Properties

Label 5586.2.a.bv
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + (\beta_1 - 2) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + (\beta_1 - 2) q^{5} + q^{6} + q^{8} + q^{9} + (\beta_1 - 2) q^{10} + ( - \beta_{2} - \beta_1 + 1) q^{11} + q^{12} + (2 \beta_{2} - 2 \beta_1 - 2) q^{13} + (\beta_1 - 2) q^{15} + q^{16} + (\beta_1 - 5) q^{17} + q^{18} - q^{19} + (\beta_1 - 2) q^{20} + ( - \beta_{2} - \beta_1 + 1) q^{22} + ( - 3 \beta_{2} - 1) q^{23} + q^{24} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + (2 \beta_{2} - 2 \beta_1 - 2) q^{26} + q^{27} + ( - \beta_{2} + 3 \beta_1 - 1) q^{29} + (\beta_1 - 2) q^{30} + ( - 2 \beta_1 - 3) q^{31} + q^{32} + ( - \beta_{2} - \beta_1 + 1) q^{33} + (\beta_1 - 5) q^{34} + q^{36} + (2 \beta_{2} - 3 \beta_1 - 1) q^{37} - q^{38} + (2 \beta_{2} - 2 \beta_1 - 2) q^{39} + (\beta_1 - 2) q^{40} + ( - \beta_{2} + 2 \beta_1 - 5) q^{41} + (4 \beta_{2} - \beta_1 + 1) q^{43} + ( - \beta_{2} - \beta_1 + 1) q^{44} + (\beta_1 - 2) q^{45} + ( - 3 \beta_{2} - 1) q^{46} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{47} + q^{48} + (\beta_{2} - 3 \beta_1 + 2) q^{50} + (\beta_1 - 5) q^{51} + (2 \beta_{2} - 2 \beta_1 - 2) q^{52} + (2 \beta_{2} - 2 \beta_1 - 3) q^{53} + q^{54} + (\beta_{2} - 6) q^{55} - q^{57} + ( - \beta_{2} + 3 \beta_1 - 1) q^{58} + (\beta_{2} - 2 \beta_1 - 2) q^{59} + (\beta_1 - 2) q^{60} + ( - 2 \beta_1 + 4) q^{61} + ( - 2 \beta_1 - 3) q^{62} + q^{64} + ( - 6 \beta_{2} + 4 \beta_1) q^{65} + ( - \beta_{2} - \beta_1 + 1) q^{66} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{67} + (\beta_1 - 5) q^{68} + ( - 3 \beta_{2} - 1) q^{69} + (4 \beta_{2} - \beta_1 - 3) q^{71} + q^{72} + ( - 3 \beta_{2} + \beta_1) q^{73} + (2 \beta_{2} - 3 \beta_1 - 1) q^{74} + (\beta_{2} - 3 \beta_1 + 2) q^{75} - q^{76} + (2 \beta_{2} - 2 \beta_1 - 2) q^{78} + (3 \beta_{2} + 3 \beta_1 - 7) q^{79} + (\beta_1 - 2) q^{80} + q^{81} + ( - \beta_{2} + 2 \beta_1 - 5) q^{82} + (2 \beta_{2} + 6 \beta_1 - 9) q^{83} + (\beta_{2} - 6 \beta_1 + 13) q^{85} + (4 \beta_{2} - \beta_1 + 1) q^{86} + ( - \beta_{2} + 3 \beta_1 - 1) q^{87} + ( - \beta_{2} - \beta_1 + 1) q^{88} + ( - \beta_{2} + 2 \beta_1 - 5) q^{89} + (\beta_1 - 2) q^{90} + ( - 3 \beta_{2} - 1) q^{92} + ( - 2 \beta_1 - 3) q^{93} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{94} + ( - \beta_1 + 2) q^{95} + q^{96} + (\beta_{2} - 2 \beta_1 - 10) q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 3 q^{8} + 3 q^{9} - 5 q^{10} + q^{11} + 3 q^{12} - 6 q^{13} - 5 q^{15} + 3 q^{16} - 14 q^{17} + 3 q^{18} - 3 q^{19} - 5 q^{20} + q^{22} - 6 q^{23} + 3 q^{24} + 4 q^{25} - 6 q^{26} + 3 q^{27} - q^{29} - 5 q^{30} - 11 q^{31} + 3 q^{32} + q^{33} - 14 q^{34} + 3 q^{36} - 4 q^{37} - 3 q^{38} - 6 q^{39} - 5 q^{40} - 14 q^{41} + 6 q^{43} + q^{44} - 5 q^{45} - 6 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{50} - 14 q^{51} - 6 