Properties

Label 576.2.i.e
Level $576$
Weight $2$
Character orbit 576.i
Analytic conductor $4.599$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,2,Mod(193,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, 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("576.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(4.59938315643\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 1) q^{3} + 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 1) q^{3} + 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - 3 q^{9} + (3 \zeta_{6} - 3) q^{11} - \zeta_{6} q^{13} + (3 \zeta_{6} - 6) q^{15} + 6 q^{17} - 4 q^{19} + ( - \zeta_{6} - 1) q^{21} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 3 \zeta_{6} + 3) q^{29} + 5 \zeta_{6} q^{31} + ( - 3 \zeta_{6} - 3) q^{33} - 3 q^{35} - 2 q^{37} + ( - \zeta_{6} + 2) q^{39} - 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 9 \zeta_{6} q^{45} + (9 \zeta_{6} - 9) q^{47} + 6 \zeta_{6} q^{49} + (12 \zeta_{6} - 6) q^{51} + 6 q^{53} - 9 q^{55} + ( - 8 \zeta_{6} + 4) q^{57} + 3 \zeta_{6} q^{59} + (13 \zeta_{6} - 13) q^{61} + ( - 3 \zeta_{6} + 3) q^{63} + ( - 3 \zeta_{6} + 3) q^{65} + 7 \zeta_{6} q^{67} + ( - 3 \zeta_{6} + 6) q^{69} + 12 q^{71} - 10 q^{73} + ( - 4 \zeta_{6} - 4) q^{75} - 3 \zeta_{6} q^{77} + ( - 11 \zeta_{6} + 11) q^{79} + 9 q^{81} + ( - 9 \zeta_{6} + 9) q^{83} + 18 \zeta_{6} q^{85} + (3 \zeta_{6} + 3) q^{87} + 6 q^{89} + q^{91} + (5 \zeta_{6} - 10) q^{93} - 12 \zeta_{6} q^{95} + (11 \zeta_{6} - 11) q^{97} + ( - 9 \zeta_{6} + 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - q^{7} - 6 q^{9} - 3 q^{11} - q^{13} - 9 q^{15} + 12 q^{17} - 8 q^{19} - 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{29} + 5 q^{31} - 9 q^{33} - 6 q^{35} - 4 q^{37} + 3 q^{39} - 3 q^{41} + q^{43} - 9 q^{45} - 9 q^{47} + 6 q^{49} + 12 q^{53} - 18 q^{55} + 3 q^{59} - 13 q^{61} + 3 q^{63} + 3 q^{65} + 7 q^{67} + 9 q^{69} + 24 q^{71} - 20 q^{73} - 12 q^{75} - 3 q^{77} + 11 q^{79} + 18 q^{81} + 9 q^{83} + 18 q^{85} + 9 q^{87} + 12 q^{89} + 2 q^{91} - 15 q^{93} - 12 q^{95} - 11 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 1.50000 2.59808i 0 −0.500000 0.866025i 0 −3.00000 0
385.1 0 1.73205i 0 1.50000 + 2.59808i 0 −0.500000 + 0.866025i 0 −3.00000 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 576.2.i.e 2
3.b odd 2 1 1728.2.i.c 2
4.b odd 2 1 576.2.i.f 2
8.b even 2 1 144.2.i.a 2
8.d odd 2 1 36.2.e.a 2
9.c even 3 1 inner 576.2.i.e 2
9.c even 3 1 5184.2.a.f 1
9.d odd 6 1 1728.2.i.c 2
9.d odd 6 1 5184.2.a.bb 1
12.b even 2 1 1728.2.i.d 2
24.f even 2 1 108.2.e.a 2
24.h odd 2 1 432.2.i.c 2
36.f odd 6 1 576.2.i.f 2
36.f odd 6 1 5184.2.a.e 1
36.h even 6 1 1728.2.i.d 2
36.h even 6 1 5184.2.a.ba 1
40.e odd 2 1 900.2.i.b 2
