Embedded newform 74.4.c.a.63.4

• Label: 74.4.c.a.63.4
• Level: $74$
• Weight: $4$
• Character: 74.63
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - x^{7} + 48x^{6} + 157x^{5} + 1944x^{4} + 3005x^{3} + 12895x^{2} - 11550x + 44100 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			63.4
Root			\(-2.17076 + 3.75987i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.63
Dual form			74.4.c.a.47.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(1.67076 - 2.89385i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(6.09877 - 10.5634i) q^{5} +6.68306 q^{6} +(9.86474 - 17.0862i) q^{7} -8.00000 q^{8} +(7.91709 + 13.7128i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 5 q^{3} - 16 q^{4} - 10 q^{5} - 20 q^{6} + 3 q^{7} - 64 q^{8} + 7 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	+	1.73205i	0.353553	+	0.612372i
							\(3\)	1.67076	−	2.89385i	0.321539	−	0.556921i	−0.659267	−	0.751909i	\(-0.729133\pi\)
0.980806	+	0.194987i	\(0.0624666\pi\)
\(5\)	6.09877	−	10.5634i	0.545491	−	0.944818i	−0.453085	−	0.891467i	\(-0.649677\pi\)
							0.998576	−	0.0533506i	\(-0.0169901\pi\)
\(7\)	9.86474	−	17.0862i	0.532646	−	0.922570i	−0.466627	−	0.884454i	\(-0.654531\pi\)
							0.999273	−	0.0381158i	\(-0.0121356\pi\)
\(11\)	14.7179			0.403421			0.201710	−	0.979445i	\(-0.435350\pi\)
							0.201710	+	0.979445i	\(0.435350\pi\)
\(13\)	−2.26239	+	3.91858i	−0.0482673	+	0.0836014i	−0.889150	−	0.457617i	\(-0.848703\pi\)
							0.840882	+	0.541218i	\(0.182037\pi\)
\(17\)	3.65552	+	6.33154i	0.0521526	+	0.0903309i	0.890923	−	0.454154i	\(-0.150058\pi\)
							−0.838771	+	0.544485i	\(0.816725\pi\)
\(19\)	−28.3160	+	49.0447i	−0.341901	+	0.592191i	−0.984786	−	0.173773i	\(-0.944404\pi\)
							0.642884	+	0.765963i	\(0.277738\pi\)
\(23\)	−7.97811			−0.0723283			−0.0361642	−	0.999346i	\(-0.511514\pi\)
							−0.0361642	+	0.999346i	\(0.511514\pi\)
\(29\)	−199.789			−1.27930			−0.639652	−	0.768665i	\(-0.720922\pi\)
							−0.639652	+	0.768665i	\(0.720922\pi\)
\(31\)	−155.473			−0.900765			−0.450382	−	0.892836i	\(-0.648712\pi\)
							−0.450382	+	0.892836i	\(0.648712\pi\)
\(37\)	−225.027	−	3.97088i	−0.999844	−	0.0176435i
							\(41\)	−74.3517	+	128.781i	−0.283214	+	0.490541i	−0.972175	−	0.234258i	\(-0.924734\pi\)
0.688960	+	0.724799i	\(0.258067\pi\)
\(43\)	212.880			0.754973			0.377486	−	0.926015i	\(-0.376789\pi\)
							0.377486	+	0.926015i	\(0.376789\pi\)
\(47\)	−96.7752			−0.300343			−0.150171	−	0.988660i	\(-0.547983\pi\)
							−0.150171	+	0.988660i	\(0.547983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.c.a.63.4	yes	8
3.2	odd	2		666.4.f.a.433.1		8
37.10	even	3	inner	74.4.c.a.47.4	✓	8
111.47	odd	6		666.4.f.a.343.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.c.a.47.4	✓	8	37.10	even	3	inner
74.4.c.a.63.4	yes	8	1.1	even	1	trivial
666.4.f.a.343.1		8	111.47	odd	6
666.4.f.a.433.1		8	3.2	odd	2