Embedded newform 74.3.g.b.51.2

• Label: 74.3.g.b.51.2
• Level: $74$
• Weight: $3$
• Character: 74.51
• Analytic conductor: $2.016$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01635395627\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 82x^{10} + 2505x^{8} + 34456x^{6} + 196096x^{4} + 262464x^{2} + 69696 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			51.2
Root			\(-1.17841i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.51
Dual form			74.3.g.b.45.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(-1.02054 + 0.589207i) q^{3} +(1.73205 - 1.00000i) q^{4} +(0.866025 + 0.232051i) q^{5} +(1.17841 - 1.17841i) q^{6} +(4.87434 + 8.44261i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-3.80567 + 6.59162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 4 q^{6} - 8 q^{7} - 24 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	+	0.366025i	−0.683013	+	0.183013i
							\(3\)	−1.02054	+	0.589207i	−0.340179	+	0.196402i	−0.660351	−	0.750957i	\(-0.729592\pi\)
0.320172	+	0.947359i	\(0.396259\pi\)
\(5\)	0.866025	+	0.232051i	0.173205	+	0.0464102i	0.344379	−	0.938831i	\(-0.388089\pi\)
							−0.171174	+	0.985241i	\(0.554756\pi\)
\(7\)	4.87434	+	8.44261i	0.696335	+	1.20609i	0.969729	+	0.244185i	\(0.0785203\pi\)
							−0.273394	+	0.961902i	\(0.588146\pi\)
\(11\)			7.70761i			0.700692i	0.936620	+	0.350346i	\(0.113936\pi\)
							−0.936620	+	0.350346i	\(0.886064\pi\)
\(13\)	8.91631	+	2.38912i	0.685870	+	0.183778i	0.584893	−	0.811110i	\(-0.301136\pi\)
							0.100977	+	0.994889i	\(0.467803\pi\)
\(17\)	−1.98546	−	7.40985i	−0.116792	−	0.435874i	0.882623	−	0.470082i	\(-0.155776\pi\)
							−0.999415	+	0.0342083i	\(0.989109\pi\)
\(19\)	9.56436	+	2.56276i	0.503388	+	0.134882i	0.501572	−	0.865116i	\(-0.332755\pi\)
							0.00181583	+	0.999998i	\(0.499422\pi\)
\(23\)	12.1462	−	12.1462i	0.528095	−	0.528095i	−0.391909	−	0.920004i	\(-0.628185\pi\)
							0.920004	+	0.391909i	\(0.128185\pi\)
\(29\)	−32.6330	−	32.6330i	−1.12528	−	1.12528i	−0.990935	−	0.134342i	\(-0.957108\pi\)
							−0.134342	−	0.990935i	\(-0.542892\pi\)
\(31\)	26.5385	+	26.5385i	0.856080	+	0.856080i	0.990874	−	0.134794i	\(-0.0430371\pi\)
							−0.134794	+	0.990874i	\(0.543037\pi\)
\(37\)	35.3499	+	10.9264i	0.955402	+	0.295309i
							\(41\)	−26.9328	+	15.5497i	−0.656898	+	0.379260i	−0.791094	−	0.611695i	\(-0.790488\pi\)
0.134196	+	0.990955i	\(0.457155\pi\)
\(43\)	15.5844	−	15.5844i	0.362428	−	0.362428i	−0.502278	−	0.864706i	\(-0.667505\pi\)
							0.864706	+	0.502278i	\(0.167505\pi\)
\(47\)	−12.3130			−0.261979			−0.130989	−	0.991384i	\(-0.541815\pi\)
							−0.130989	+	0.991384i	\(0.541815\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.3.g.b.51.2	yes	12
37.8	odd	12	inner	74.3.g.b.45.2	✓	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.3.g.b.45.2	✓	12	37.8	odd	12	inner
74.3.g.b.51.2	yes	12	1.1	even	1	trivial