Embedded newform 42.4.f.a.5.7

• Label: 42.4.f.a.5.7
• 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			5.7
Root			\(0.339489 - 2.98073i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.5
Dual form			42.4.f.a.17.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(0.588012 + 5.16277i) q^{3} +(2.00000 + 3.46410i) q^{4} +(4.27911 - 7.41164i) q^{5} +(-4.14431 + 9.53020i) q^{6} +(-6.41772 + 17.3728i) q^{7} +8.00000i q^{8} +(-26.3085 + 6.07155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 80 q^{7} + 18 q^{9}+O(q^{10}) \)

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
							\(3\)	0.588012	+	5.16277i	0.113163	+	0.993576i
\(5\)	4.27911	−	7.41164i	0.382736	−	0.662917i								−0.608717	−	0.793388i	\(-0.708315\pi\)
							0.991452	+	0.130470i	\(0.0416487\pi\)
\(7\)	−6.41772	+	17.3728i	−0.346524	+	0.938041i
							\(11\)	53.8609	−	31.0966i	1.47633	−	0.852361i	0.476690	−	0.879072i	\(-0.341837\pi\)
0.999643	+	0.0267105i	\(0.00850323\pi\)
\(13\)		−	61.7061i		−	1.31648i	−0.752810	−	0.658238i	\(-0.771302\pi\)
							0.752810	−	0.658238i	\(-0.228698\pi\)
\(17\)	−13.3365	−	23.0995i	−0.190270	−	0.329557i	0.755070	−	0.655644i	\(-0.227603\pi\)
							−0.945340	+	0.326088i	\(0.894270\pi\)
\(19\)	−58.6213	−	33.8450i	−0.707824	−	0.408662i	0.102431	−	0.994740i	\(-0.467338\pi\)
							−0.810255	+	0.586078i	\(0.800671\pi\)
\(23\)	45.8287	+	26.4592i	0.415476	+	0.239875i	0.693140	−	0.720803i	\(-0.256227\pi\)
							−0.277664	+	0.960678i	\(0.589560\pi\)
\(29\)		−	55.1116i		−	0.352895i	−0.984310	−	0.176448i	\(-0.943539\pi\)
							0.984310	−	0.176448i	\(-0.0564606\pi\)
\(31\)	−134.080	+	77.4108i	−0.776819	+	0.448497i	−0.835302	−	0.549792i	\(-0.814707\pi\)
							0.0584827	+	0.998288i	\(0.481374\pi\)
\(37\)	−157.940	+	273.560i	−0.701761	+	1.21549i	0.266087	+	0.963949i	\(0.414269\pi\)
							−0.967848	+	0.251536i	\(0.919064\pi\)
\(41\)	−210.211			−0.800719			−0.400360	−	0.916358i	\(-0.631115\pi\)
							−0.400360	+	0.916358i	\(0.631115\pi\)
\(43\)	351.939			1.24815			0.624073	−	0.781366i	\(-0.285477\pi\)
							0.624073	+	0.781366i	\(0.285477\pi\)
\(47\)	−115.819	+	200.605i	−0.359446	+	0.622579i	−0.987868	−	0.155294i	\(-0.950368\pi\)
							0.628422	+	0.777872i	\(0.283701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.f.a.5.7	yes	16
3.2	odd	2	inner	42.4.f.a.5.1	✓	16
4.3	odd	2		336.4.bc.e.257.4		16
7.2	even	3		294.4.d.a.293.1		16
7.3	odd	6	inner	42.4.f.a.17.1	yes	16
7.4	even	3		294.4.f.a.227.4		16
7.5	odd	6		294.4.d.a.293.8		16
7.6	odd	2		294.4.f.a.215.6		16
12.11	even	2		336.4.bc.e.257.7		16
21.2	odd	6		294.4.d.a.293.16		16
21.5	even	6		294.4.d.a.293.9		16
21.11	odd	6		294.4.f.a.227.6		16
21.17	even	6	inner	42.4.f.a.17.7	yes	16
21.20	even	2		294.4.f.a.215.4		16
28.3	even	6		336.4.bc.e.17.7		16
84.59	odd	6		336.4.bc.e.17.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.f.a.5.1	✓	16	3.2	odd	2	inner
42.4.f.a.5.7	yes	16	1.1	even	1	trivial
42.4.f.a.17.1	yes	16	7.3	odd	6	inner
42.4.f.a.17.7	yes	16	21.17	even	6	inner
294.4.d.a.293.1		16	7.2	even	3
294.4.d.a.293.8		16	7.5	odd	6
294.4.d.a.293.9		16	21.5	even	6
294.4.d.a.293.16		16	21.2	odd	6
294.4.f.a.215.4		16	21.20	even	2
294.4.f.a.215.6		16	7.6	odd	2
294.4.f.a.227.4		16	7.4	even	3
294.4.f.a.227.6		16	21.11	odd	6
336.4.bc.e.17.4		16	84.59	odd	6
336.4.bc.e.17.7		16	28.3	even	6
336.4.bc.e.257.4		16	4.3	odd	2
336.4.bc.e.257.7		16	12.11	even	2