Properties

Label 588.1.p.a
Level $588$
Weight $1$
Character orbit 588.p
Analytic conductor $0.293$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 588.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.293450227428\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.588.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.1075850221824.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{9} + q^{13} -\zeta_{6} q^{19} + \zeta_{6}^{2} q^{25} - q^{27} + \zeta_{6}^{2} q^{31} + \zeta_{6} q^{37} -\zeta_{6}^{2} q^{39} - q^{43} - q^{57} + 2 \zeta_{6} q^{61} -\zeta_{6}^{2} q^{67} + \zeta_{6}^{2} q^{73} + \zeta_{6} q^{75} + \zeta_{6} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{93} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{9} + 2q^{13} - q^{19} - q^{25} - 2q^{27} - q^{31} + q^{37} + q^{39} - 2q^{43} - 2q^{57} + 2q^{61} + q^{67} - q^{73} + q^{75} + q^{79} - q^{81} + q^{93} - 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
569.1 0 0.500000 + 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.1.p.a 2
3.b odd 2 1 CM 588.1.p.a 2
4.b odd 2 1 2352.1.bn.a 2
7.b odd 2 1 84.1.p.a 2
7.c even 3 1 588.1.c.a 1
7.c even 3 1 inner 588.1.p.a 2
7.d odd 6 1 84.1.p.a 2
7.d odd 6 1 588.1.c.b 1
12.b even 2 1 2352.1.bn.a 2
21.c even 2 1 84.1.p.a 2
21.g even 6 1 84.1.p.a 2
21.g even 6 1 588.1.c.b 1
21.h odd 6 1 588.1.c.a 1
21.h odd 6 1 inner 588.1.p.a 2
28.d even 2 1 336.1.bn.a 2
28.f even 6 1 336.1.bn.a 2
28.f even 6 1 2352.1.d.a 1
28.g odd 6 1 2352.1.d.b 1
28.g odd 6 1 2352.1.bn.a 2
35.c odd 2 1 2100.1.bn.c 2
35.f even 4 2 2100.1.bh.a 4
35.i odd 6 1 2100.1.bn.c 2
35.k even 12 2 2100.1.bh.a 4
56.e even 2 1 1344.1.bn.a 2
56.h odd 2 1 1344.1.bn.b 2
56.j odd 6 1 1344.1.bn.b 2
56.m even 6 1 1344.1.bn.a 2
63.i even 6 1 2268.1.m.a 2
63.k odd 6 1 2268.1.bh.b 2
63.l odd 6 1 2268.1.m.a 2
63.l odd 6 1 2268.1.bh.b 2
63.o even 6 1 2268.1.m.a 2
63.o even 6 1 2268.1.bh.b 2
63.s even 6 1 2268.1.bh.b 2
63.t odd 6 1 2268.1.m.a 2
84.h odd 2 1 336.1.bn.a 2
84.j odd 6 1 336.1.bn.a 2
84.j odd 6 1 2352.1.d.a 1
84.n even 6 1 2352.1.d.b 1
84.n even 6 1 2352.1.bn.a 2
105.g even 2 1 2100.1.bn.c 2
105.k odd 4 2 2100.1.bh.a 4
105.p even 6 1 2100.1.bn.c 2
105.w odd 12 2 2100.1.bh.a 4
168.e odd 2 1 1344.1.bn.a 2
168.i even 2 1 1344.1.bn.b 2
168.ba even 6 1 1344.1.bn.b 2
168.be odd 6 1 1344.1.bn.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 7.b odd 2 1
84.1.p.a 2 7.d odd 6 1
84.1.p.a 2 21.c even 2 1
84.1.p.a 2 21.g even 6 1
336.1.bn.a 2 28.d even 2 1
336.1.bn.a 2 28.f even 6 1
336.1.bn.a 2 84.h odd 2 1
336.1.bn.a 2 84.j odd 6 1
588.1.c.a 1 7.c even 3 1
588.1.c.a 1 21.h odd 6 1
588.1.c.b 1 7.d odd 6 1
588.1.c.b 1 21.g even 6 1
588.1.p.a 2 1.a even 1 1 trivial
588.1.p.a 2 3.b odd 2 1 CM
588.1.p.a 2 7.c even 3 1 inner
588.1.p.a 2 21.h odd 6 1 inner
1344.1.bn.a 2 56.e even 2 1
1344.1.bn.a 2 56.m even 6 1
1344.1.bn.a 2 168.e odd 2 1
1344.1.bn.a 2 168.be odd 6 1
1344.1.bn.b 2 56.h odd 2 1
1344.1.bn.b 2 56.j odd 6 1
1344.1.bn.b 2 168.i even 2 1
1344.1.bn.b 2 168.ba even 6 1
2100.1.bh.a 4 35.f even 4 2
2100.1.bh.a 4 35.k even 12 2
2100.1.bh.a 4 105.k odd 4 2
2100.1.bh.a 4 105.w odd 12 2
2100.1.bn.c 2 35.c odd 2 1
2100.1.bn.c 2 35.i odd 6 1
2100.1.bn.c 2 105.g even 2 1
2100.1.bn.c 2 105.p even 6 1
2268.1.m.a 2 63.i even 6 1
2268.1.m.a 2 63.l odd 6 1
2268.1.m.a 2 63.o even 6 1
2268.1.m.a 2 63.t odd 6 1
2268.1.bh.b 2 63.k odd 6 1
2268.1.bh.b 2 63.l odd 6 1
2268.1.bh.b 2 63.o even 6 1
2268.1.bh.b 2 63.s even 6 1
2352.1.d.a 1 28.f even 6 1
2352.1.d.a 1 84.j odd 6 1
2352.1.d.b 1 28.g odd 6 1
2352.1.d.b 1 84.n even 6 1
2352.1.bn.a 2 4.b odd 2 1
2352.1.bn.a 2 12.b even 2 1
2352.1.bn.a 2 28.g odd 6 1
2352.1.bn.a 2 84.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1 + T + T^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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