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Properties

Label 1728.2.i.k
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
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] = [1728,2,Mod(577,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), degree \(2\), 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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{5} + (\beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{13}+ \cdots + ( - 2 \beta_{2} - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 16 q^{19} - 6 q^{23} + 8 q^{25} - 10 q^{29} - 10 q^{31} - 4 q^{35} - 16 q^{37} - 14 q^{41} + 10 q^{43} + 2 q^{47} + 16 q^{53} + 4 q^{55} - 14 q^{59} - 6 q^{61} - 2 q^{65} + 10 q^{67} + 8 q^{71} - 10 q^{77} - 22 q^{79} + 6 q^{83} + 32 q^{89} - 52 q^{91} - 8 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
0 0 0 0.500000 − 0.866025i 0 −1.72474 − 2.98735i 0 0 0
577.2 0 0 0 0.500000 − 0.866025i 0 0.724745 + 1.25529i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −1.72474 + 2.98735i 0 0 0
1153.2 0 0 0 0.500000 + 0.866025i 0 0.724745 − 1.25529i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.k 4
3.b odd 2 1 576.2.i.i 4
4.b odd 2 1 1728.2.i.m 4
8.b even 2 1 864.2.i.c 4
8.d odd 2 1 864.2.i.e 4
9.c even 3 1 inner 1728.2.i.k 4
9.c even 3 1 5184.2.a.bn 2
9.d odd 6 1 576.2.i.i 4
9.d odd 6 1 5184.2.a.by 2
12.b even 2 1 576.2.i.m 4
24.f even 2 1 288.2.i.c ✓ 4
24.h odd 2 1 288.2.i.e yes 4
36.f odd 6 1 1728.2.i.m 4
36.f odd 6 1 5184.2.a.bj 2
36.h even 6 1 576.2.i.m 4
36.h even 6 1 5184.2.a.bu 2
72.j odd 6 1 288.2.i.e yes 4
72.j odd 6 1 2592.2.a.n 2
72.l even 6 1 288.2.i.c ✓ 4
72.l even 6 1 2592.2.a.j 2
72.n even 6 1 864.2.i.c 4
72.n even 6 1 2592.2.a.s 2
72.p odd 6 1 864.2.i.e 4
72.p odd 6 1 2592.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c ✓ 4 24.f even 2 1
288.2.i.c ✓ 4 72.l even 6 1
288.2.i.e yes 4 24.h odd 2 1
288.2.i.e yes 4 72.j odd 6 1
576.2.i.i 4 3.b odd 2 1
576.2.i.i 4 9.d odd 6 1
576.2.i.m 4 12.b even 2 1
576.2.i.m 4 36.h even 6 1
864.2.i.c 4 8.b even 2 1
864.2.i.c 4 72.n even 6 1
864.2.i.e 4 8.d odd 2 1
864.2.i.e 4 72.p odd 6 1
1728.2.i.k 4 1.a even 1 1 trivial
1728.2.i.k 4 9.c even 3 1 inner
1728.2.i.m 4 4.b odd 2 1
1728.2.i.m 4 36.f odd 6 1
2592.2.a.j 2 72.l even 6 1
2592.2.a.n 2 72.j odd 6 1
2592.2.a.o 2 72.p odd 6 1
2592.2.a.s 2 72.n even 6 1
5184.2.a.bj 2 36.f odd 6 1
5184.2.a.bn 2 9.c even 3 1
5184.2.a.bu 2 36.h even 6 1
5184.2.a.by 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 9T_{7}^{2} - 10T_{7} + 25 \) 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
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 14 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + \cdots + 225 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + \cdots + 13225 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 529 \) Copy content Toggle raw display
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