Embedded newform 76.4.f.a.27.16

• Label: 76.4.f.a.27.16
• 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			27.16
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.147992 + 2.82455i) q^{2} +(-2.98979 + 5.17847i) q^{3} +(-7.95620 + 0.836024i) q^{4} +(-1.90510 + 3.29973i) q^{5} +(-15.0693 - 7.67845i) q^{6} -2.74987i q^{7} +(-3.53885 - 22.3490i) q^{8} +(-4.37772 - 7.58243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.147992	+	2.82455i	0.0523231	+	0.998630i
							\(3\)	−2.98979	+	5.17847i	−0.575386	+	0.996597i	0.420614	+	0.907240i	\(0.361815\pi\)
−0.996000	+	0.0893577i	\(0.971519\pi\)
\(5\)	−1.90510	+	3.29973i	−0.170398	+	0.295137i	−0.938559	−	0.345119i	\(-0.887839\pi\)
							0.768161	+	0.640256i	\(0.221172\pi\)
\(7\)		−	2.74987i		−	0.148479i	−0.997240	−	0.0742395i	\(-0.976347\pi\)
							0.997240	−	0.0742395i	\(-0.0236529\pi\)
\(11\)		−	15.1389i		−	0.414958i	−0.978239	−	0.207479i	\(-0.933474\pi\)
							0.978239	−	0.207479i	\(-0.0665259\pi\)
\(13\)	−24.8365	+	14.3394i	−0.529878	+	0.305925i	−0.740967	−	0.671542i	\(-0.765632\pi\)
							0.211089	+	0.977467i	\(0.432299\pi\)
\(17\)	−11.8965	+	20.6054i	−0.169725	+	0.293973i	−0.938323	−	0.345759i	\(-0.887621\pi\)
							0.768598	+	0.639732i	\(0.220955\pi\)
\(19\)	−82.0880	+	10.9801i	−0.991172	+	0.132579i
							\(23\)	−57.0658	+	32.9469i	−0.517349	+	0.298692i	−0.735849	−	0.677145i	\(-0.763217\pi\)
0.218500	+	0.975837i	\(0.429884\pi\)
\(29\)	−144.532	+	83.4459i	−0.925483	+	0.534328i	−0.885380	−	0.464868i	\(-0.846102\pi\)
							−0.0401027	+	0.999196i	\(0.512769\pi\)
\(31\)	89.3278			0.517540			0.258770	−	0.965939i	\(-0.416683\pi\)
							0.258770	+	0.965939i	\(0.416683\pi\)
\(37\)			208.064i			0.924475i	0.886756	+	0.462238i	\(0.152953\pi\)
							−0.886756	+	0.462238i	\(0.847047\pi\)
\(41\)	−30.2818	−	17.4832i	−0.115347	−	0.0665955i	0.441217	−	0.897401i	\(-0.354547\pi\)
							−0.556563	+	0.830805i	\(0.687880\pi\)
\(43\)	298.492	+	172.334i	1.05859	+	0.611179i	0.925043	−	0.379863i	\(-0.124029\pi\)
							0.133551	+	0.991042i	\(0.457362\pi\)
\(47\)	118.179	−	68.2307i	0.366770	−	0.211755i	−0.305277	−	0.952264i	\(-0.598749\pi\)
							0.672046	+	0.740509i	\(0.265416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.16	✓	56
4.3	odd	2	inner	76.4.f.a.27.25	yes	56
19.12	odd	6	inner	76.4.f.a.31.25	yes	56
76.31	even	6	inner	76.4.f.a.31.16	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.16	✓	56	1.1	even	1	trivial
76.4.f.a.27.25	yes	56	4.3	odd	2	inner
76.4.f.a.31.16	yes	56	76.31	even	6	inner
76.4.f.a.31.25	yes	56	19.12	odd	6	inner