Newform orbit 588.4.i.e

• Label: 588.4.i.e
• Level: $588$
• Weight: $4$
• Character orbit: 588.i
• Analytic conductor: $34.693$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(34.6931230834\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 3 \zeta_{6} + 3) q^{3} - 18 \zeta_{6} q^{5} - 9 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 18 q^{5} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

1.50000

−

2.59808i

0

−9.00000

−

15.5885i

0

0

0

−4.50000

−

7.79423i

0

373.1

0

1.50000

+

2.59808i

0

−9.00000

+

15.5885i

0

0

0

−4.50000

+

7.79423i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.4.i.e		2
3.b	odd	2	1		1764.4.k.o		2
7.b	odd	2	1		588.4.i.d		2
7.c	even	3	1		588.4.a.c		1
7.c	even	3	1	inner	588.4.i.e		2
7.d	odd	6	1		12.4.a.a	✓	1
7.d	odd	6	1		588.4.i.d		2
21.c	even	2	1		1764.4.k.b		2
21.g	even	6	1		36.4.a.a		1
21.g	even	6	1		1764.4.k.b		2
21.h	odd	6	1		1764.4.a.b		1
21.h	odd	6	1		1764.4.k.o		2
28.f	even	6	1		48.4.a.a		1
28.g	odd	6	1		2352.4.a.bk		1
35.i	odd	6	1		300.4.a.b		1
35.k	even	12	2		300.4.d.e		2
56.j	odd	6	1		192.4.a.f		1
56.m	even	6	1		192.4.a.l		1
63.i	even	6	1		324.4.e.a		2
63.k	odd	6	1		324.4.e.h		2
63.s	even	6	1		324.4.e.a		2
63.t	odd	6	1		324.4.e.h		2
77.i	even	6	1		1452.4.a.d		1
84.j	odd	6	1		144.4.a.g		1
91.s	odd	6	1		2028.4.a.c		1
91.bb	even	12	2		2028.4.b.c		2
105.p	even	6	1		900.4.a.g		1
105.w	odd	12	2		900.4.d.c		2
112.v	even	12	2		768.4.d.j		2
112.x	odd	12	2		768.4.d.g		2
140.s	even	6	1		1200.4.a.be		1
140.x	odd	12	2		1200.4.f.d		2
168.ba	even	6	1		576.4.a.b		1
168.be	odd	6	1		576.4.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.4.a.a	✓	1	7.d	odd	6	1
36.4.a.a		1	21.g	even	6	1
48.4.a.a		1	28.f	even	6	1
144.4.a.g		1	84.j	odd	6	1
192.4.a.f		1	56.j	odd	6	1
192.4.a.l		1	56.m	even	6	1
300.4.a.b		1	35.i	odd	6	1
300.4.d.e		2	35.k	even	12	2
324.4.e.a		2	63.i	even	6	1
324.4.e.a		2	63.s	even	6	1
324.4.e.h		2	63.k	odd	6	1
324.4.e.h		2	63.t	odd	6	1
576.4.a.a		1	168.be	odd	6	1
576.4.a.b		1	168.ba	even	6	1
588.4.a.c		1	7.c	even	3	1
588.4.i.d		2	7.b	odd	2	1
588.4.i.d		2	7.d	odd	6	1
588.4.i.e		2	1.a	even	1	1	trivial
588.4.i.e		2	7.c	even	3	1	inner
768.4.d.g		2	112.x	odd	12	2
768.4.d.j		2	112.v	even	12	2
900.4.a.g		1	105.p	even	6	1
900.4.d.c		2	105.w	odd	12	2
1200.4.a.be		1	140.s	even	6	1
1200.4.f.d		2	140.x	odd	12	2
1452.4.a.d		1	77.i	even	6	1
1764.4.a.b		1	21.h	odd	6	1
1764.4.k.b		2	21.c	even	2	1
1764.4.k.b		2	21.g	even	6	1
1764.4.k.o		2	3.b	odd	2	1
1764.4.k.o		2	21.h	odd	6	1
2028.4.a.c		1	91.s	odd	6	1
2028.4.b.c		2	91.bb	even	12	2
2352.4.a.bk		1	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 18T_{5} + 324 \) acting on \(S_{4}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 3T + 9 \)
$5$	\( T^{2} + 18T + 324 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 36T + 1296 \)