Properties

Label 1728.3.e.h
Level $1728$
Weight $3$
Character orbit 1728.e
Analytic conductor $47.085$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,3,Mod(1025,1728)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1025");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 216)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - 3 q^{7} + 7 \beta q^{11} - 7 q^{13} - 5 \beta q^{17} + 19 q^{19} - \beta q^{23} + 17 q^{25} + 18 \beta q^{29} - 10 q^{31} - 3 \beta q^{35} - 63 q^{37} + 18 \beta q^{41} + 50 q^{43} - 15 \beta q^{47} - 40 q^{49} - 26 \beta q^{53} - 56 q^{55} - 35 \beta q^{59} - 79 q^{61} - 7 \beta q^{65} - 77 q^{67} + 28 \beta q^{71} - 17 q^{73} - 21 \beta q^{77} - 11 q^{79} + 14 \beta q^{83} + 40 q^{85} + 15 \beta q^{89} + 21 q^{91} + 19 \beta q^{95} - 97 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{7} - 14 q^{13} + 38 q^{19} + 34 q^{25} - 20 q^{31} - 126 q^{37} + 100 q^{43} - 80 q^{49} - 112 q^{55} - 158 q^{61} - 154 q^{67} - 34 q^{73} - 22 q^{79} + 80 q^{85} + 42 q^{91} - 194 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
1.41421i
1.41421i
0 0 0 2.82843i 0 −3.00000 0 0 0
1025.2 0 0 0 2.82843i 0 −3.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.e.h 2
3.b odd 2 1 inner 1728.3.e.h 2
4.b odd 2 1 1728.3.e.j 2
8.b even 2 1 216.3.e.a 2
8.d odd 2 1 432.3.e.f 2
12.b even 2 1 1728.3.e.j 2
24.f even 2 1 432.3.e.f 2
24.h odd 2 1 216.3.e.a 2
72.j odd 6 2 648.3.m.c 4
72.l even 6 2 1296.3.q.g 4
72.n even 6 2 648.3.m.c 4
72.p odd 6 2 1296.3.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.a 2 8.b even 2 1
216.3.e.a 2 24.h odd 2 1
432.3.e.f 2 8.d odd 2 1
432.3.e.f 2 24.f even 2 1
648.3.m.c 4 72.j odd 6 2
648.3.m.c 4 72.n even 6 2
1296.3.q.g 4 72.l even 6 2
1296.3.q.g 4 72.p odd 6 2
1728.3.e.h 2 1.a even 1 1 trivial
1728.3.e.h 2 3.b odd 2 1 inner
1728.3.e.j 2 4.b odd 2 1
1728.3.e.j 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{7} + 3 \) 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
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 392 \) Copy content Toggle raw display
$13$ \( (T + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 200 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 2592 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T + 63)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2592 \) Copy content Toggle raw display
$43$ \( (T - 50)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1800 \) Copy content Toggle raw display
$53$ \( T^{2} + 5408 \) Copy content Toggle raw display
$59$ \( T^{2} + 9800 \) Copy content Toggle raw display
$61$ \( (T + 79)^{2} \) Copy content Toggle raw display
$67$ \( (T + 77)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 6272 \) Copy content Toggle raw display
$73$ \( (T + 17)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1568 \) Copy content Toggle raw display
$89$ \( T^{2} + 1800 \) Copy content Toggle raw display
$97$ \( (T + 97)^{2} \) Copy content Toggle raw display
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