Properties

Label 588.3.p.b
Level $588$
Weight $3$
Character orbit 588.p
Analytic conductor $16.022$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
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: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -3 \zeta_{6} q^{3} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -3 \zeta_{6} q^{3} + ( -9 + 9 \zeta_{6} ) q^{9} + 22 q^{13} + ( 26 - 26 \zeta_{6} ) q^{19} -25 \zeta_{6} q^{25} + 27 q^{27} -46 \zeta_{6} q^{31} + ( -26 + 26 \zeta_{6} ) q^{37} -66 \zeta_{6} q^{39} -22 q^{43} -78 q^{57} + ( 74 - 74 \zeta_{6} ) q^{61} -122 \zeta_{6} q^{67} -46 \zeta_{6} q^{73} + ( -75 + 75 \zeta_{6} ) q^{75} + ( 142 - 142 \zeta_{6} ) q^{79} -81 \zeta_{6} q^{81} + ( -138 + 138 \zeta_{6} ) q^{93} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 9q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 9q^{9} + 44q^{13} + 26q^{19} - 25q^{25} + 54q^{27} - 46q^{31} - 26q^{37} - 66q^{39} - 44q^{43} - 156q^{57} + 74q^{61} - 122q^{67} - 46q^{73} - 75q^{75} + 142q^{79} - 81q^{81} - 138q^{93} - 4q^{97} + O(q^{100}) \)

Character values

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

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

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 −1.50000 + 2.59808i 0 0 0 0 0 −4.50000 7.79423i 0
569.1 0 −1.50000 2.59808i 0 0 0 0 0 −4.50000 + 7.79423i 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 588.3.p.b 2
3.b odd 2 1 CM 588.3.p.b 2
7.b odd 2 1 588.3.p.c 2
7.c even 3 1 588.3.c.c 1
7.c even 3 1 inner 588.3.p.b 2
7.d odd 6 1 12.3.c.a 1
7.d odd 6 1 588.3.p.c 2
21.c even 2 1 588.3.p.c 2
21.g even 6 1 12.3.c.a 1
21.g even 6 1 588.3.p.c 2
21.h odd 6 1 588.3.c.c 1
21.h odd 6 1 inner 588.3.p.b 2
28.f even 6 1 48.3.e.a 1
35.i odd 6 1 300.3.g.b 1
35.k even 12 2 300.3.b.a 2
56.j odd 6 1 192.3.e.b 1
56.m even 6 1 192.3.e.a 1
63.i even 6 1 324.3.g.b 2
63.k odd 6 1 324.3.g.b 2
63.s even 6 1 324.3.g.b 2
63.t odd 6 1 324.3.g.b 2
77.i even 6 1 1452.3.e.b 1
84.j odd 6 1 48.3.e.a 1
105.p even 6 1 300.3.g.b 1
105.w odd 12 2 300.3.b.a 2
112.v even 12 2 768.3.h.b 2
112.x odd 12 2 768.3.h.a 2
140.s even 6 1 1200.3.l.b 1
140.x odd 12 2 1200.3.c.c 2
168.ba even 6 1 192.3.e.b 1
168.be odd 6 1 192.3.e.a 1
231.k odd 6 1 1452.3.e.b 1
252.n even 6 1 1296.3.q.b 2
252.r odd 6 1 1296.3.q.b 2
252.bj even 6 1 1296.3.q.b 2
252.bn odd 6 1 1296.3.q.b 2
336.bo even 12 2 768.3.h.a 2
336.br odd 12 2 768.3.h.b 2
420.be odd 6 1 1200.3.l.b 1
420.br even 12 2 1200.3.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 7.d odd 6 1
12.3.c.a 1 21.g even 6 1
48.3.e.a 1 28.f even 6 1
48.3.e.a 1 84.j odd 6 1
192.3.e.a 1 56.m even 6 1
192.3.e.a 1 168.be odd 6 1
192.3.e.b 1 56.j odd 6 1
192.3.e.b 1 168.ba even 6 1
300.3.b.a 2 35.k even 12 2
300.3.b.a 2 105.w odd 12 2
300.3.g.b 1 35.i odd 6 1
300.3.g.b 1 105.p even 6 1
324.3.g.b 2 63.i even 6 1
324.3.g.b 2 63.k odd 6 1
324.3.g.b 2 63.s even 6 1
324.3.g.b 2 63.t odd 6 1
588.3.c.c 1 7.c even 3 1
588.3.c.c 1 21.h odd 6 1
588.3.p.b 2 1.a even 1 1 trivial
588.3.p.b 2 3.b odd 2 1 CM
588.3.p.b 2 7.c even 3 1 inner
588.3.p.b 2 21.h odd 6 1 inner
588.3.p.c 2 7.b odd 2 1
588.3.p.c 2 7.d odd 6 1
588.3.p.c 2 21.c even 2 1
588.3.p.c 2 21.g even 6 1
768.3.h.a 2 112.x odd 12 2
768.3.h.a 2 336.bo even 12 2
768.3.h.b 2 112.v even 12 2
768.3.h.b 2 336.br odd 12 2
1200.3.c.c 2 140.x odd 12 2
1200.3.c.c 2 420.br even 12 2
1200.3.l.b 1 140.s even 6 1
1200.3.l.b 1 420.be odd 6 1
1296.3.q.b 2 252.n even 6 1
1296.3.q.b 2 252.r odd 6 1
1296.3.q.b 2 252.bj even 6 1
1296.3.q.b 2 252.bn odd 6 1
1452.3.e.b 1 77.i even 6 1
1452.3.e.b 1 231.k odd 6 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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -22 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 676 - 26 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 2116 + 46 T + T^{2} \)
$37$ \( 676 + 26 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 22 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 5476 - 74 T + T^{2} \)
$67$ \( 14884 + 122 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 2116 + 46 T + T^{2} \)
$79$ \( 20164 - 142 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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