Embedded newform 74.4.e.a.11.4

• Label: 74.4.e.a.11.4
• 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			11.4
Root			\(3.11082i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.11
Dual form			74.4.e.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(1.55541 - 2.69405i) q^{3} +(2.00000 - 3.46410i) q^{4} +(-12.2401 - 7.06683i) q^{5} +6.22164i q^{6} +(-13.5058 + 23.3927i) q^{7} +8.00000i q^{8} +(8.66140 + 15.0020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

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
							\(3\)	1.55541	−	2.69405i	0.299339	−	0.518470i	−0.676646	−	0.736308i	\(-0.736567\pi\)
0.975985	+	0.217839i	\(0.0699006\pi\)
\(5\)	−12.2401	−	7.06683i	−1.09479	−	0.632077i	−0.159941	−	0.987127i	\(-0.551131\pi\)
							−0.934847	+	0.355050i	\(0.884464\pi\)
\(7\)	−13.5058	+	23.3927i	−0.729244	+	1.26309i	0.227959	+	0.973671i	\(0.426795\pi\)
							−0.957203	+	0.289417i	\(0.906539\pi\)
\(11\)	−40.4701			−1.10929			−0.554645	−	0.832087i	\(-0.687146\pi\)
							−0.554645	+	0.832087i	\(0.687146\pi\)
\(13\)	−12.6060	−	7.27806i	−0.268943	−	0.155275i	0.359464	−	0.933159i	\(-0.382960\pi\)
							−0.628407	+	0.777884i	\(0.716293\pi\)
\(17\)	−56.7472	+	32.7630i	−0.809601	+	0.467423i	−0.846817	−	0.531884i	\(-0.821484\pi\)
							0.0372163	+	0.999307i	\(0.488151\pi\)
\(19\)	−39.7468	−	22.9478i	−0.479923	−	0.277084i	0.240461	−	0.970659i	\(-0.422701\pi\)
							−0.720384	+	0.693575i	\(0.756035\pi\)
\(23\)		−	57.6414i		−	0.522568i	−0.965262	−	0.261284i	\(-0.915854\pi\)
							0.965262	−	0.261284i	\(-0.0841460\pi\)
\(29\)		−	224.796i		−	1.43943i	−0.694269	−	0.719716i	\(-0.744272\pi\)
							0.694269	−	0.719716i	\(-0.255728\pi\)
\(31\)			89.6760i			0.519558i	0.965668	+	0.259779i	\(0.0836496\pi\)
							−0.965668	+	0.259779i	\(0.916350\pi\)
\(37\)	108.954	+	196.931i	0.484107	+	0.875009i
							\(41\)	−36.5413	+	63.2914i	−0.139190	+	0.241084i	−0.927190	−	0.374591i	\(-0.877783\pi\)
0.788000	+	0.615675i	\(0.211117\pi\)
\(43\)			333.173i			1.18159i	0.806821	+	0.590796i	\(0.201186\pi\)
							−0.806821	+	0.590796i	\(0.798814\pi\)
\(47\)	−291.705			−0.905311			−0.452655	−	0.891686i	\(-0.649523\pi\)
							−0.452655	+	0.891686i	\(0.649523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.11.4	✓	20
3.2	odd	2		666.4.s.d.307.9		20
37.27	even	6	inner	74.4.e.a.27.4	yes	20
111.101	odd	6		666.4.s.d.397.9		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.4	✓	20	1.1	even	1	trivial
74.4.e.a.27.4	yes	20	37.27	even	6	inner
666.4.s.d.307.9		20	3.2	odd	2
666.4.s.d.397.9		20	111.101	odd	6