Embedded newform 76.2.f.a.31.3

• Label: 76.2.f.a.31.3
• Level: $76$
• Weight: $2$
• Character: 76.31
• Analytic conductor: $0.607$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.606863055362\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 3*x^15 + 6*x^14 - 9*x^13 + 12*x^12 - 9*x^11 + 3*x^10 + 6*x^9 - 10*x^8 + 12*x^7 + 12*x^6 - 72*x^5 + 192*x^4 - 288*x^3 + 384*x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.3
Root			\(1.34543 + 0.435684i\) of defining polynomial
Character	\(\chi\)	\(=\)	76.31
Dual form			76.2.f.a.27.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05003 + 0.947334i) q^{2} +(-0.982349 - 1.70148i) q^{3} +(0.205118 - 1.98945i) q^{4} +(-0.349646 - 0.605604i) q^{5} +(2.64336 + 0.855988i) q^{6} -3.80025i q^{7} +(1.66930 + 2.28330i) q^{8} +(-0.430019 + 0.744815i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05003	+	0.947334i	−0.742482	+	0.669866i
							\(3\)	−0.982349	−	1.70148i	−0.567160	−	0.982349i	−0.996845	−	0.0793705i	\(-0.974709\pi\)
0.429686	−	0.902979i	\(-0.358624\pi\)
\(5\)	−0.349646	−	0.605604i	−0.156366	−	0.270834i	0.777189	−	0.629267i	\(-0.216645\pi\)
							−0.933556	+	0.358432i	\(0.883311\pi\)
\(7\)		−	3.80025i		−	1.43636i	−0.695858	−	0.718179i	\(-0.744976\pi\)
							0.695858	−	0.718179i	\(-0.255024\pi\)
\(11\)			2.16607i			0.653094i	0.945181	+	0.326547i	\(0.105885\pi\)
							−0.945181	+	0.326547i	\(0.894115\pi\)
\(13\)	1.16473	+	0.672457i	0.323038	+	0.186506i	0.652746	−	0.757577i	\(-0.273617\pi\)
							−0.329708	+	0.944083i	\(0.606950\pi\)
\(17\)	−1.89546	−	3.28303i	−0.459716	−	0.796251i	0.539230	−	0.842159i	\(-0.318715\pi\)
							−0.998946	+	0.0459074i	\(0.985382\pi\)
\(19\)	−1.62181	+	4.04595i	−0.372069	+	0.928205i
							\(23\)	4.89133	+	2.82401i	1.01991	+	0.588846i	0.914078	−	0.405537i	\(-0.132916\pi\)
0.105834	+	0.994384i	\(0.466249\pi\)
\(29\)	8.65155	+	4.99498i	1.60655	+	0.927544i	0.990134	+	0.140125i	\(0.0447506\pi\)
							0.616419	+	0.787418i	\(0.288583\pi\)
\(31\)	7.76382			1.39442			0.697212	−	0.716865i	\(-0.254424\pi\)
							0.697212	+	0.716865i	\(0.254424\pi\)
\(37\)		−	1.31241i		−	0.215759i	−0.994164	−	0.107879i	\(-0.965594\pi\)
							0.994164	−	0.107879i	\(-0.0344060\pi\)
\(41\)	−7.58097	+	4.37688i	−1.18395	+	0.683553i	−0.956925	−	0.290336i	\(-0.906233\pi\)
							−0.227024	+	0.973889i	\(0.572900\pi\)
\(43\)	5.35195	−	3.08995i	0.816165	−	0.471213i	−0.0329270	−	0.999458i	\(-0.510483\pi\)
							0.849092	+	0.528245i	\(0.177150\pi\)
\(47\)	−2.06084	−	1.18983i	−0.300605	−	0.173554i	0.342110	−	0.939660i	\(-0.388859\pi\)
							−0.642715	+	0.766106i	\(0.722192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.2.f.a.31.3	yes	16
3.2	odd	2		684.2.r.a.487.6		16
4.3	odd	2	inner	76.2.f.a.31.1	yes	16
8.3	odd	2		1216.2.n.f.639.2		16
8.5	even	2		1216.2.n.f.639.7		16
12.11	even	2		684.2.r.a.487.8		16
19.8	odd	6	inner	76.2.f.a.27.1	✓	16
57.8	even	6		684.2.r.a.559.8		16
76.27	even	6	inner	76.2.f.a.27.3	yes	16
152.27	even	6		1216.2.n.f.255.7		16
152.141	odd	6		1216.2.n.f.255.2		16
228.179	odd	6		684.2.r.a.559.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.f.a.27.1	✓	16	19.8	odd	6	inner
76.2.f.a.27.3	yes	16	76.27	even	6	inner
76.2.f.a.31.1	yes	16	4.3	odd	2	inner
76.2.f.a.31.3	yes	16	1.1	even	1	trivial
684.2.r.a.487.6		16	3.2	odd	2
684.2.r.a.487.8		16	12.11	even	2
684.2.r.a.559.6		16	228.179	odd	6
684.2.r.a.559.8		16	57.8	even	6
1216.2.n.f.255.2		16	152.141	odd	6
1216.2.n.f.255.7		16	152.27	even	6
1216.2.n.f.639.2		16	8.3	odd	2
1216.2.n.f.639.7		16	8.5	even	2