Embedded newform 74.4.e.a.27.8

• Label: 74.4.e.a.27.8
• Level: $74$
• Weight: $4$
• Character: 74.27
• 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			27.8
Root			\(-1.08183i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.27
Dual form			74.4.e.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(-0.540917 - 0.936896i) q^{3} +(2.00000 + 3.46410i) q^{4} +(2.09330 - 1.20857i) q^{5} -2.16367i q^{6} +(17.5687 + 30.4298i) q^{7} +8.00000i q^{8} +(12.9148 - 22.3691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−0.540917	−	0.936896i	−0.104100	−	0.180306i	0.809270	−	0.587436i	\(-0.199863\pi\)
−0.913370	+	0.407131i	\(0.866529\pi\)
\(5\)	2.09330	−	1.20857i	0.187230	−	0.108097i	−0.403455	−	0.914999i	\(-0.632191\pi\)
							0.590685	+	0.806902i	\(0.298857\pi\)
\(7\)	17.5687	+	30.4298i	0.948619	+	1.64306i	0.748338	+	0.663318i	\(0.230852\pi\)
							0.200281	+	0.979739i	\(0.435815\pi\)
\(11\)	30.7561			0.843029			0.421515	−	0.906822i	\(-0.361499\pi\)
							0.421515	+	0.906822i	\(0.361499\pi\)
\(13\)	−21.4108	+	12.3615i	−0.456790	+	0.263728i	−0.710694	−	0.703501i	\(-0.751619\pi\)
							0.253903	+	0.967230i	\(0.418286\pi\)
\(17\)	−52.4721	−	30.2948i	−0.748609	−	0.432209i	0.0765824	−	0.997063i	\(-0.475599\pi\)
							−0.825191	+	0.564854i	\(0.808933\pi\)
\(19\)	−31.8446	+	18.3855i	−0.384507	+	0.221996i	−0.679778	−	0.733418i	\(-0.737924\pi\)
							0.295270	+	0.955414i	\(0.404590\pi\)
\(23\)		−	195.332i		−	1.77084i	−0.464787	−	0.885422i	\(-0.653869\pi\)
							0.464787	−	0.885422i	\(-0.346131\pi\)
\(29\)			21.4124i			0.137110i	0.997647	+	0.0685548i	\(0.0218388\pi\)
							−0.997647	+	0.0685548i	\(0.978161\pi\)
\(31\)		−	179.129i		−	1.03782i	−0.854828	−	0.518911i	\(-0.826337\pi\)
							0.854828	−	0.518911i	\(-0.173663\pi\)
\(37\)	3.36467	−	225.037i	0.0149499	−	0.999888i
							\(41\)	153.955	+	266.659i	0.586434	+	1.01573i	0.994695	+	0.102869i	\(0.0328021\pi\)
−0.408261	+	0.912865i	\(0.633865\pi\)
\(43\)		−	395.356i		−	1.40212i	−0.713102	−	0.701060i	\(-0.752711\pi\)
							0.713102	−	0.701060i	\(-0.247289\pi\)
\(47\)	−475.963			−1.47716			−0.738578	−	0.674168i	\(-0.764502\pi\)
							−0.738578	+	0.674168i	\(0.764502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.27.8	yes	20
3.2	odd	2		666.4.s.d.397.3		20
37.11	even	6	inner	74.4.e.a.11.8	✓	20
111.11	odd	6		666.4.s.d.307.3		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.8	✓	20	37.11	even	6	inner
74.4.e.a.27.8	yes	20	1.1	even	1	trivial
666.4.s.d.307.3		20	111.11	odd	6
666.4.s.d.397.3		20	3.2	odd	2