Properties

Label 59.19.b.a
Level $59$
Weight $19$
Character orbit 59.b
Self dual yes
Analytic conductor $121.178$
Analytic rank $0$
Dimension $1$
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] = [59,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 19 \)
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: \(121.177821249\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 10810 q^{3} + 262144 q^{4} - 1985254 q^{5} - 50982910 q^{7} - 270564389 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10810 q^{3} + 262144 q^{4} - 1985254 q^{5} - 50982910 q^{7} - 270564389 q^{9} + 2833776640 q^{12} - 21460595740 q^{15} + 68719476736 q^{16} - 216651752350 q^{17} - 62437037542 q^{19} - 520422424576 q^{20} - 551125257100 q^{21} + 126536178891 q^{25} - 7112816531180 q^{27} - 13364863959040 q^{28} + 28956785336138 q^{29} + 101214026009140 q^{35} - 70926831190016 q^{36} + 652243002714578 q^{41} + 537139035519806 q^{45} + 742857543516160 q^{48} + 970843514157651 q^{49} - 23\!\cdots\!00 q^{51}+ \cdots + 12\!\cdots\!68 q^{95}+O(q^{100}) \) 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
0 10810.0 262144. −1.98525e6 0 −5.09829e7 0 −2.70564e8 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 59.19.b.a 1
59.b odd 2 1 CM 59.19.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.19.b.a 1 1.a even 1 1 trivial
59.19.b.a 1 59.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{19}^{\mathrm{new}}(59, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3} - 10810 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 10810 \) Copy content Toggle raw display
$5$ \( T + 1985254 \) Copy content Toggle raw display
$7$ \( T + 50982910 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 216651752350 \) Copy content Toggle raw display
$19$ \( T + 62437037542 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 28956785336138 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 652243002714578 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 5739806619558650 \) Copy content Toggle raw display
$59$ \( T + 8662995818654939 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 56\!\cdots\!62 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 18\!\cdots\!62 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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