q^{52} - 9 q^{53} + 3 q^{54} - 17 q^{55} - 3 q^{57} - q^{58} - 7 q^{59} - 5 q^{60} + 10 q^{61} - 11 q^{62} + 3 q^{64} - 2 q^{65} + q^{66} - 12 q^{67} - 14 q^{68} - 6 q^{69} - 6 q^{71} + 3 q^{72} - 2 q^{73} - 4 q^{74} + 4 q^{75} - 3 q^{76} - 6 q^{78} - 15 q^{79} - 5 q^{80} + 3 q^{81} - 14 q^{82} - 19 q^{83} + 34 q^{85} + 6 q^{86} - q^{87} + q^{88} - 14 q^{89} - 5 q^{90} - 6 q^{92} - 11 q^{93} + 4 q^{94} + 5 q^{95} + 3 q^{96} - 31 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
−1.65544
−0.210756
2.86620
1.00000 1.00000 1.00000 −3.65544 1.00000 0 1.00000 1.00000 −3.65544
1.2 1.00000 1.00000 1.00000 −2.21076 1.00000 0 1.00000 1.00000 −2.21076
1.3 1.00000 1.00000 1.00000 0.866198 1.00000 0 1.00000 1.00000 0.866198
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(19\) \(1\)

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 5586.2.a.bv 3
7.b odd 2 1 5586.2.a.bu 3
7.c even 3 2 798.2.j.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.i 6 7.c even 3 2
5586.2.a.bu 3 7.b odd 2 1
5586.2.a.bv 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{3} + 5T_{5}^{2} + 3T_{5} - 7 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 17T_{11} + 21 \) Copy content Toggle raw display
\( T_{13}^{3} + 6T_{13}^{2} - 20T_{13} - 88 \) Copy content Toggle raw display
\( T_{17}^{3} + 14T_{17}^{2} + 60T_{17} + 74 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 5 T^{2} + 3 T - 7 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 17 T + 21 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 20 T - 88 \) Copy content Toggle raw display
$17$ \( T^{3} + 14 T^{2} + 60 T + 74 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 54 T - 302 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} - 41 T + 43 \) Copy content Toggle raw display
$31$ \( T^{3} + 11 T^{2} + 19 T - 7 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 44 T - 194 \) Copy content Toggle raw display
$41$ \( T^{3} + 14 T^{2} + 46 T + 42 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 92 T + 606 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 96 T - 144 \) Copy content Toggle raw display
$53$ \( T^{3} + 9 T^{2} - 5 T - 101 \) Copy content Toggle raw display
$59$ \( T^{3} + 7 T^{2} - 3 T - 63 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} + 12 T + 56 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} + 16 T - 32 \) Copy content Toggle raw display
$71$ \( T^{3} + 6 T^{2} - 92 T + 206 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} - 56 T - 196 \) Copy content Toggle raw display
$79$ \( T^{3} + 15 T^{2} - 81 T - 1067 \) Copy content Toggle raw display
$83$ \( T^{3} + 19 T^{2} - 157 T - 3279 \) Copy content Toggle raw display
$89$ \( T^{3} + 14 T^{2} + 46 T + 42 \) Copy content Toggle raw display
$97$ \( T^{3} + 31 T^{2} + 301 T + 873 \) Copy content Toggle raw display
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