Properties

Label 588.4.a.j
Level $588$
Weight $4$
Character orbit 588.a
Self dual yes
Analytic conductor $34.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,4,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.6931230834\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.136768.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 23x^{2} + 18x + 119 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} - \beta_1 q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - \beta_1 q^{5} + 9 q^{9} + ( - \beta_{3} + 3 \beta_1) q^{11} + 3 \beta_{3} q^{13} + 3 \beta_1 q^{15} + ( - 3 \beta_{3} - 3 \beta_{2} - \beta_1 - 12) q^{17} + ( - 3 \beta_{3} + 5 \beta_{2} + 6 \beta_1 - 48) q^{19} + (\beta_{3} + 6 \beta_{2} + 3 \beta_1 + 48) q^{23} + (6 \beta_{3} - 6 \beta_{2} + 81) q^{25} - 27 q^{27} + (\beta_{3} - 9 \beta_{2} + 24) q^{29} + (3 \beta_{3} - 9 \beta_{2} + 6 \beta_1 - 12) q^{31} + (3 \beta_{3} - 9 \beta_1) q^{33} + ( - 6 \beta_{3} + 24 \beta_{2} - 6 \beta_1 + 64) q^{37} - 9 \beta_{3} q^{39} + ( - 3 \beta_{3} + 15 \beta_{2} - \beta_1 - 252) q^{41} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1 - 28) q^{43} - 9 \beta_1 q^{45} + (3 \beta_{3} + 3 \beta_{2} - 10 \beta_1 - 216) q^{47} + (9 \beta_{3} + 9 \beta_{2} + 3 \beta_1 + 36) q^{51} + ( - 10 \beta_{3} - 18 \beta_{2} - 24 \beta_1 - 162) q^{53} + ( - 21 \beta_{3} - 5 \beta_{2} + 6 \beta_1 - 588) q^{55} + (9 \beta_{3} - 15 \beta_{2} - 18 \beta_1 + 144) q^{57} + (21 \beta_{3} - 15 \beta_{2} - 10 \beta_1 - 84) q^{59} + (15 \beta_{3} - 14 \beta_{2} + 30 \beta_1 - 240) q^{61} + (9 \beta_{3} + 69 \beta_{2} - 18 \beta_1 - 90) q^{65} + (30 \beta_{3} - 48 \beta_{2} - 6 \beta_1 + 180) q^{67} + ( - 3 \beta_{3} - 18 \beta_{2} - 9 \beta_1 - 144) q^{69} + (9 \beta_{3} - 30 \beta_{2} + 3 \beta_1 - 336) q^{71} + ( - 18 \beta_{3} + 49 \beta_{2} + 30 \beta_1 - 168) q^{73} + ( - 18 \beta_{3} + 18 \beta_{2} - 243) q^{75} + ( - 24 \beta_{3} + 96 \beta_{2} + 36 \beta_1 - 496) q^{79} + 81 q^{81} + (18 \beta_{3} + 18 \beta_{2} - 8 \beta_1 - 780) q^{83} + ( - 24 \beta_{3} - 66 \beta_{2} + 30 \beta_1 + 170) q^{85} + ( - 3 \beta_{3} + 27 \beta_{2} - 72) q^{87} + ( - 15 \beta_{3} - 69 \beta_{2} - 11 \beta_1 - 540) q^{89} + ( - 9 \beta_{3} + 27 \beta_{2} - 18 \beta_1 + 36) q^{93} + ( - 10 \beta_{3} - 48 \beta_{2} + 66 \beta_1 - 936) q^{95} + (48 \beta_{3} - 49 \beta_{2} + 18 \beta_1 - 504) q^{97} + ( - 9 \beta_{3} + 27 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 48 q^{17} - 192 q^{19} + 192 q^{23} + 324 q^{25} - 108 q^{27} + 96 q^{29} - 48 q^{31} + 256 q^{37} - 1008 q^{41} - 112 q^{43} - 864 q^{47} + 144 q^{51} - 648 q^{53} - 2352 q^{55} + 576 q^{57} - 336 q^{59} - 960 q^{61} - 360 q^{65} + 720 q^{67} - 576 q^{69} - 1344 q^{71} - 672 q^{73} - 972 q^{75} - 1984 q^{79} + 324 q^{81} - 3120 q^{83} + 680 q^{85} - 288 q^{87} - 2160 q^{89} + 144 q^{93} - 3744 q^{95} - 2016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 23x^{2} + 18x + 119 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 3\nu^{2} + 25\nu + 2 ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{3} + 21\nu^{2} + 77\nu - 140 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 7\nu^{2} + 7\nu - 68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 7\beta _1 + 14 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{3} - 8\beta_{2} + 7\beta _1 + 350 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 7\beta_{2} + 14\beta _1 + 92 ) / 4 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.93153
