Embedded newform 74.4.e.a.27.9

• Label: 74.4.e.a.27.9
• 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			27.9
Root			\(5.89146i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.27
Dual form			74.4.e.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(2.94573 + 5.10216i) q^{3} +(2.00000 + 3.46410i) q^{4} +(-12.2374 + 7.06527i) q^{5} +11.7829i q^{6} +(0.0476865 + 0.0825955i) q^{7} +8.00000i q^{8} +(-3.85468 + 6.67649i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 40 q^{4} - 18 q^{5} - 2 q^{7} - 76 q^{9}+O(q^{10}) \)

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
							\(3\)	2.94573	+	5.10216i	0.566906	+	0.981911i	0.996870	+	0.0790643i	\(0.0251932\pi\)
−0.429963	+	0.902846i	\(0.641473\pi\)
\(5\)	−12.2374	+	7.06527i	−1.09455	+	0.631937i	−0.934784	−	0.355218i	\(-0.884407\pi\)
							−0.159764	+	0.987155i	\(0.551073\pi\)
\(7\)	0.0476865	+	0.0825955i	0.00257483	+	0.00445974i	0.867310	−	0.497768i	\(-0.165847\pi\)
							−0.864735	+	0.502228i	\(0.832514\pi\)
\(11\)	1.95565			0.0536047			0.0268023	−	0.999641i	\(-0.491468\pi\)
							0.0268023	+	0.999641i	\(0.491468\pi\)
\(13\)	48.4136	−	27.9516i	1.03289	−	0.596337i	0.115076	−	0.993357i	\(-0.463289\pi\)
							0.917810	+	0.397020i	\(0.129956\pi\)
\(17\)	94.2875	+	54.4369i	1.34518	+	0.776640i	0.987562	−	0.157228i	\(-0.0502557\pi\)
							0.357618	+	0.933868i	\(0.383589\pi\)
\(19\)	−16.5704	+	9.56693i	−0.200080	+	0.115516i	−0.596693	−	0.802470i	\(-0.703519\pi\)
							0.396613	+	0.917986i	\(0.370186\pi\)
\(23\)			112.313i			1.01821i	0.860704	+	0.509105i	\(0.170024\pi\)
							−0.860704	+	0.509105i	\(0.829976\pi\)
\(29\)		−	166.296i		−	1.06484i	−0.846480	−	0.532421i	\(-0.821282\pi\)
							0.846480	−	0.532421i	\(-0.178718\pi\)
\(31\)		−	237.659i		−	1.37693i	−0.725270	−	0.688465i	\(-0.758285\pi\)
							0.725270	−	0.688465i	\(-0.241715\pi\)
\(37\)	−203.978	−	95.1113i	−0.906316	−	0.422600i
							\(41\)	−79.1501	−	137.092i	−0.301492	−	0.522199i	0.674982	−	0.737834i	\(-0.264151\pi\)
−0.976474	+	0.215635i	\(0.930818\pi\)
\(43\)		−	72.7216i		−	0.257905i	−0.991651	−	0.128953i	\(-0.958838\pi\)
							0.991651	−	0.128953i	\(-0.0411616\pi\)
\(47\)	−0.209175			−0.000649175			−0.000324588	−	1.00000i	\(-0.500103\pi\)
							−0.000324588		1.00000i	\(0.500103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.e.a.27.9	yes	20
3.2	odd	2		666.4.s.d.397.4		20
37.11	even	6	inner	74.4.e.a.11.9	✓	20
111.11	odd	6		666.4.s.d.307.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.e.a.11.9	✓	20	37.11	even	6	inner
74.4.e.a.27.9	yes	20	1.1	even	1	trivial
666.4.s.d.307.4		20	111.11	odd	6
666.4.s.d.397.4		20	3.2	odd	2