Properties

Label 39.19.d.c
Level $39$
Weight $19$
Character orbit 39.d
Self dual yes
Analytic conductor $80.101$
Analytic rank $0$
Dimension $2$
CM discriminant -39
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,19,Mod(38,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.38");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 39.d (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: \(80.1005937068\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3^{3}\cdot 17 \)
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 \(\beta = 459\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 19683 q^{3} + 369899 q^{4} + 2796 \beta q^{5} + 19683 \beta q^{6} + 107755 \beta q^{8} + 387420489 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 19683 q^{3} + 369899 q^{4} + 2796 \beta q^{5} + 19683 \beta q^{6} + 107755 \beta q^{8} + 387420489 q^{9} + 1767192228 q^{10} + 5877052 \beta q^{11} + 7280722017 q^{12} - 10604499373 q^{13} + 55033668 \beta q^{15} - 28861009991 q^{16} + 387420489 \beta q^{18} + 1034237604 \beta q^{20} + 3714549577236 q^{22} + 2120941665 \beta q^{24} + 1126372203863 q^{25} - 10604499373 \beta q^{26} + 7625597484987 q^{27} + 34783644623724 q^{30} - 57108336711 \beta q^{32} + 115678014516 \beta q^{33} + 143306451460611 q^{36} - 208728361158759 q^{39} + 190423798528140 q^{40} + 610631662148 \beta q^{41} - 765985469578630 q^{43} + 2173915657748 \beta q^{44} + 1083227687244 \beta q^{45} - 1734909557356 \beta q^{47} - 568071259652853 q^{48} + 16\!\cdots\!49 q^{49} + \cdots + 22\!\cdots\!28 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 39366 q^{3} + 739798 q^{4} + 774840978 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 39366 q^{3} + 739798 q^{4} + 774840978 q^{9} + 3534384456 q^{10} + 14561444034 q^{12} - 21208998746 q^{13} - 57722019982 q^{16} + 7429099154472 q^{22} + 2252744407726 q^{25} + 15251194969974 q^{27} + 69567289247448 q^{30} + 286612902921222 q^{36} - 417456722317518 q^{39} + 380847597056280 q^{40} - 15\!\cdots\!60 q^{43}+ \cdots - 21\!\cdots\!16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
−1.73205
1.73205
−795.011 19683.0 369899. −2.22285e6 −1.56482e7 0 −8.56664e7 3.87420e8 1.76719e9
38.2 795.011 19683.0 369899. 2.22285e6 1.56482e7 0 8.56664e7 3.87420e8 1.76719e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.19.d.c 2
3.b odd 2 1 inner 39.19.d.c 2
13.b even 2 1 inner 39.19.d.c 2
39.d odd 2 1 CM 39.19.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.19.d.c 2 1.a even 1 1 trivial
39.19.d.c 2 3.b odd 2 1 inner
39.19.d.c 2 13.b even 2 1 inner
39.19.d.c 2 39.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 632043 \) acting on \(S_{19}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 632043 \) Copy content Toggle raw display
$3$ \( (T - 19683)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4941069469488 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 21\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( (T + 10604499373)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 23\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( (T + 765985469578630)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 19\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 21\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( (T + 16\!\cdots\!70)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 28\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 57\!\cdots\!50)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 18\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{2} - 13\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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