Properties

Label 1728.4.c.d
Level $1728$
Weight $4$
Character orbit 1728.c
Analytic conductor $101.955$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(1727,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1727");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not 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: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 12 \beta q^{5} - 17 \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 12 \beta q^{5} - 17 \beta q^{7} + 36 q^{11} - 19 q^{13} + 60 \beta q^{17} + \beta q^{19} - 108 q^{23} - 307 q^{25} + 120 \beta q^{29} + 54 \beta q^{31} - 612 q^{35} + 109 q^{37} - 216 \beta q^{41} + 270 \beta q^{43} - 468 q^{47} - 524 q^{49} - 72 \beta q^{53} - 432 \beta q^{55} - 36 q^{59} + 145 q^{61} + 228 \beta q^{65} + 181 \beta q^{67} - 576 q^{71} - 809 q^{73} - 612 \beta q^{77} + 287 \beta q^{79} - 360 q^{83} + 2160 q^{85} + 156 \beta q^{89} + 323 \beta q^{91} + 36 q^{95} + 955 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 72 q^{11} - 38 q^{13} - 216 q^{23} - 614 q^{25} - 1224 q^{35} + 218 q^{37} - 936 q^{47} - 1048 q^{49} - 72 q^{59} + 290 q^{61} - 1152 q^{71} - 1618 q^{73} - 720 q^{83} + 4320 q^{85} + 72 q^{95} + 1910 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 20.7846i 0 29.4449i 0 0 0
1727.2 0 0 0 20.7846i 0 29.4449i 0 0 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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.c.d 2
3.b odd 2 1 1728.4.c.a 2
4.b odd 2 1 1728.4.c.a 2
8.b even 2 1 432.4.c.a 2
8.d odd 2 1 432.4.c.d yes 2
12.b even 2 1 inner 1728.4.c.d 2
24.f even 2 1 432.4.c.a 2
24.h odd 2 1 432.4.c.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.4.c.a 2 8.b even 2 1
432.4.c.a 2 24.f even 2 1
432.4.c.d yes 2 8.d odd 2 1
432.4.c.d yes 2 24.h odd 2 1
1728.4.c.a 2 3.b odd 2 1
1728.4.c.a 2 4.b odd 2 1
1728.4.c.d 2 1.a even 1 1 trivial
1728.4.c.d 2 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 432 \) Copy content Toggle raw display
\( T_{7}^{2} + 867 \) Copy content Toggle raw display
\( T_{11} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 432 \) Copy content Toggle raw display
$7$ \( T^{2} + 867 \) Copy content Toggle raw display
$11$ \( (T - 36)^{2} \) Copy content Toggle raw display
$13$ \( (T + 19)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 10800 \) Copy content Toggle raw display
$19$ \( T^{2} + 3 \) Copy content Toggle raw display
$23$ \( (T + 108)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 43200 \) Copy content Toggle raw display
$31$ \( T^{2} + 8748 \) Copy content Toggle raw display
$37$ \( (T - 109)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 139968 \) Copy content Toggle raw display
$43$ \( T^{2} + 218700 \) Copy content Toggle raw display
$47$ \( (T + 468)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 15552 \) Copy content Toggle raw display
$59$ \( (T + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T - 145)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 98283 \) Copy content Toggle raw display
$71$ \( (T + 576)^{2} \) Copy content Toggle raw display
$73$ \( (T + 809)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 247107 \) Copy content Toggle raw display
$83$ \( (T + 360)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 73008 \) Copy content Toggle raw display
$97$ \( (T - 955)^{2} \) Copy content Toggle raw display
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