↑

Properties

Label 3584.2.m.be
Level $3584$
Weight $2$
Character orbit 3584.m
Analytic conductor $28.618$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3584,2,Mod(897,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.897");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.m (of order \(4\), degree \(2\), 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: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{3} - \beta_1 + 1) q^{5} - \beta_1 q^{7} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{3} - \beta_1 + 1) q^{5} - \beta_1 q^{7} + 3 \beta_1 q^{9} + (2 \beta_1 - 2) q^{11} + (\beta_{2} - 3 \beta_1 - 3) q^{13} + ( - \beta_{3} - \beta_{2} + 6) q^{15} + (\beta_{3} + \beta_{2}) q^{17} - \beta_{2} q^{19} - \beta_{3} q^{21} + ( - \beta_{3} + \beta_{2} - 6 \beta_1) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{25} + (2 \beta_{2} + \beta_1 + 1) q^{29} + (\beta_{3} + \beta_{2} - 4) q^{31} + (2 \beta_{3} + 2 \beta_{2}) q^{33} + (\beta_{2} - \beta_1 - 1) q^{35} + ( - 2 \beta_{3} + 3 \beta_1 - 3) q^{37} + ( - 3 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{39} + (\beta_{3} - \beta_{2}) q^{41} + ( - 2 \beta_{3} - 4 \beta_1 + 4) q^{43} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{45} + ( - \beta_{3} - \beta_{2} + 8) q^{47} - q^{49} + ( - 6 \beta_1 - 6) q^{51} + ( - \beta_1 + 1) q^{53} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{55} + 6 \beta_1 q^{57} + (\beta_{3} - 4 \beta_1 + 4) q^{59} + (\beta_{2} + 7 \beta_1 + 7) q^{61} + 3 q^{63} + (4 \beta_{3} + 4 \beta_{2} - 12) q^{65} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{67} + ( - 6 \beta_{3} - 6 \beta_1 + 6) q^{69} + 4 \beta_1 q^{71} - 6 \beta_1 q^{73} + ( - 3 \beta_{3} - 12 \beta_1 + 12) q^{75} + (2 \beta_1 + 2) q^{77} + 4 q^{79} + 9 q^{81} + (\beta_{2} + 4 \beta_1 + 4) q^{83} + (2 \beta_{3} + 6 \beta_1 - 6) q^{85} + (\beta_{3} - \beta_{2} - 12 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{89} + (\beta_{3} + 3 \beta_1 - 3) q^{91} + (4 \beta_{2} - 6 \beta_1 - 6) q^{93} + ( - \beta_{3} - \beta_{2} + 6) q^{95} + (3 \beta_{3} + 3 \beta_{2} + 8) q^{97} + ( - 6 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 8 q^{11} - 12 q^{13} + 24 q^{15} + 4 q^{29} - 16 q^{31} - 4 q^{35} - 12 q^{37} + 16 q^{43} + 12 q^{45} + 32 q^{47} - 4 q^{49} - 24 q^{51} + 4 q^{53} + 16 q^{59} + 28 q^{61} + 12 q^{63} - 48 q^{65} - 16 q^{67} + 24 q^{69} + 48 q^{75} + 8 q^{77} + 16 q^{79} + 36 q^{81} + 16 q^{83} - 24 q^{85} - 12 q^{91} - 24 q^{93} + 24 q^{95} + 32 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1 \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 −1.73205 + 1.73205i 0 −0.732051 − 0.732051i 0 1.00000i 0 − 3.00000i 0
897.2 0 1.73205 − 1.73205i 0 2.73205 + 2.73205i 0 1.00000i 0 − 3.00000i 0
2689.1 0 −1.73205 − 1.73205i 0 −0.732051 + 0.732051i 0 − 1.00000i 0 3.00000i 0
2689.2 0 1.73205 + 1.73205i 0 2.73205 − 2.73205i 0 − 1.00000i 0 3.00000i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.m.be yes 4
4.b odd 2 1 3584.2.m.bf yes 4
8.b even 2 1 3584.2.m.bd yes 4
8.d odd 2 1 3584.2.m.bc ✓ 4
16.e even 4 1 3584.2.m.bd yes 4
16.e even 4 1 inner 3584.2.m.be yes 4
16.f odd 4 1 3584.2.m.bc ✓ 4
16.f odd 4 1 3584.2.m.bf yes 4
32.g even 8 2 7168.2.a.v 4
32.h odd 8 2 7168.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.bc ✓ 4 8.d odd 2 1
3584.2.m.bc ✓ 4 16.f odd 4 1
3584.2.m.bd yes 4 8.b even 2 1
3584.2.m.bd yes 4 16.e even 4 1
3584.2.m.be yes 4 1.a even 1 1 trivial
3584.2.m.be yes 4 16.e even 4 1 inner
3584.2.m.bf yes 4 4.b odd 2 1
3584.2.m.bf yes 4 16.f odd 4 1
7168.2.a.u 4 32.h odd 8 2
7168.2.a.v 4 32.g even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{4} + 36 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{3} + 8T_{5}^{2} + 16T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 8 \) Copy content Toggle raw display
\( T_{13}^{4} + 12T_{13}^{3} + 72T_{13}^{2} + 144T_{13} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 36 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( (T^{2} - 16 T + 52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$61$ \( T^{4} - 28 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$89$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 44)^{2} \) Copy content Toggle raw display
show more
show less