Properties

Label 39.13.d.d
Level $39$
Weight $13$
Character orbit 39.d
Self dual yes
Analytic conductor $35.646$
Analytic rank $0$
Dimension $2$
CM discriminant -39
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.38");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 13 \)
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: \(35.6457588738\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{39}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 39 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3^{2} \)
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 \(\beta = 9\sqrt{39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 729 q^{3} - 937 q^{4} - 552 \beta q^{5} - 729 \beta q^{6} - 5033 \beta q^{8} + 531441 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 729 q^{3} - 937 q^{4} - 552 \beta q^{5} - 729 \beta q^{6} - 5033 \beta q^{8} + 531441 q^{9} - 1743768 q^{10} - 57224 \beta q^{11} + 683073 q^{12} - 4826809 q^{13} + 402408 \beta q^{15} - 12061295 q^{16} + 531441 \beta q^{18} + 517224 \beta q^{20} - 180770616 q^{22} + 3669057 \beta q^{24} + 718419311 q^{25} - 4826809 \beta q^{26} - 387420489 q^{27} + 1271206872 q^{30} + 8553873 \beta q^{32} + 41716296 \beta q^{33} - 497960217 q^{36} + 3518743761 q^{39} + 8776384344 q^{40} - 168934744 \beta q^{41} - 10591541998 q^{43} + 53618888 \beta q^{44} - 293355432 \beta q^{45} - 164287864 \beta q^{47} + 8792684055 q^{48} + 13841287201 q^{49} + 718419311 \beta q^{50} + 4522720033 q^{52} - 387420489 \beta q^{54} + 99785380032 q^{55} + 538420056 \beta q^{59} - 377056296 \beta q^{60} - 89162579378 q^{61} + 76424749127 q^{64} + 2664398568 \beta q^{65} + 131781779064 q^{66} - 1836353464 \beta q^{71} - 2674742553 \beta q^{72} - 523727677719 q^{75} + 3518743761 \beta q^{78} - 171789411458 q^{79} + 6657834840 \beta q^{80} + 282429536481 q^{81} - 533664856296 q^{82} - 2883782504 \beta q^{83} - 10591541998 \beta q^{86} + 909818510328 q^{88} - 6400396024 \beta q^{89} - 926709809688 q^{90} - 518985362376 q^{94} - 6235773417 \beta q^{96} + 13841287201 \beta q^{98} - 30411179784 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1458 q^{3} - 1874 q^{4} + 1062882 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1458 q^{3} - 1874 q^{4} + 1062882 q^{9} - 3487536 q^{10} + 1366146 q^{12} - 9653618 q^{13} - 24122590 q^{16} - 361541232 q^{22} + 1436838622 q^{25} - 774840978 q^{27} + 2542413744 q^{30} - 995920434 q^{36} + 7037487522 q^{39} + 17552768688 q^{40} - 21183083996 q^{43} + 17585368110 q^{48} + 27682574402 q^{49} + 9045440066 q^{52} + 199570760064 q^{55} - 178325158756 q^{61} + 152849498254 q^{64} + 263563558128 q^{66} - 1047455355438 q^{75} - 343578822916 q^{79} + 564859072962 q^{81} - 1067329712592 q^{82} + 1819637020656 q^{88} - 1853419619376 q^{90} - 1037970724752 q^{94}+O(q^{100}) \) 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
−6.24500
6.24500
−56.2050 −729.000 −937.000 31025.2 40973.4 0 282880. 531441. −1.74377e6
38.2 56.2050 −729.000 −937.000 −31025.2 −40973.4 0 −282880. 531441. −1.74377e6
\(n\): 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 39.13.d.d 2
3.b odd 2 1 inner 39.13.d.d 2
13.b even 2 1 inner 39.13.d.d 2
39.d odd 2 1 CM 39.13.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.13.d.d 2 1.a even 1 1 trivial
39.13.d.d 2 3.b odd 2 1 inner
39.13.d.d 2 13.b even 2 1 inner
39.13.d.d 2 39.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3159 \) acting on \(S_{13}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3159 \) Copy content Toggle raw display
$3$ \( (T + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 962559936 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 10344417729984 \) Copy content Toggle raw display
$13$ \( (T + 4826809)^{2} \) 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
$41$ \( T^{2} - 90\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T + 10591541998)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 85\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 91\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T + 89162579378)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 10\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 171789411458)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 26\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{2} - 12\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
show more
show less