Properties

Label 59.3.b.a
Level $59$
Weight $3$
Character orbit 59.b
Self dual yes
Analytic conductor $1.608$
Analytic rank $0$
Dimension $3$
CM discriminant -59
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,3,Mod(58,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 59.b (of order \(2\), degree \(1\), minimal)

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: \(1.60763355973\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1593.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 9x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{3} + 4 q^{4} + ( - 2 \beta_{2} - \beta_1) q^{5} + (\beta_{2} - 4 \beta_1) q^{7} + ( - 3 \beta_{2} + 4 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{3} + 4 q^{4} + ( - 2 \beta_{2} - \beta_1) q^{5} + (\beta_{2} - 4 \beta_1) q^{7} + ( - 3 \beta_{2} + 4 \beta_1 + 9) q^{9} + (4 \beta_{2} + 4 \beta_1) q^{12} + (6 \beta_{2} - 5 \beta_1 - 29) q^{15} + 16 q^{16} - 25 q^{17} + ( - 2 \beta_{2} + 11 \beta_1) q^{19} + ( - 8 \beta_{2} - 4 \beta_1) q^{20} + ( - 3 \beta_{2} - 11 \beta_1 - 17) q^{21} + ( - 11 \beta_{2} + 5 \beta_1 + 25) q^{25} + (9 \beta_{2} + 9 \beta_1 - 5) q^{27} + (4 \beta_{2} - 16 \beta_1) q^{28} + (13 \beta_{2} - 4 \beta_1) q^{29} + (\beta_{2} + 20 \beta_1 + 11) q^{35} + ( - 12 \beta_{2} + 16 \beta_1 + 36) q^{36} + (13 \beta_{2} - 19 \beta_1) q^{41} + ( - 29 \beta_{2} - 29 \beta_1 + 31) q^{45} + (16 \beta_{2} + 16 \beta_1) q^{48} + (22 \beta_{2} - \beta_1 + 49) q^{49} + ( - 25 \beta_{2} - 25 \beta_1) q^{51} + ( - 11 \beta_{2} + 29 \beta_1) q^{53} + (6 \beta_{2} + 31 \beta_1 + 55) q^{57} - 59 q^{59} + (24 \beta_{2} - 20 \beta_1 - 116) q^{60} + ( - 17 \beta_{2} - 17 \beta_1 - 110) q^{63} + 64 q^{64} - 100 q^{68} + 83 q^{71} + (58 \beta_{2} + 29 \beta_1 - 86) q^{75} + ( - 8 \beta_{2} + 44 \beta_1) q^{76} + ( - 23 \beta_{2} - 31 \beta_1) q^{79} + ( - 32 \beta_{2} - 16 \beta_1) q^{80} + ( - 5 \beta_{2} - 5 \beta_1 + 81) q^{81} + ( - 12 \beta_{2} - 44 \beta_1 - 68) q^{84} + (50 \beta_{2} + 25 \beta_1) q^{85} + ( - 39 \beta_{2} + \beta_1 + 115) q^{87} + (\beta_{2} - 55 \beta_1 - 46) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{4} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12 q^{4} + 27 q^{9} - 87 q^{15} + 48 q^{16} - 75 q^{17} - 51 q^{21} + 75 q^{25} - 15 q^{27} + 33 q^{35} + 108 q^{36} + 93 q^{45} + 147 q^{49} + 165 q^{57} - 177 q^{59} - 348 q^{60} - 330 q^{63} + 192 q^{64} - 300 q^{68} + 249 q^{71} - 258 q^{75} + 243 q^{81} - 204 q^{84} + 345 q^{87} - 138 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 9x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
−0.844760
−2.48705
3.33181
0 −5.28638 4.00000 9.72800 0 −1.06258 0 18.9458 0
58.2 0 0.185421 4.00000 −2.85789 0 12.6207 0 −8.96562 0
58.3 0 5.10096 4.00000 −6.87011 0 −11.5581 0 17.0198 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
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.3.b.a 3
3.b odd 2 1 531.3.c.a 3
4.b odd 2 1 944.3.h.b 3
59.b odd 2 1 CM 59.3.b.a 3
177.d even 2 1 531.3.c.a 3
236.c even 2 1 944.3.h.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.3.b.a 3 1.a even 1 1 trivial
59.3.b.a 3 59.b odd 2 1 CM
531.3.c.a 3 3.b odd 2 1
531.3.c.a 3 177.d even 2 1
944.3.h.b 3 4.b odd 2 1
944.3.h.b 3 236.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(59, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 27T + 5 \) Copy content Toggle raw display
$5$ \( T^{3} - 75T - 191 \) Copy content Toggle raw display
$7$ \( T^{3} - 147T - 155 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( (T + 25)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 1083T - 443 \) Copy content Toggle raw display
$23$ \( T^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 2523T + 23497 \) Copy content Toggle raw display
$31$ \( T^{3} \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 5043 T - 137783 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} \) Copy content Toggle raw display
$53$ \( T^{3} - 8427 T + 190825 \) Copy content Toggle raw display
$59$ \( (T + 59)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( (T - 83)^{3} \) Copy content Toggle raw display
$73$ \( T^{3} \) Copy content Toggle raw display
$79$ \( T^{3} - 18723 T + 288853 \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
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