Properties

Label 592.2.a.i
Level $592$
Weight $2$
Character orbit 592.a
Self dual yes
Analytic conductor $4.727$
Analytic rank $1$
Dimension $3$
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] = [592,2,Mod(1,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.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: \(4.72714379966\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 296)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1 - 2) q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1 - 2) q^{7} + (\beta_1 + 1) q^{9} + 3 \beta_1 q^{11} + ( - 3 \beta_1 + 1) q^{13} + (\beta_{2} - \beta_1 - 3) q^{15} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + (2 \beta_{2} - 2 \beta_1 - 2) q^{19} + (3 \beta_{2} + \beta_1) q^{21} + (\beta_1 - 3) q^{23} + ( - 2 \beta_{2} + \beta_1 - 2) q^{25} + (2 \beta_{2} - 2 \beta_1 + 1) q^{27} + (\beta_1 - 3) q^{29} + ( - \beta_{2} - 6) q^{31} + ( - 6 \beta_1 - 3) q^{33} + ( - 4 \beta_{2} + 2) q^{35} - q^{37} + ( - \beta_{2} + 6 \beta_1 + 2) q^{39} + (\beta_{2} - 2 \beta_1 - 5) q^{41} + ( - 4 \beta_{2} - 2 \beta_1) q^{43} + (\beta_{2} + \beta_1 + 1) q^{45} + ( - \beta_{2} - 3 \beta_1 - 4) q^{47} + ( - 5 \beta_{2} + 3 \beta_1 + 1) q^{49} + ( - 2 \beta_1 + 6) q^{51} + ( - 3 \beta_{2} + 5 \beta_1 - 2) q^{53} + (3 \beta_1 + 3) q^{55} + (4 \beta_{2} + 2 \beta_1 - 2) q^{57} + (4 \beta_{2} + 2 \beta_1 + 2) q^{59} + ( - 3 \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - 2 \beta_1 - 4) q^{63} + (\beta_{2} - 3 \beta_1 - 3) q^{65} + ( - 5 \beta_{2} + 6 \beta_1) q^{67} + (3 \beta_{2} - 2 \beta_1 + 2) q^{69} + (\beta_{2} - 3 \beta_1 + 2) q^{71} + ( - 3 \beta_1 - 2) q^{73} + 7 q^{75} + ( - 3 \beta_{2} - 3 \beta_1 - 6) q^{77} + (2 \beta_{2} - \beta_1 + 1) q^{79} + (\beta_{2} - \beta_1 - 8) q^{81} + (3 \beta_{2} - 5 \beta_1 + 4) q^{83} + (2 \beta_{2} - 4) q^{85} + (3 \beta_{2} - 2 \beta_1 + 2) q^{87} + ( - 4 \beta_{2} + 4) q^{89} + (4 \beta_{2} + 2 \beta_1 + 4) q^{91} + (5 \beta_{2} + \beta_1 + 9) q^{93} + ( - 6 \beta_{2} + 4) q^{95} + (6 \beta_{2} + 2 \beta_1 + 2) q^{97} + (3 \beta_{2} + 3 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - q^{5} - 7 q^{7} + 3 q^{9} + 3 q^{13} - 10 q^{15} - 4 q^{17} - 8 q^{19} - 3 q^{21} - 9 q^{23} - 4 q^{25} + q^{27} - 9 q^{29} - 17 q^{31} - 9 q^{33} + 10 q^{35} - 3 q^{37} + 7 q^{39} - 16 q^{41} + 4 q^{43} + 2 q^{45} - 11 q^{47} + 8 q^{49} + 18 q^{51} - 3 q^{53} + 9 q^{55} - 10 q^{57} + 2 q^{59} + 15 q^{61} - 12 q^{63} - 10 q^{65} + 5 q^{67} + 3 q^{69} + 5 q^{71} - 6 q^{73} + 21 q^{75} - 15 q^{77} + q^{79} - 25 q^{81} + 9 q^{83} - 14 q^{85} + 3 q^{87} + 16 q^{89} + 8 q^{91} + 22 q^{93} + 18 q^{95} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.11491
−1.86081
−0.254102
0 −2.47283 0 1.47283 0 −2.64207 0 3.11491 0
1.2 0 −1.46260 0 0.462598 0 0.323404 0 −0.860806 0
1.3 0 1.93543 0 −2.93543 0 −4.68133 0 0.745898 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(37\) \( +1 \)

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 592.2.a.i 3
3.b odd 2 1 5328.2.a.bn 3
4.b odd 2 1 296.2.a.c 3
8.b even 2 1 2368.2.a.be 3
8.d odd 2 1 2368.2.a.bb 3
12.b even 2 1 2664.2.a.p 3
20.d odd 2 1 7400.2.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.a.c 3 4.b odd 2 1
592.2.a.i 3 1.a even 1 1 trivial
2368.2.a.bb 3 8.d odd 2 1
2368.2.a.be 3 8.b even 2 1
2664.2.a.p 3 12.b even 2 1
5328.2.a.bn 3 3.b odd 2 1
7400.2.a.k 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(592))\):

\( T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 5T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 5T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} + 7 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 36T - 27 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} + \cdots + 62 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$23$ \( T^{3} + 9 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$29$ \( T^{3} + 9 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$31$ \( T^{3} + 17 T^{2} + \cdots + 148 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 16 T^{2} + \cdots + 47 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$47$ \( T^{3} + 11 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} + \cdots + 292 \) Copy content Toggle raw display
$59$ \( T^{3} - 2 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( T^{3} - 15 T^{2} + \cdots + 52 \) Copy content Toggle raw display
$67$ \( T^{3} - 5 T^{2} + \cdots + 944 \) Copy content Toggle raw display
$71$ \( T^{3} - 5 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} + \cdots - 37 \) Copy content Toggle raw display
$79$ \( T^{3} - T^{2} + \cdots + 32 \) Copy content Toggle raw display
$83$ \( T^{3} - 9 T^{2} + \cdots - 112 \) Copy content Toggle raw display
$89$ \( T^{3} - 16T^{2} + 64 \) Copy content Toggle raw display
$97$ \( T^{3} - 244T + 256 \) Copy content Toggle raw display
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