Properties

Label 1728.3.e.o
Level $1728$
Weight $3$
Character orbit 1728.e
Analytic conductor $47.085$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 864)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 3) q^{7} + ( - 2 \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 3) q^{13} - \beta_1 q^{17} + (2 \beta_{3} + 15) q^{19} + ( - \beta_{2} - 5 \beta_1) q^{23} + ( - 4 \beta_{3} - 19) q^{25} - 2 \beta_1 q^{29} + 18 q^{31} + ( - 14 \beta_{2} - 7 \beta_1) q^{35} + ( - 3 \beta_{3} - 7) q^{37} + (4 \beta_{2} + 6 \beta_1) q^{41} + ( - 4 \beta_{3} + 6) q^{43} + (11 \beta_{2} + \beta_1) q^{47} + (6 \beta_{3} + 32) q^{49} + ( - 14 \beta_{2} - 6 \beta_1) q^{53} + (8 \beta_{3} + 60) q^{55} + ( - 8 \beta_{2} + 5 \beta_1) q^{59} + (9 \beta_{3} + 13) q^{61} + ( - 14 \beta_{2} - 7 \beta_1) q^{65} + ( - 4 \beta_{3} + 51) q^{67} + (10 \beta_{2} - 4 \beta_1) q^{71} + (2 \beta_{3} + 15) q^{73} + (21 \beta_{2} + 15 \beta_1) q^{77} + ( - 3 \beta_{3} + 81) q^{79} + ( - 14 \beta_{2} + 2 \beta_1) q^{83} + 28 q^{85} + (8 \beta_{2} - 13 \beta_1) q^{89} + (6 \beta_{3} + 81) q^{91} + (37 \beta_{2} + 23 \beta_1) q^{95} + (6 \beta_{3} - 45) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{7} - 12 q^{13} + 60 q^{19} - 76 q^{25} + 72 q^{31} - 28 q^{37} + 24 q^{43} + 128 q^{49} + 240 q^{55} + 52 q^{61} + 204 q^{67} + 60 q^{73} + 324 q^{79} + 112 q^{85} + 324 q^{91} - 180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{3} - 6\zeta_{8}^{2} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\zeta_{8}^{3} - 4\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\zeta_{8}^{3} + 6\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( 2\beta_{3} - 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( -\beta_{2} - 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -2\beta_{3} - 3\beta_{2} ) / 24 \) 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
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 8.82843i 0 −11.4853 0 0 0
1025.2 0 0 0 3.17157i 0 5.48528 0 0 0
1025.3 0 0 0 3.17157i 0 5.48528 0 0 0
1025.4 0 0 0 8.82843i 0 −11.4853 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.o 4
3.b odd 2 1 inner 1728.3.e.o 4
4.b odd 2 1 1728.3.e.r 4
8.b even 2 1 864.3.e.b 4
8.d odd 2 1 864.3.e.d yes 4
12.b even 2 1 1728.3.e.r 4
24.f even 2 1 864.3.e.d yes 4
24.h odd 2 1 864.3.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.e.b 4 8.b even 2 1
864.3.e.b 4 24.h odd 2 1
864.3.e.d yes 4 8.d odd 2 1
864.3.e.d yes 4 24.f even 2 1
1728.3.e.o 4 1.a even 1 1 trivial
1728.3.e.o 4 3.b odd 2 1 inner
1728.3.e.r 4 4.b odd 2 1
1728.3.e.r 4 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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 88T_{5}^{2} + 784 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} - 63 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T - 63)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 216T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T - 63)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$19$ \( (T^{2} - 30 T - 63)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1944 T^{2} + 685584 \) Copy content Toggle raw display
$29$ \( T^{4} + 352 T^{2} + 12544 \) Copy content Toggle raw display
$31$ \( (T - 18)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T - 599)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2656 T^{2} + \cdots + 1597696 \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T - 1116)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 7128 T^{2} + \cdots + 12194064 \) Copy content Toggle raw display
$53$ \( T^{4} + 10336 T^{2} + \cdots + 6635776 \) Copy content Toggle raw display
$59$ \( T^{4} + 8856 T^{2} + \cdots + 6906384 \) Copy content Toggle raw display
$61$ \( (T^{2} - 26 T - 5663)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 102 T + 1449)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 10368 T^{2} + \cdots + 16257024 \) Copy content Toggle raw display
$73$ \( (T^{2} - 30 T - 63)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 162 T + 5913)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 14688 T^{2} + \cdots + 49787136 \) Copy content Toggle raw display
$89$ \( T^{4} + 25624 T^{2} + \cdots + 414736 \) Copy content Toggle raw display
$97$ \( (T^{2} + 90 T - 567)^{2} \) Copy content Toggle raw display
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