Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.293450227428\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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: $D_6$
Artin field: Galois closure of 6.0.2420208.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{9} + q^{13} + q^{19} + q^{25} - q^{27} + q^{31} - q^{37} - q^{39} - q^{43} - q^{57} - 2 q^{61} - q^{67} + q^{73} - q^{75} - q^{79} + q^{81} - q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0
0 −1.00000 0 0 0 0 0 1.00000 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 \(\Q(\sqrt{-3}) \)

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 1 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 1 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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