Embedded newform 76.2.f.a.31.4

• Label: 76.2.f.a.31.4
• 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			31.4
Root			\(-0.835469 + 1.14105i\) of defining polynomial
Character	\(\chi\)	\(=\)	76.31
Dual form			76.2.f.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.570443 - 1.29406i) q^{2} +(-0.637123 - 1.10353i) q^{3} +(-1.34919 + 1.47638i) q^{4} +(-1.60333 - 2.77705i) q^{5} +(-1.06459 + 1.45398i) q^{6} +1.25044i q^{7} +(2.68016 + 0.903746i) q^{8} +(0.688149 - 1.19191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.570443	−	1.29406i	−0.403364	−	0.915040i
							\(3\)	−0.637123	−	1.10353i	−0.367843	−	0.637123i	0.621385	−	0.783505i	\(-0.286570\pi\)
−0.989228	+	0.146382i	\(0.953237\pi\)
\(5\)	−1.60333	−	2.77705i	−0.717030	−	1.24193i	−0.962171	−	0.272445i	\(-0.912168\pi\)
							0.245141	−	0.969487i	\(-0.421166\pi\)
\(7\)			1.25044i			0.472622i	0.971677	+	0.236311i	\(0.0759384\pi\)
							−0.971677	+	0.236311i	\(0.924062\pi\)
\(11\)		−	2.11093i		−	0.636469i	−0.948012	−	0.318234i	\(-0.896910\pi\)
							0.948012	−	0.318234i	\(-0.103090\pi\)
\(13\)	2.12978	+	1.22963i	0.590695	+	0.341038i	0.765372	−	0.643588i	\(-0.222555\pi\)
							−0.174678	+	0.984626i	\(0.555888\pi\)
\(17\)	0.765026	+	1.32506i	0.185546	+	0.321375i	0.943760	−	0.330630i	\(-0.107261\pi\)
							−0.758214	+	0.652005i	\(0.773928\pi\)
\(19\)	3.76307	−	2.19984i	0.863308	−	0.504678i
							\(23\)	7.61951	+	4.39913i	1.58878	+	0.917282i	0.993509	+	0.113756i	\(0.0362881\pi\)
0.595270	+	0.803526i	\(0.297045\pi\)
\(29\)	−5.20937	−	3.00763i	−0.967356	−	0.558503i	−0.0689265	−	0.997622i	\(-0.521957\pi\)
							−0.898429	+	0.439119i	\(0.855291\pi\)
\(31\)	−7.78947			−1.39903			−0.699515	−	0.714618i	\(-0.746601\pi\)
							−0.699515	+	0.714618i	\(0.746601\pi\)
\(37\)			9.97599i			1.64004i	0.572334	+	0.820021i	\(0.306038\pi\)
							−0.572334	+	0.820021i	\(0.693962\pi\)
\(41\)	1.09450	−	0.631908i	0.170932	−	0.0986875i	−0.412093	−	0.911142i	\(-0.635202\pi\)
							0.583025	+	0.812454i	\(0.301869\pi\)
\(43\)	5.04619	−	2.91342i	0.769537	−	0.444292i	−0.0631725	−	0.998003i	\(-0.520122\pi\)
							0.832709	+	0.553710i	\(0.186789\pi\)
\(47\)	−6.12910	−	3.53864i	−0.894022	−	0.516164i	−0.0187658	−	0.999824i	\(-0.505974\pi\)
							−0.875256	+	0.483660i	\(0.839307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.2.f.a.31.4	yes	16
3.2	odd	2		684.2.r.a.487.5		16
4.3	odd	2	inner	76.2.f.a.31.7	yes	16
8.3	odd	2		1216.2.n.f.639.3		16
8.5	even	2		1216.2.n.f.639.6		16
12.11	even	2		684.2.r.a.487.2		16
19.8	odd	6	inner	76.2.f.a.27.7	yes	16
57.8	even	6		684.2.r.a.559.2		16
76.27	even	6	inner	76.2.f.a.27.4	✓	16
152.27	even	6		1216.2.n.f.255.6		16
152.141	odd	6		1216.2.n.f.255.3		16
228.179	odd	6		684.2.r.a.559.5		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.f.a.27.4	✓	16	76.27	even	6	inner
76.2.f.a.27.7	yes	16	19.8	odd	6	inner
76.2.f.a.31.4	yes	16	1.1	even	1	trivial
76.2.f.a.31.7	yes	16	4.3	odd	2	inner
684.2.r.a.487.2		16	12.11	even	2
684.2.r.a.487.5		16	3.2	odd	2
684.2.r.a.559.2		16	57.8	even	6
684.2.r.a.559.5		16	228.179	odd	6
1216.2.n.f.255.3		16	152.141	odd	6
1216.2.n.f.255.6		16	152.27	even	6
1216.2.n.f.639.3		16	8.3	odd	2
1216.2.n.f.639.6		16	8.5	even	2