Embedded newform 42.4.f.a.5.6

• Label: 42.4.f.a.5.6
• 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			5.6
Root			\(-0.0204843 + 2.99993i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.5
Dual form			42.4.f.a.17.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{2} +(-0.0354799 - 5.19603i) q^{3} +(2.00000 + 3.46410i) q^{4} +(5.27257 - 9.13236i) q^{5} +(5.13458 - 9.03527i) q^{6} +(17.7029 + 5.44135i) q^{7} +8.00000i q^{8} +(-26.9975 + 0.368709i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 80 q^{7} + 18 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.0354799	−	5.19603i	−0.00682811	−	0.999977i
\(5\)	5.27257	−	9.13236i	0.471593	−	0.816823i								−0.527879	−	0.849320i	\(-0.677012\pi\)
							0.999472	+	0.0324964i	\(0.0103457\pi\)
\(7\)	17.7029	+	5.44135i	0.955865	+	0.293806i
							\(11\)	−26.6918	+	15.4105i	−0.731626	+	0.422405i	−0.819017	−	0.573770i	\(-0.805481\pi\)
0.0873906	+	0.996174i	\(0.472147\pi\)
\(13\)			19.8400i			0.423280i	0.977348	+	0.211640i	\(0.0678804\pi\)
							−0.977348	+	0.211640i	\(0.932120\pi\)
\(17\)	46.3453	+	80.2724i	0.661199	+	1.14523i	0.980301	+	0.197510i	\(0.0632854\pi\)
							−0.319102	+	0.947720i	\(0.603381\pi\)
\(19\)	−118.901	−	68.6474i	−1.43567	−	0.828884i	−0.438125	−	0.898914i	\(-0.644357\pi\)
							−0.997545	+	0.0700296i	\(0.977691\pi\)
\(23\)	−37.6697	−	21.7486i	−0.341508	−	0.197170i	0.319431	−	0.947610i	\(-0.396508\pi\)
							−0.660939	+	0.750440i	\(0.729842\pi\)
\(29\)			134.318i			0.860079i	0.902810	+	0.430039i	\(0.141500\pi\)
							−0.902810	+	0.430039i	\(0.858500\pi\)
\(31\)	144.963	−	83.6945i	0.839876	−	0.484903i	−0.0173460	−	0.999850i	\(-0.505522\pi\)
							0.857222	+	0.514947i	\(0.172188\pi\)
\(37\)	191.747	−	332.115i	0.851972	−	1.47566i	−0.0274532	−	0.999623i	\(-0.508740\pi\)
							0.879426	−	0.476036i	\(-0.157927\pi\)
\(41\)	107.887			0.410954			0.205477	−	0.978662i	\(-0.434125\pi\)
							0.205477	+	0.978662i	\(0.434125\pi\)
\(43\)	−285.480			−1.01245			−0.506225	−	0.862402i	\(-0.668959\pi\)
							−0.506225	+	0.862402i	\(0.668959\pi\)
\(47\)	120.906	−	209.416i	0.375234	−	0.649924i	−0.615128	−	0.788427i	\(-0.710896\pi\)
							0.990362	+	0.138503i	\(0.0442291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.f.a.5.6	yes	16
3.2	odd	2	inner	42.4.f.a.5.4	✓	16
4.3	odd	2		336.4.bc.e.257.5		16
7.2	even	3		294.4.d.a.293.7		16
7.3	odd	6	inner	42.4.f.a.17.4	yes	16
7.4	even	3		294.4.f.a.227.1		16
7.5	odd	6		294.4.d.a.293.2		16
7.6	odd	2		294.4.f.a.215.7		16
12.11	even	2		336.4.bc.e.257.2		16
21.2	odd	6		294.4.d.a.293.10		16
21.5	even	6		294.4.d.a.293.15		16
21.11	odd	6		294.4.f.a.227.7		16
21.17	even	6	inner	42.4.f.a.17.6	yes	16
21.20	even	2		294.4.f.a.215.1		16
28.3	even	6		336.4.bc.e.17.2		16
84.59	odd	6		336.4.bc.e.17.5		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.f.a.5.4	✓	16	3.2	odd	2	inner
42.4.f.a.5.6	yes	16	1.1	even	1	trivial
42.4.f.a.17.4	yes	16	7.3	odd	6	inner
42.4.f.a.17.6	yes	16	21.17	even	6	inner
294.4.d.a.293.2		16	7.5	odd	6
294.4.d.a.293.7		16	7.2	even	3
294.4.d.a.293.10		16	21.2	odd	6
294.4.d.a.293.15		16	21.5	even	6
294.4.f.a.215.1		16	21.20	even	2
294.4.f.a.215.7		16	7.6	odd	2
294.4.f.a.227.1		16	7.4	even	3
294.4.f.a.227.7		16	21.11	odd	6
336.4.bc.e.17.2		16	28.3	even	6
336.4.bc.e.17.5		16	84.59	odd	6
336.4.bc.e.257.2		16	12.11	even	2
336.4.bc.e.257.5		16	4.3	odd	2