Embedded newform 42.4.f.a.5.2

• Label: 42.4.f.a.5.2
• 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			5.2
Root			\(2.30541 + 1.91966i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.5
Dual form			42.4.f.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73205 - 1.00000i) q^{2} +(-3.99309 + 3.32495i) q^{3} +(2.00000 + 3.46410i) q^{4} +(9.90442 - 17.1550i) q^{5} +(10.2412 - 1.76589i) q^{6} +(18.4277 - 1.84901i) q^{7} -8.00000i q^{8} +(4.88947 - 26.5536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 80 q^{7} + 18 q^{9}+O(q^{10}) \)

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
							\(3\)	−3.99309	+	3.32495i	−0.768470	+	0.639886i
\(5\)	9.90442	−	17.1550i	0.885879	−	1.53439i								0.0411754	−	0.999152i	\(-0.486890\pi\)
							0.844703	−	0.535235i	\(-0.179777\pi\)
\(7\)	18.4277	−	1.84901i	0.995004	−	0.0998373i
							\(11\)	4.28742	−	2.47535i	0.117519	−	0.0678495i	−0.440088	−	0.897954i	\(-0.645053\pi\)
0.557607	+	0.830105i	\(0.311720\pi\)
\(13\)		−	17.7414i		−	0.378505i	−0.981928	−	0.189253i	\(-0.939394\pi\)
							0.981928	−	0.189253i	\(-0.0606065\pi\)
\(17\)	0.947671	+	1.64141i	0.0135202	+	0.0234177i	0.872706	−	0.488245i	\(-0.162363\pi\)
							−0.859186	+	0.511663i	\(0.829030\pi\)
\(19\)	83.9968	+	48.4956i	1.01422	+	0.585561i	0.912425	−	0.409245i	\(-0.134208\pi\)
							0.101796	+	0.994805i	\(0.467541\pi\)
\(23\)	−135.935	−	78.4822i	−1.23237	−	0.711508i	−0.264845	−	0.964291i	\(-0.585321\pi\)
							−0.967523	+	0.252783i	\(0.918654\pi\)
\(29\)			92.0138i			0.589191i	0.955622	+	0.294595i	\(0.0951849\pi\)
							−0.955622	+	0.294595i	\(0.904815\pi\)
\(31\)	67.3521	−	38.8857i	0.390219	−	0.225293i	−0.292036	−	0.956407i	\(-0.594333\pi\)
							0.682255	+	0.731114i	\(0.260999\pi\)
\(37\)	−124.469	+	215.587i	−0.553044	+	0.957901i	0.445009	+	0.895526i	\(0.353201\pi\)
							−0.998053	+	0.0623743i	\(0.980133\pi\)
\(41\)	343.701			1.30920			0.654598	−	0.755977i	\(-0.272838\pi\)
							0.654598	+	0.755977i	\(0.272838\pi\)
\(43\)	−24.5859			−0.0871934			−0.0435967	−	0.999049i	\(-0.513882\pi\)
							−0.0435967	+	0.999049i	\(0.513882\pi\)
\(47\)	−235.248	+	407.461i	−0.730093	+	1.26456i	0.226750	+	0.973953i	\(0.427190\pi\)
							−0.956843	+	0.290606i	\(0.906143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.f.a.5.2	✓	16
3.2	odd	2	inner	42.4.f.a.5.5	yes	16
4.3	odd	2		336.4.bc.e.257.6		16
7.2	even	3		294.4.d.a.293.12		16
7.3	odd	6	inner	42.4.f.a.17.5	yes	16
7.4	even	3		294.4.f.a.227.8		16
7.5	odd	6		294.4.d.a.293.13		16
7.6	odd	2		294.4.f.a.215.3		16
12.11	even	2		336.4.bc.e.257.8		16
21.2	odd	6		294.4.d.a.293.5		16
21.5	even	6		294.4.d.a.293.4		16
21.11	odd	6		294.4.f.a.227.3		16
21.17	even	6	inner	42.4.f.a.17.2	yes	16
21.20	even	2		294.4.f.a.215.8		16
28.3	even	6		336.4.bc.e.17.8		16
84.59	odd	6		336.4.bc.e.17.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.f.a.5.2	✓	16	1.1	even	1	trivial
42.4.f.a.5.5	yes	16	3.2	odd	2	inner
42.4.f.a.17.2	yes	16	21.17	even	6	inner
42.4.f.a.17.5	yes	16	7.3	odd	6	inner
294.4.d.a.293.4		16	21.5	even	6
294.4.d.a.293.5		16	21.2	odd	6
294.4.d.a.293.12		16	7.2	even	3
294.4.d.a.293.13		16	7.5	odd	6
294.4.f.a.215.3		16	7.6	odd	2
294.4.f.a.215.8		16	21.20	even	2
294.4.f.a.227.3		16	21.11	odd	6
294.4.f.a.227.8		16	7.4	even	3
336.4.bc.e.17.6		16	84.59	odd	6
336.4.bc.e.17.8		16	28.3	even	6
336.4.bc.e.257.6		16	4.3	odd	2
336.4.bc.e.257.8		16	12.11	even	2