Embedded newform 42.4.e.c.25.1

• Label: 42.4.e.c.25.1
• Level: $42$
• Weight: $4$
• Character: 42.25
• Analytic conductor: $2.478$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47808022024\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{1345})\)
Defining polynomial:	\( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.1
Root			\(-8.91856 - 15.4474i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.25
Dual form			42.4.e.c.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-10.4186 - 18.0455i) q^{5} +6.00000 q^{6} +(-18.3371 - 2.59808i) q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 5 q^{5} + 24 q^{6} + 32 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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
							\(3\)	−1.50000	+	2.59808i	−0.288675	+	0.500000i
\(5\)	−10.4186	−	18.0455i	−0.931864	−	1.61404i								−0.780132	−	0.625615i	\(-0.784848\pi\)
							−0.151732	−	0.988422i	\(-0.548485\pi\)
\(7\)	−18.3371	−	2.59808i	−0.990111	−	0.140283i
							\(11\)	−7.58144	+	13.1314i	−0.207808	+	0.359934i	−0.951024	−	0.309118i	\(-0.899966\pi\)
0.743216	+	0.669052i	\(0.233300\pi\)
\(13\)	2.16288			0.0461442			0.0230721	−	0.999734i	\(-0.492655\pi\)
							0.0230721	+	0.999734i	\(0.492655\pi\)
\(17\)	59.6742	−	103.359i	0.851361	−	1.47460i	−0.0286202	−	0.999590i	\(-0.509111\pi\)
							0.879981	−	0.475009i	\(-0.157555\pi\)
\(19\)	16.7557	+	29.0217i	0.202317	+	0.350423i	0.949274	−	0.314449i	\(-0.101820\pi\)
							−0.746958	+	0.664871i	\(0.768486\pi\)
\(23\)	−0.325758	−	0.564230i	−0.00295327	−	0.00511522i	0.864545	−	0.502555i	\(-0.167607\pi\)
							−0.867498	+	0.497440i	\(0.834273\pi\)
\(29\)	−163.208			−1.04507			−0.522535	−	0.852618i	\(-0.675014\pi\)
							−0.522535	+	0.852618i	\(0.675014\pi\)
\(31\)	111.663	−	193.406i	0.646943	−	1.12054i	−0.336906	−	0.941538i	\(-0.609380\pi\)
							0.983849	−	0.179000i	\(-0.0572863\pi\)
\(37\)	−84.2670	−	145.955i	−0.374417	−	0.648509i	0.615823	−	0.787885i	\(-0.288824\pi\)
							−0.990240	+	0.139376i	\(0.955490\pi\)
\(41\)	−323.023			−1.23043			−0.615216	−	0.788359i	\(-0.710931\pi\)
							−0.615216	+	0.788359i	\(0.710931\pi\)
\(43\)	221.557			0.785746			0.392873	−	0.919593i	\(-0.371481\pi\)
							0.392873	+	0.919593i	\(0.371481\pi\)
\(47\)	−254.023	−	439.980i	−0.788362	−	1.36548i	−0.926970	−	0.375136i	\(-0.877596\pi\)
							0.138608	−	0.990347i	\(-0.455737\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.e.c.25.1	✓	4
3.2	odd	2		126.4.g.g.109.2		4
4.3	odd	2		336.4.q.j.193.1		4
7.2	even	3	inner	42.4.e.c.37.1	yes	4
7.3	odd	6		294.4.a.m.1.1		2
7.4	even	3		294.4.a.n.1.2		2
7.5	odd	6		294.4.e.l.79.2		4
7.6	odd	2		294.4.e.l.67.2		4
21.2	odd	6		126.4.g.g.37.2		4
21.5	even	6		882.4.g.bf.667.1		4
21.11	odd	6		882.4.a.v.1.1		2
21.17	even	6		882.4.a.z.1.2		2
21.20	even	2		882.4.g.bf.361.1		4
28.3	even	6		2352.4.a.ca.1.1		2
28.11	odd	6		2352.4.a.bq.1.2		2
28.23	odd	6		336.4.q.j.289.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.c.25.1	✓	4	1.1	even	1	trivial
42.4.e.c.37.1	yes	4	7.2	even	3	inner
126.4.g.g.37.2		4	21.2	odd	6
126.4.g.g.109.2		4	3.2	odd	2
294.4.a.m.1.1		2	7.3	odd	6
294.4.a.n.1.2		2	7.4	even	3
294.4.e.l.67.2		4	7.6	odd	2
294.4.e.l.79.2		4	7.5	odd	6
336.4.q.j.193.1		4	4.3	odd	2
336.4.q.j.289.1		4	28.23	odd	6
882.4.a.v.1.1		2	21.11	odd	6
882.4.a.z.1.2		2	21.17	even	6
882.4.g.bf.361.1		4	21.20	even	2
882.4.g.bf.667.1		4	21.5	even	6
2352.4.a.bq.1.2		2	28.11	odd	6
2352.4.a.ca.1.1		2	28.3	even	6