Embedded newform 74.4.e.a.11.2

• Label: 74.4.e.a.11.2
• Level: $74$
• Weight: $4$
• Character: 74.11
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 346*x^18 + 50697*x^16 + 4104768*x^14 + 200532432*x^12 + 6039270720*x^10 + 109290291168*x^8 + 1091662316544*x^6 + 4894436404224*x^4 + 5352885383680*x^2 + 1118416232704
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.2
Root			\(-5.89468i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.11
Dual form			74.4.e.a.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(-2.94734 + 5.10494i) q^{3} +(2.00000 - 3.46410i) q^{4} +(-12.8210 - 7.40221i) q^{5} -11.7894i q^{6} +(7.99249 - 13.8434i) q^{7} +8.00000i q^{8} +(-3.87362 - 6.70931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73205	+	1.00000i	−0.612372	+	0.353553i
							\(3\)	−2.94734	+	5.10494i	−0.567216	+	0.982446i	0.429624	+	0.903008i	\(0.358646\pi\)
−0.996840	+	0.0794386i	\(0.974687\pi\)
\(5\)	−12.8210	−	7.40221i	−1.14675	−	0.662074i	−0.198654	−	0.980070i	\(-0.563657\pi\)
							−0.948092	+	0.317996i	\(0.896990\pi\)
\(7\)	7.99249	−	13.8434i	0.431554	−	0.747473i	−0.565454	−	0.824780i	\(-0.691299\pi\)
							0.997007	+	0.0773072i	\(0.0246322\pi\)
\(11\)	61.9658			1.69849			0.849245	−	0.527998i	\(-0.177057\pi\)
							0.849245	+	0.527998i	\(0.177057\pi\)
\(13\)	−62.4556	−	36.0587i	−1.33247	−	0.769299i	−0.346789	−	0.937943i	\(-0.612728\pi\)
							−0.985677	+	0.168644i	\(0.946061\pi\)
\(17\)	98.0374	−	56.6019i	1.39868	−	0.807528i	0.404426	−	0.914571i	\(-0.367471\pi\)
							0.994254	+	0.107042i	\(0.0341380\pi\)
\(19\)	−133.568	−	77.1157i	−1.61277	−	0.931134i	−0.988724	−	0.149747i	\(-0.952154\pi\)
							−0.624047	−	0.781387i	\(-0.714513\pi\)
\(23\)		−	98.0431i		−	0.888844i	−0.895818	−	0.444422i	\(-0.853409\pi\)
							0.895818	−	0.444422i	\(-0.146591\pi\)
\(29\)			57.1816i			0.366150i	0.983099	+	0.183075i	\(0.0586052\pi\)
							−0.983099	+	0.183075i	\(0.941395\pi\)
\(31\)			21.7446i			0.125982i	0.998014	+	0.0629910i	\(0.0200640\pi\)
							−0.998014	+	0.0629910i	\(0.979936\pi\)
\(37\)	49.0355	−	219.655i	0.217875	−	0.975977i
							\(41\)	−133.514	+	231.253i	−0.508570	+	0.880869i	0.491381	+	0.870945i	\(0.336492\pi\)
−0.999951	+	0.00992393i	\(0.996841\pi\)
\(43\)		−	284.962i		−	1.01061i	−0.862940	−	0.505306i	\(-0.831380\pi\)
							0.862940	−	0.505306i	\(-0.168620\pi\)
\(47\)	−158.989			−0.493423			−0.246712	−	0.969089i	\(-0.579350\pi\)
							−0.246712	+	0.969089i	\(0.579350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.11.2	✓	20
3.2	odd	2		666.4.s.d.307.10		20
37.27	even	6	inner	74.4.e.a.27.2	yes	20
111.101	odd	6		666.4.s.d.397.10		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.2	✓	20	1.1	even	1	trivial
74.4.e.a.27.2	yes	20	37.27	even	6	inner
666.4.s.d.307.10		20	3.2	odd	2
666.4.s.d.397.10		20	111.101	odd	6