Embedded newform 76.6.f.a.27.2

• Label: 76.6.f.a.27.2
• Level: $76$
• Weight: $6$
• Character: 76.27
• Analytic conductor: $12.189$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.1891703058\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.2
Character	\(\chi\)	\(=\)	76.27
Dual form			76.6.f.a.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.64509 + 0.364630i) q^{2} +(2.01363 - 3.48771i) q^{3} +(31.7341 - 4.11673i) q^{4} +(54.6845 - 94.7163i) q^{5} +(-10.0954 + 20.4227i) q^{6} -177.092i q^{7} +(-177.641 + 34.8105i) q^{8} +(113.391 + 196.398i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.64509	+	0.364630i	−0.997920	+	0.0644580i
							\(3\)	2.01363	−	3.48771i	0.129174	−	0.223737i	−0.794183	−	0.607679i	\(-0.792101\pi\)
0.923357	+	0.383943i	\(0.125434\pi\)
\(5\)	54.6845	−	94.7163i	0.978225	−	1.69434i	0.309373	−	0.950941i	\(-0.399881\pi\)
							0.668852	−	0.743395i	\(-0.266786\pi\)
\(7\)		−	177.092i		−	1.36601i	−0.730413	−	0.683006i	\(-0.760672\pi\)
							0.730413	−	0.683006i	\(-0.239328\pi\)
\(11\)			61.3620i			0.152904i	0.997073	+	0.0764518i	\(0.0243591\pi\)
							−0.997073	+	0.0764518i	\(0.975641\pi\)
\(13\)	252.490	−	145.775i	0.414368	−	0.239235i	−0.278297	−	0.960495i	\(-0.589770\pi\)
							0.692665	+	0.721260i	\(0.256437\pi\)
\(17\)	269.971	−	467.603i	0.226566	−	0.392424i	−0.730222	−	0.683210i	\(-0.760583\pi\)
							0.956788	+	0.290786i	\(0.0939168\pi\)
\(19\)	−1472.04	+	556.059i	−0.935481	+	0.353376i
							\(23\)	−2540.20	+	1466.59i	−1.00126	+	0.578080i	−0.908622	−	0.417619i	\(-0.862865\pi\)
−0.0926422	+	0.995699i	\(0.529531\pi\)
\(29\)	3743.38	−	2161.24i	0.826549	−	0.477208i	−0.0261207	−	0.999659i	\(-0.508315\pi\)
							0.852670	+	0.522451i	\(0.174982\pi\)
\(31\)	5976.45			1.11696			0.558482	−	0.829516i	\(-0.311384\pi\)
							0.558482	+	0.829516i	\(0.311384\pi\)
\(37\)		−	2104.57i		−	0.252732i	−0.991984	−	0.126366i	\(-0.959669\pi\)
							0.991984	−	0.126366i	\(-0.0403313\pi\)
\(41\)	8334.82	+	4812.11i	0.774349	+	0.447071i	0.834424	−	0.551123i	\(-0.185801\pi\)
							−0.0600749	+	0.998194i	\(0.519134\pi\)
\(43\)	−16202.5	−	9354.53i	−1.33632	−	0.771526i	−0.350063	−	0.936726i	\(-0.613840\pi\)
							−0.986260	+	0.165200i	\(0.947173\pi\)
\(47\)	1389.93	−	802.479i	0.0917804	−	0.0529894i	−0.453407	−	0.891303i	\(-0.649792\pi\)
							0.545188	+	0.838314i	\(0.316458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.27.2	✓	96
4.3	odd	2	inner	76.6.f.a.27.18	yes	96
19.12	odd	6	inner	76.6.f.a.31.18	yes	96
76.31	even	6	inner	76.6.f.a.31.2	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.2	✓	96	1.1	even	1	trivial
76.6.f.a.27.18	yes	96	4.3	odd	2	inner
76.6.f.a.31.2	yes	96	76.31	even	6	inner
76.6.f.a.31.18	yes	96	19.12	odd	6	inner