40.k even 4 2 900.2.s.b 4
56.e even 2 1 1764.2.j.b 2
56.k odd 6 1 1764.2.i.a 2
56.k odd 6 1 1764.2.l.c 2
56.m even 6 1 1764.2.i.c 2
56.m even 6 1 1764.2.l.a 2
72.j odd 6 1 432.2.i.c 2
72.j odd 6 1 1296.2.a.b 1
72.l even 6 1 108.2.e.a 2
72.l even 6 1 324.2.a.a 1
72.n even 6 1 144.2.i.a 2
72.n even 6 1 1296.2.a.k 1
72.p odd 6 1 36.2.e.a 2
72.p odd 6 1 324.2.a.c 1
120.m even 2 1 2700.2.i.b 2
120.q odd 4 2 2700.2.s.b 4
168.e odd 2 1 5292.2.j.a 2
168.v even 6 1 5292.2.i.c 2
168.v even 6 1 5292.2.l.a 2
168.be odd 6 1 5292.2.i.a 2
168.be odd 6 1 5292.2.l.c 2
360.z odd 6 1 900.2.i.b 2
360.z odd 6 1 8100.2.a.j 1
360.bd even 6 1 2700.2.i.b 2
360.bd even 6 1 8100.2.a.g 1
360.bo even 12 2 900.2.s.b 4
360.bo even 12 2 8100.2.d.h 2
360.bt odd 12 2 2700.2.s.b 4
360.bt odd 12 2 8100.2.d.c 2
504.u odd 6 1 5292.2.i.a 2
504.ba odd 6 1 1764.2.i.a 2
504.be even 6 1 1764.2.j.b 2
504.bf even 6 1 1764.2.l.a 2
504.bt even 6 1 5292.2.l.a 2
504.ce odd 6 1 1764.2.l.c 2
504.cm odd 6 1 5292.2.l.c 2
504.co odd 6 1 5292.2.j.a 2
504.cy even 6 1 5292.2.i.c 2
504.cz even 6 1 1764.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 8.d odd 2 1
36.2.e.a 2 72.p odd 6 1
108.2.e.a 2 24.f even 2 1
108.2.e.a 2 72.l even 6 1
144.2.i.a 2 8.b even 2 1
144.2.i.a 2 72.n even 6 1
324.2.a.a 1 72.l even 6 1
324.2.a.c 1 72.p odd 6 1
432.2.i.c 2 24.h odd 2 1
432.2.i.c 2 72.j odd 6 1
576.2.i.e 2 1.a even 1 1 trivial
576.2.i.e 2 9.c even 3 1 inner
576.2.i.f 2 4.b odd 2 1
576.2.i.f 2 36.f odd 6 1
900.2.i.b 2 40.e odd 2 1
900.2.i.b 2 360.z odd 6 1
900.2.s.b 4 40.k even 4 2
900.2.s.b 4 360.bo even 12 2
1296.2.a.b 1 72.j odd 6 1
1296.2.a.k 1 72.n even 6 1
1728.2.i.c 2 3.b odd 2 1
1728.2.i.c 2 9.d odd 6 1
1728.2.i.d 2 12.b even 2 1
1728.2.i.d 2 36.h even 6 1
1764.2.i.a 2 56.k odd 6 1
1764.2.i.a 2 504.ba odd 6 1
1764.2.i.c 2 56.m even 6 1
1764.2.i.c 2 504.cz even 6 1
1764.2.j.b 2 56.e even 2 1
1764.2.j.b 2 504.be even 6 1
1764.2.l.a 2 56.m even 6 1
1764.2.l.a 2 504.bf even 6 1
1764.2.l.c 2 56.k odd 6 1
1764.2.l.c 2 504.ce odd 6 1
2700.2.i.b 2 120.m even 2 1
2700.2.i.b 2 360.bd even 6 1
2700.2.s.b 4 120.q odd 4 2
2700.2.s.b 4 360.bt odd 12 2
5184.2.a.e 1 36.f odd 6 1
5184.2.a.f 1 9.c even 3 1
5184.2.a.ba 1 36.h even 6 1
5184.2.a.bb 1 9.d odd 6 1
5292.2.i.a 2 168.be odd 6 1
5292.2.i.a 2 504.u odd 6 1
5292.2.i.c 2 168.v even 6 1
5292.2.i.c 2 504.cy even 6 1
5292.2.j.a 2 168.e odd 2 1
5292.2.j.a 2 504.co odd 6 1
5292.2.l.a 2 168.v even 6 1
5292.2.l.a 2 504.bt even 6 1
5292.2.l.c 2 168.be odd 6 1
5292.2.l.c 2 504.cm odd 6 1
8100.2.a.g 1 360.bd even 6 1
8100.2.a.j 1 360.z odd 6 1
8100.2.d.c 2 360.bt odd 12 2
8100.2.d.h 2 360.bo even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
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