Properties

Label 588.3.c.c
Level $588$
Weight $3$
Character orbit 588.c
Self dual yes
Analytic conductor $16.022$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 9q^{9} + 22q^{13} - 26q^{19} + 25q^{25} + 27q^{27} + 46q^{31} + 26q^{37} + 66q^{39} - 22q^{43} - 78q^{57} - 74q^{61} + 122q^{67} + 46q^{73} + 75q^{75} - 142q^{79} + 81q^{81} + 138q^{93} - 2q^{97} + O(q^{100}) \)

Character values

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

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

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 3.00000 0 0 0 0 0 9.00000 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 588.3.c.c 1
3.b odd 2 1 CM 588.3.c.c 1
7.b odd 2 1 12.3.c.a 1
7.c even 3 2 588.3.p.b 2
7.d odd 6 2 588.3.p.c 2
21.c even 2 1 12.3.c.a 1
21.g even 6 2 588.3.p.c 2
21.h odd 6 2 588.3.p.b 2
28.d even 2 1 48.3.e.a 1
35.c odd 2 1 300.3.g.b 1
35.f even 4 2 300.3.b.a 2
56.e even 2 1 192.3.e.a 1
56.h odd 2 1 192.3.e.b 1
63.l odd 6 2 324.3.g.b 2
63.o even 6 2 324.3.g.b 2
77.b even 2 1 1452.3.e.b 1
84.h odd 2 1 48.3.e.a 1
105.g even 2 1 300.3.g.b 1
105.k odd 4 2 300.3.b.a 2
112.j even 4 2 768.3.h.b 2
112.l odd 4 2 768.3.h.a 2
140.c even 2 1 1200.3.l.b 1
140.j odd 4 2 1200.3.c.c 2
168.e odd 2 1 192.3.e.a 1
168.i even 2 1 192.3.e.b 1
231.h odd 2 1 1452.3.e.b 1
252.s odd 6 2 1296.3.q.b 2
252.bi even 6 2 1296.3.q.b 2
336.v odd 4 2 768.3.h.b 2
336.y even 4 2 768.3.h.a 2
420.o odd 2 1 1200.3.l.b 1
420.w even 4 2 1200.3.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 7.b odd 2 1
12.3.c.a 1 21.c even 2 1
48.3.e.a 1 28.d even 2 1
48.3.e.a 1 84.h odd 2 1
192.3.e.a 1 56.e even 2 1
192.3.e.a 1 168.e odd 2 1
192.3.e.b 1 56.h odd 2 1
192.3.e.b 1 168.i even 2 1
300.3.b.a 2 35.f even 4 2
300.3.b.a 2 105.k odd 4 2
300.3.g.b 1 35.c odd 2 1
300.3.g.b 1 105.g even 2 1
324.3.g.b 2 63.l odd 6 2
324.3.g.b 2 63.o even 6 2
588.3.c.c 1 1.a even 1 1 trivial
588.3.c.c 1 3.b odd 2 1 CM
588.3.p.b 2 7.c even 3 2
588.3.p.b 2 21.h odd 6 2
588.3.p.c 2 7.d odd 6 2
588.3.p.c 2 21.g even 6 2
768.3.h.a 2 112.l odd 4 2
768.3.h.a 2 336.y even 4 2
768.3.h.b 2 112.j even 4 2
768.3.h.b 2 336.v odd 4 2
1200.3.c.c 2 140.j odd 4 2
1200.3.c.c 2 420.w even 4 2
1200.3.l.b 1 140.c even 2 1
1200.3.l.b 1 420.o odd 2 1
1296.3.q.b 2 252.s odd 6 2
1296.3.q.b 2 252.bi even 6 2
1452.3.e.b 1 77.b even 2 1
1452.3.e.b 1 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\):

\( T_{5} \)
\( T_{13} - 22 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -22 + T \)
$17$ \( T \)
$19$ \( 26 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -46 + T \)
$37$ \( -26 + T \)
$41$ \( T \)
$43$ \( 22 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 74 + T \)
$67$ \( -122 + T \)
$71$ \( T \)
$73$ \( -46 + T \)
$79$ \( 142 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 2 + T \)
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