Embedded newform 74.4.e.a.27.6

• Label: 74.4.e.a.27.6
• 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			27.6
Root			\(-7.81115i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.27
Dual form			74.4.e.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(-3.90558 - 6.76466i) q^{3} +(2.00000 + 3.46410i) q^{4} +(-18.8983 + 10.9109i) q^{5} -15.6223i q^{6} +(-2.61919 - 4.53657i) q^{7} +8.00000i q^{8} +(-17.0071 + 29.4571i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

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
							\(3\)	−3.90558	−	6.76466i	−0.751629	−	1.30186i	−0.947033	−	0.321136i	\(-0.895935\pi\)
0.195404	−	0.980723i	\(-0.437398\pi\)
\(5\)	−18.8983	+	10.9109i	−1.69031	+	0.975903i	−0.736055	+	0.676921i	\(0.763314\pi\)
							−0.954259	+	0.298982i	\(0.903353\pi\)
\(7\)	−2.61919	−	4.53657i	−0.141423	−	0.244952i	0.786610	−	0.617450i	\(-0.211834\pi\)
							−0.928033	+	0.372499i	\(0.878501\pi\)
\(11\)	−25.5382			−0.700006			−0.350003	−	0.936749i	\(-0.613819\pi\)
							−0.350003	+	0.936749i	\(0.613819\pi\)
\(13\)	−12.0397	+	6.95110i	−0.256861	+	0.148299i	−0.622902	−	0.782300i	\(-0.714046\pi\)
							0.366041	+	0.930599i	\(0.380713\pi\)
\(17\)	−83.9431	−	48.4646i	−1.19760	−	0.691434i	−0.237580	−	0.971368i	\(-0.576354\pi\)
							−0.960019	+	0.279934i	\(0.909687\pi\)
\(19\)	83.0586	−	47.9539i	1.00289	−	0.579020i	0.0937890	−	0.995592i	\(-0.470102\pi\)
							0.909103	+	0.416572i	\(0.136769\pi\)
\(23\)		−	90.5468i		−	0.820883i	−0.911887	−	0.410441i	\(-0.865375\pi\)
							0.911887	−	0.410441i	\(-0.134625\pi\)
\(29\)			264.487i			1.69359i	0.531923	+	0.846793i	\(0.321470\pi\)
							−0.531923	+	0.846793i	\(0.678530\pi\)
\(31\)			75.6990i			0.438579i	0.975660	+	0.219289i	\(0.0703739\pi\)
							−0.975660	+	0.219289i	\(0.929626\pi\)
\(37\)	−167.022	+	150.853i	−0.742115	+	0.670272i
							\(41\)	−74.8270	−	129.604i	−0.285025	−	0.493677i	0.687590	−	0.726099i	\(-0.258669\pi\)
−0.972615	+	0.232421i	\(0.925335\pi\)
\(43\)			480.395i			1.70371i	0.523778	+	0.851855i	\(0.324522\pi\)
							−0.523778	+	0.851855i	\(0.675478\pi\)
\(47\)	−413.725			−1.28400			−0.642001	−	0.766704i	\(-0.721895\pi\)
							−0.642001	+	0.766704i	\(0.721895\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.27.6	yes	20
3.2	odd	2		666.4.s.d.397.5		20
37.11	even	6	inner	74.4.e.a.11.6	✓	20
111.11	odd	6		666.4.s.d.307.5		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.6	✓	20	37.11	even	6	inner
74.4.e.a.27.6	yes	20	1.1	even	1	trivial
666.4.s.d.307.5		20	111.11	odd	6
666.4.s.d.397.5		20	3.2	odd	2