Embedded newform 74.4.e.a.11.7

• Label: 74.4.e.a.11.7
• 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			11.7
Root			\(5.08624i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.11
Dual form			74.4.e.a.27.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(-2.54312 + 4.40481i) q^{3} +(2.00000 - 3.46410i) q^{4} +(9.68300 + 5.59048i) q^{5} +10.1725i q^{6} +(-10.2076 + 17.6800i) q^{7} -8.00000i q^{8} +(0.565092 + 0.978768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.54312	+	4.40481i	−0.489423	+	0.847706i	−0.999926	−	0.0121700i	\(-0.996126\pi\)
0.510502	+	0.859876i	\(0.329459\pi\)
\(5\)	9.68300	+	5.59048i	0.866074	+	0.500028i	0.866042	−	0.499972i	\(-0.166656\pi\)
							3.23462e−5		1.00000i	\(0.499990\pi\)
\(7\)	−10.2076	+	17.6800i	−0.551157	+	0.954632i	0.447034	+	0.894517i	\(0.352480\pi\)
							−0.998191	+	0.0601155i	\(0.980853\pi\)
\(11\)	15.2149			0.417041			0.208521	−	0.978018i	\(-0.433135\pi\)
							0.208521	+	0.978018i	\(0.433135\pi\)
\(13\)	43.1298	+	24.9010i	0.920158	+	0.531253i	0.883685	−	0.468081i	\(-0.155055\pi\)
							0.0364723	+	0.999335i	\(0.488388\pi\)
\(17\)	24.3603	−	14.0644i	0.347543	−	0.200654i	−0.316059	−	0.948739i	\(-0.602360\pi\)
							0.663603	+	0.748085i	\(0.269027\pi\)
\(19\)	−68.6065	−	39.6100i	−0.828390	−	0.478271i	0.0249111	−	0.999690i	\(-0.492070\pi\)
							−0.853301	+	0.521418i	\(0.825403\pi\)
\(23\)		−	119.876i		−	1.08678i	−0.839480	−	0.543390i	\(-0.817141\pi\)
							0.839480	−	0.543390i	\(-0.182859\pi\)
\(29\)		−	162.913i		−	1.04318i	−0.853197	−	0.521589i	\(-0.825340\pi\)
							0.853197	−	0.521589i	\(-0.174660\pi\)
\(31\)			119.304i			0.691215i	0.938379	+	0.345608i	\(0.112327\pi\)
							−0.938379	+	0.345608i	\(0.887673\pi\)
\(37\)	224.549	−	15.1957i	0.997718	−	0.0675178i
							\(41\)	−57.4205	+	99.4552i	−0.218721	+	0.378836i	−0.954417	−	0.298475i	\(-0.903522\pi\)
0.735696	+	0.677312i	\(0.236855\pi\)
\(43\)		−	382.387i		−	1.35613i	−0.735002	−	0.678064i	\(-0.762819\pi\)
							0.735002	−	0.678064i	\(-0.237181\pi\)
\(47\)	504.346			1.56524			0.782622	−	0.622497i	\(-0.213882\pi\)
							0.782622	+	0.622497i	\(0.213882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.11.7	✓	20
3.2	odd	2		666.4.s.d.307.2		20
37.27	even	6	inner	74.4.e.a.27.7	yes	20
111.101	odd	6		666.4.s.d.397.2		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.7	✓	20	1.1	even	1	trivial
74.4.e.a.27.7	yes	20	37.27	even	6	inner
666.4.s.d.307.2		20	3.2	odd	2
666.4.s.d.397.2		20	111.101	odd	6