2.89590
−2.51732
−3.31012
0 −3.00000 0 −19.1403 0 0 0 9.00000 0
1.2 0 −3.00000 0 −8.16940 0 0 0 9.00000 0
1.3 0 −3.00000 0 10.6550 0 0 0 9.00000 0
1.4 0 −3.00000 0 16.6547 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.a.j 4
3.b odd 2 1 1764.4.a.bc 4
4.b odd 2 1 2352.4.a.cq 4
7.b odd 2 1 588.4.a.k yes 4
7.c even 3 2 588.4.i.l 8
7.d odd 6 2 588.4.i.k 8
21.c even 2 1 1764.4.a.ba 4
21.g even 6 2 1764.4.k.bd 8
21.h odd 6 2 1764.4.k.bb 8
28.d even 2 1 2352.4.a.cl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.j 4 1.a even 1 1 trivial
588.4.a.k yes 4 7.b odd 2 1
588.4.i.k 8 7.d odd 6 2
588.4.i.l 8 7.c even 3 2
1764.4.a.ba 4 21.c even 2 1
1764.4.a.bc 4 3.b odd 2 1
1764.4.k.bb 8 21.h odd 6 2
1764.4.k.bd 8 21.g even 6 2
2352.4.a.cl 4 28.d even 2 1
2352.4.a.cq 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 412T_{5}^{2} + 576T_{5} + 27748 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 412 T^{2} + 576 T + 27748 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 4136 T^{2} + 82944 T + 733072 \) Copy content Toggle raw display
$13$ \( T^{4} - 7092 T^{2} + \cdots + 10008036 \) Copy content Toggle raw display
$17$ \( T^{4} + 48 T^{3} - 9772 T^{2} + \cdots - 4865084 \) Copy content Toggle raw display
$19$ \( T^{4} + 192 T^{3} + \cdots - 41971136 \) Copy content Toggle raw display
$23$ \( T^{4} - 192 T^{3} + \cdots - 40554608 \) Copy content Toggle raw display
$29$ \( T^{4} - 96 T^{3} - 11696 T^{2} + \cdots + 38719552 \) Copy content Toggle raw display
$31$ \( T^{4} + 48 T^{3} + \cdots - 189895104 \) Copy content Toggle raw display
$37$ \( T^{4} - 256 T^{3} + \cdots - 1479272192 \) Copy content Toggle raw display
$41$ \( T^{4} + 1008 T^{3} + \cdots + 1611829828 \) Copy content Toggle raw display
$43$ \( T^{4} + 112 T^{3} + \cdots - 789373952 \) Copy content Toggle raw display
$47$ \( T^{4} + 864 T^{3} + \cdots - 1168478144 \) Copy content Toggle raw display
$53$ \( T^{4} + 648 T^{3} + \cdots - 20504773616 \) Copy content Toggle raw display
$59$ \( T^{4} + 336 T^{3} + \cdots + 18986185792 \) Copy content Toggle raw display
$61$ \( T^{4} + 960 T^{3} + \cdots - 106656271196 \) Copy content Toggle raw display
$67$ \( T^{4} - 720 T^{3} + \cdots + 14336621568 \) Copy content Toggle raw display
$71$ \( T^{4} + 1344 T^{3} + \cdots - 17989567344 \) Copy content Toggle raw display
$73$ \( T^{4} + 672 T^{3} + \cdots - 90986816444 \) Copy content Toggle raw display
$79$ \( T^{4} + 1984 T^{3} + \cdots - 1013049875456 \) Copy content Toggle raw display
$83$ \( T^{4} + 3120 T^{3} + \cdots + 74256064768 \) Copy content Toggle raw display
$89$ \( T^{4} + 2160 T^{3} + \cdots - 311467391228 \) Copy content Toggle raw display
$97$ \( T^{4} + 2016 T^{3} + \cdots - 580580611196 \) Copy content Toggle raw display
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