Properties

Label 1620.3.l.f
Level $1620$
Weight $3$
Character orbit 1620.l
Analytic conductor $44.142$
Analytic rank $0$
Dimension $24$
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] = [1620,3,Mod(973,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.973");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.l (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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{11} + 18 q^{17} + 12 q^{23} + 6 q^{25} + 36 q^{35} - 42 q^{37} - 72 q^{41} + 78 q^{47} - 156 q^{53} + 66 q^{55} + 96 q^{61} + 132 q^{65} + 78 q^{67} - 156 q^{71} + 240 q^{77} - 132 q^{83} - 96 q^{85} + 84 q^{91} + 168 q^{95} + 90 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
973.1 0 0 0 −4.98831 0.341776i 0 7.68868 7.68868i 0 0 0
973.2 0 0 0 −3.97841 3.02858i 0 −7.24392 + 7.24392i 0 0 0
973.3 0 0 0 −3.87041 + 3.16543i 0 3.43863 3.43863i 0 0 0
973.4 0 0 0 −3.12092 + 3.90639i 0 −9.07494 + 9.07494i 0 0 0
973.5 0 0 0 −2.01955 + 4.57399i 0 2.94505 2.94505i 0 0 0
973.6 0 0 0 −1.81205 4.66009i 0 −0.519255 + 0.519255i 0 0 0
973.7 0 0 0 0.555312 4.96907i 0 3.23469 3.23469i 0 0 0
973.8 0 0 0 2.81568 + 4.13182i 0 −4.37804 + 4.37804i 0 0 0
973.9 0 0 0 3.08973 3.93110i 0 −2.44070 + 2.44070i 0 0 0
973.10 0 0 0 3.37743 + 3.68687i 0 7.75406 7.75406i 0 0 0
973.11 0 0 0 4.95548 + 0.665771i 0 1.80677 1.80677i 0 0 0
973.12 0 0 0 4.99601 0.199651i 0 −3.21104 + 3.21104i 0 0 0
1297.1 0 0 0 −4.98831 + 0.341776i 0 7.68868 + 7.68868i 0 0 0
1297.2 0 0 0 −3.97841 + 3.02858i 0 −7.24392 7.24392i 0 0 0
1297.3 0 0 0 −3.87041 3.16543i 0 3.43863 + 3.43863i 0 0 0
1297.4 0 0 0 −3.12092 3.90639i 0 −9.07494 9.07494i 0 0 0
1297.5 0 0 0 −2.01955 4.57399i 0 2.94505 + 2.94505i 0 0 0
1297.6 0 0 0 −1.81205 + 4.66009i 0 −0.519255 0.519255i 0 0 0
1297.7 0 0 0 0.555312 + 4.96907i 0 3.23469 + 3.23469i 0 0 0
1297.8 0 0 0 2.81568 4.13182i 0 −4.37804 4.37804i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 973.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.l.f 24
3.b odd 2 1 1620.3.l.g 24
5.c odd 4 1 inner 1620.3.l.f 24
9.c even 3 2 180.3.u.a 48
9.d odd 6 2 540.3.v.a 48
15.e even 4 1 1620.3.l.g 24
45.k odd 12 2 180.3.u.a 48
45.l even 12 2 540.3.v.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.u.a 48 9.c even 3 2
180.3.u.a 48 45.k odd 12 2
540.3.v.a 48 9.d odd 6 2
540.3.v.a 48 45.l even 12 2
1620.3.l.f 24 1.a even 1 1 trivial
1620.3.l.f 24 5.c odd 4 1 inner
1620.3.l.g 24 3.b odd 2 1
1620.3.l.g 24 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{24} - 310 T_{7}^{21} + 34980 T_{7}^{20} - 44088 T_{7}^{19} + 48050 T_{7}^{18} - 4294854 T_{7}^{17} + 309277434 T_{7}^{16} - 560018860 T_{7}^{15} + 622491612 T_{7}^{14} + \cdots + 69\!\cdots\!44 \) Copy content Toggle raw display
\( T_{11}^{12} + 6 T_{11}^{11} - 735 T_{11}^{10} - 3870 T_{11}^{9} + 171969 T_{11}^{8} + 667344 T_{11}^{7} - 16173989 T_{11}^{6} - 37660398 T_{11}^{5} + 649389456 T_{11}^{4} + 623750130 T_{11}^{3} + \cdots + 807941620 \) Copy content Toggle raw display