Embedded newform 74.4.e.a.27.10

• Label: 74.4.e.a.27.10
• 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			27.10
Root			\(7.08776i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.27
Dual form			74.4.e.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(3.54388 + 6.13818i) q^{3} +(2.00000 + 3.46410i) q^{4} +(13.1273 - 7.57908i) q^{5} +14.1755i q^{6} +(-5.28959 - 9.16184i) q^{7} +8.00000i q^{8} +(-11.6182 + 20.1233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

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
							\(3\)	3.54388	+	6.13818i	0.682020	+	1.18129i	0.974363	+	0.224980i	\(0.0722317\pi\)
−0.292343	+	0.956313i	\(0.594435\pi\)
\(5\)	13.1273	−	7.57908i	1.17415	−	0.677893i	0.219493	−	0.975614i	\(-0.429560\pi\)
							0.954653	+	0.297721i	\(0.0962264\pi\)
\(7\)	−5.28959	−	9.16184i	−0.285611	−	0.494693i	0.687146	−	0.726519i	\(-0.258863\pi\)
							−0.972757	+	0.231826i	\(0.925530\pi\)
\(11\)	−26.3884			−0.723310			−0.361655	−	0.932312i	\(-0.617788\pi\)
							−0.361655	+	0.932312i	\(0.617788\pi\)
\(13\)	−38.4353	+	22.1906i	−0.820002	+	0.473428i	−0.850417	−	0.526109i	\(-0.823650\pi\)
							0.0304151	+	0.999537i	\(0.490317\pi\)
\(17\)	−65.3877	−	37.7516i	−0.932873	−	0.538595i	−0.0451541	−	0.998980i	\(-0.514378\pi\)
							−0.887719	+	0.460385i	\(0.847711\pi\)
\(19\)	90.0514	−	51.9912i	1.08733	−	0.627768i	0.154463	−	0.987999i	\(-0.450635\pi\)
							0.932863	+	0.360230i	\(0.117302\pi\)
\(23\)		−	43.3057i		−	0.392603i	−0.980544	−	0.196302i	\(-0.937107\pi\)
							0.980544	−	0.196302i	\(-0.0628931\pi\)
\(29\)		−	78.7083i		−	0.503992i	−0.967728	−	0.251996i	\(-0.918913\pi\)
							0.967728	−	0.251996i	\(-0.0810870\pi\)
\(31\)			264.403i			1.53188i	0.642913	+	0.765940i	\(0.277726\pi\)
							−0.642913	+	0.765940i	\(0.722274\pi\)
\(37\)	−85.8315	−	208.053i	−0.381368	−	0.924423i
							\(41\)	−89.5746	−	155.148i	−0.341200	−	0.590976i	0.643456	−	0.765483i	\(-0.277500\pi\)
−0.984656	+	0.174508i	\(0.944167\pi\)
\(43\)			208.041i			0.737813i	0.929466	+	0.368907i	\(0.120268\pi\)
							−0.929466	+	0.368907i	\(0.879732\pi\)
\(47\)	529.657			1.64379			0.821897	−	0.569636i	\(-0.192916\pi\)
							0.821897	+	0.569636i	\(0.192916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.27.10	yes	20
3.2	odd	2		666.4.s.d.397.1		20
37.11	even	6	inner	74.4.e.a.11.10	✓	20
111.11	odd	6		666.4.s.d.307.1		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.10	✓	20	37.11	even	6	inner
74.4.e.a.27.10	yes	20	1.1	even	1	trivial
666.4.s.d.307.1		20	111.11	odd	6
666.4.s.d.397.1		20	3.2	odd	2