0.984583i	\(0.555967\pi\)
−0.765213	+	0.643777i	\(0.777366\pi\)
\(5\)	6.65041	−	11.5188i	0.594831	−	1.03028i	−0.398740	−	0.917064i	\(-0.630552\pi\)
							0.993571	−	0.113213i	\(-0.0361142\pi\)
\(7\)		−	4.05088i		−	0.218727i	−0.994002	−	0.109364i	\(-0.965119\pi\)
							0.994002	−	0.109364i	\(-0.0348813\pi\)
\(11\)		−	56.7832i		−	1.55643i	−0.627996	−	0.778217i	\(-0.716125\pi\)
							0.627996	−	0.778217i	\(-0.283875\pi\)
\(13\)	−0.249360	+	0.143968i	−0.00532001	+	0.00307151i	−0.502658	−	0.864486i	\(-0.667644\pi\)
							0.497338	+	0.867557i	\(0.334311\pi\)
\(17\)	26.3870	−	45.7035i	0.376457	−	0.652043i	−0.614087	−	0.789239i	\(-0.710475\pi\)
							0.990544	+	0.137195i	\(0.0438088\pi\)
\(19\)	−82.3338	+	8.95232i	−0.994141	+	0.108095i
							\(23\)	96.2060	−	55.5446i	0.872189	−	0.503558i	0.00411382	−	0.999992i	\(-0.498691\pi\)
0.868075	+	0.496433i	\(0.165357\pi\)
\(29\)	−85.4732	+	49.3480i	−0.547309	+	0.315989i	−0.748036	−	0.663658i	\(-0.769003\pi\)
							0.200727	+	0.979647i	\(0.435670\pi\)
\(31\)	−136.866			−0.792961			−0.396481	−	0.918043i	\(-0.629769\pi\)
							−0.396481	+	0.918043i	\(0.629769\pi\)
\(37\)		−	259.739i		−	1.15408i	−0.816717	−	0.577039i	\(-0.804208\pi\)
							0.816717	−	0.577039i	\(-0.195792\pi\)
\(41\)	−38.5859	−	22.2776i	−0.146978	−	0.0848579i	0.424707	−	0.905331i	\(-0.360377\pi\)
							−0.571686	+	0.820473i	\(0.693710\pi\)
\(43\)	−299.816	−	173.099i	−1.06329	−	0.613892i	−0.136951	−	0.990578i	\(-0.543730\pi\)
							−0.926341	+	0.376686i	\(0.877064\pi\)
\(47\)	−159.549	+	92.1155i	−0.495161	+	0.285881i	−0.726713	−	0.686941i	\(-0.758953\pi\)
							0.231552	+	0.972823i	\(0.425620\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.13	yes	56
4.3	odd	2	inner	76.4.f.a.27.3	✓	56
19.12	odd	6	inner	76.4.f.a.31.3	yes	56
76.31	even	6	inner	76.4.f.a.31.13	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.3	✓	56	4.3	odd	2	inner
76.4.f.a.27.13	yes	56	1.1	even	1	trivial
76.4.f.a.31.3	yes	56	19.12	odd	6	inner
76.4.f.a.31.13	yes	56	76.31	even	6	inner