Embedded newform 42.4.f.a.5.8

• Label: 42.4.f.a.5.8
• Level: $42$
• Weight: $4$
• Character: 42.5
• Analytic conductor: $2.478$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	42.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47808022024\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 - x^14 - 2*x^13 + 9*x^12 - 24*x^11 + 714*x^10 - 1940*x^9 - 2834*x^8 - 17460*x^7 + 57834*x^6 - 17496*x^5 + 59049*x^4 - 118098*x^3 - 531441*x^2 - 9565938*x + 43046721
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.8
Root			\(2.99617 + 0.151487i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.5
Dual form			42.4.f.a.17.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(5.18952 - 0.262384i) q^{3} +(2.00000 + 3.46410i) q^{4} +(-2.24534 + 3.88904i) q^{5} +(9.25090 + 4.73506i) q^{6} +(-9.71288 - 15.7690i) q^{7} +8.00000i q^{8} +(26.8623 - 2.72329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 80 q^{7} + 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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
							\(3\)	5.18952	−	0.262384i	0.998724	−	0.0504958i
\(5\)	−2.24534	+	3.88904i	−0.200829	+	0.347846i								−0.948796	−	0.315890i	\(-0.897697\pi\)
							0.747967	+	0.663736i	\(0.231030\pi\)
\(7\)	−9.71288	−	15.7690i	−0.524446	−	0.851443i
							\(11\)	−20.2835	+	11.7107i	−0.555974	+	0.320992i	−0.751528	−	0.659701i	\(-0.770683\pi\)
0.195554	+	0.980693i	\(0.437350\pi\)
\(13\)			5.91384i			0.126170i	0.998008	+	0.0630848i	\(0.0200939\pi\)
							−0.998008	+	0.0630848i	\(0.979906\pi\)
\(17\)	−58.0418	−	100.531i	−0.828071	−	1.43426i	−0.899549	−	0.436819i	\(-0.856105\pi\)
							0.0714778	−	0.997442i	\(-0.477228\pi\)
\(19\)	8.02533	+	4.63343i	0.0969019	+	0.0559464i	0.547668	−	0.836696i	\(-0.315516\pi\)
							−0.450766	+	0.892642i	\(0.648849\pi\)
\(23\)	−107.721	−	62.1928i	−0.976583	−	0.563830i	−0.0753461	−	0.997157i	\(-0.524006\pi\)
							−0.901237	+	0.433327i	\(0.857339\pi\)
\(29\)			207.807i			1.33065i	0.746555	+	0.665324i	\(0.231707\pi\)
							−0.746555	+	0.665324i	\(0.768293\pi\)
\(31\)	122.764	−	70.8780i	0.711262	−	0.410647i	−0.100266	−	0.994961i	\(-0.531970\pi\)
							0.811528	+	0.584314i	\(0.198636\pi\)
\(37\)	−149.838	+	259.526i	−0.665761	+	1.15313i	0.313317	+	0.949648i	\(0.398560\pi\)
							−0.979078	+	0.203483i	\(0.934774\pi\)
\(41\)	508.379			1.93647			0.968237	−	0.250032i	\(-0.0804412\pi\)
							0.968237	+	0.250032i	\(0.0804412\pi\)
\(43\)	391.127			1.38712			0.693562	−	0.720397i	\(-0.256041\pi\)
							0.693562	+	0.720397i	\(0.256041\pi\)
\(47\)	40.2575	−	69.7281i	0.124940	−	0.216402i	−0.796770	−	0.604283i	\(-0.793460\pi\)
							0.921709	+	0.387881i	\(0.126793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.f.a.5.8	yes	16
3.2	odd	2	inner	42.4.f.a.5.3	✓	16
4.3	odd	2		336.4.bc.e.257.1		16
7.2	even	3		294.4.d.a.293.3		16
7.3	odd	6	inner	42.4.f.a.17.3	yes	16
7.4	even	3		294.4.f.a.227.2		16
7.5	odd	6		294.4.d.a.293.6		16
7.6	odd	2		294.4.f.a.215.5		16
12.11	even	2		336.4.bc.e.257.3		16
21.2	odd	6		294.4.d.a.293.14		16
21.5	even	6		294.4.d.a.293.11		16
21.11	odd	6		294.4.f.a.227.5		16
21.17	even	6	inner	42.4.f.a.17.8	yes	16
21.20	even	2		294.4.f.a.215.2		16
28.3	even	6		336.4.bc.e.17.3		16
84.59	odd	6		336.4.bc.e.17.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.f.a.5.3	✓	16	3.2	odd	2	inner
42.4.f.a.5.8	yes	16	1.1	even	1	trivial
42.4.f.a.17.3	yes	16	7.3	odd	6	inner
42.4.f.a.17.8	yes	16	21.17	even	6	inner
294.4.d.a.293.3		16	7.2	even	3
294.4.d.a.293.6		16	7.5	odd	6
294.4.d.a.293.11		16	21.5	even	6
294.4.d.a.293.14		16	21.2	odd	6
294.4.f.a.215.2		16	21.20	even	2
294.4.f.a.215.5		16	7.6	odd	2
294.4.f.a.227.2		16	7.4	even	3
294.4.f.a.227.5		16	21.11	odd	6
336.4.bc.e.17.1		16	84.59	odd	6
336.4.bc.e.17.3		16	28.3	even	6
336.4.bc.e.257.1		16	4.3	odd	2
336.4.bc.e.257.3		16	12.11	even	2