Properties

Label 1620.4.i.d
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
Analytic rank $0$
Dimension $2$
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] = [1620,4,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.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: \(95.5830942093\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q - 5 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 5 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} + ( - 60 \zeta_{6} + 60) q^{11} - 86 \zeta_{6} q^{13} + 18 q^{17} + 44 q^{19} - 48 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + ( - 186 \zeta_{6} + 186) q^{29} - 176 \zeta_{6} q^{31} - 80 q^{35} + 254 q^{37} - 186 \zeta_{6} q^{41} + ( - 100 \zeta_{6} + 100) q^{43} + (168 \zeta_{6} - 168) q^{47} + 87 \zeta_{6} q^{49} - 498 q^{53} - 300 q^{55} + 252 \zeta_{6} q^{59} + ( - 58 \zeta_{6} + 58) q^{61} + (430 \zeta_{6} - 430) q^{65} + 1036 \zeta_{6} q^{67} + 168 q^{71} + 506 q^{73} - 960 \zeta_{6} q^{77} + (272 \zeta_{6} - 272) q^{79} + (948 \zeta_{6} - 948) q^{83} - 90 \zeta_{6} q^{85} - 1014 q^{89} - 1376 q^{91} - 220 \zeta_{6} q^{95} + ( - 766 \zeta_{6} + 766) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{5} + 16 q^{7} + 60 q^{11} - 86 q^{13} + 36 q^{17} + 88 q^{19} - 48 q^{23} - 25 q^{25} + 186 q^{29} - 176 q^{31} - 160 q^{35} + 508 q^{37} - 186 q^{41} + 100 q^{43} - 168 q^{47} + 87 q^{49} - 996 q^{53} - 600 q^{55} + 252 q^{59} + 58 q^{61} - 430 q^{65} + 1036 q^{67} + 336 q^{71} + 1012 q^{73} - 960 q^{77} - 272 q^{79} - 948 q^{83} - 90 q^{85} - 2028 q^{89} - 2752 q^{91} - 220 q^{95} + 766 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.50000 4.33013i 0 8.00000 13.8564i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 0 8.00000 + 13.8564i 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 1620.4.i.d 2
3.b odd 2 1 1620.4.i.j 2
9.c even 3 1 20.4.a.a 1
9.c even 3 1 inner 1620.4.i.d 2
9.d odd 6 1 180.4.a.a 1
9.d odd 6 1 1620.4.i.j 2
36.f odd 6 1 80.4.a.c 1
36.h even 6 1 720.4.a.k 1
45.h odd 6 1 900.4.a.m 1
45.j even 6 1 100.4.a.a 1
45.k odd 12 2 100.4.c.a 2
45.l even 12 2 900.4.d.k 2
63.g even 3 1 980.4.i.e 2
63.h even 3 1 980.4.i.e 2
63.k odd 6 1 980.4.i.n 2
63.l odd 6 1 980.4.a.c 1
63.t odd 6 1 980.4.i.n 2
72.n even 6 1 320.4.a.d 1
72.p odd 6 1 320.4.a.k 1
99.h odd 6 1 2420.4.a.d 1
144.v odd 12 2 1280.4.d.c 2
144.x even 12 2 1280.4.d.n 2
180.p odd 6 1 400.4.a.o 1
180.x even 12 2 400.4.c.j 2
360.z odd 6 1 1600.4.a.p 1
360.bk even 6 1 1600.4.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 9.c even 3 1
80.4.a.c 1 36.f odd 6 1
100.4.a.a 1 45.j even 6 1
100.4.c.a 2 45.k odd 12 2
180.4.a.a 1 9.d odd 6 1
320.4.a.d 1 72.n even 6 1
320.4.a.k 1 72.p odd 6 1
400.4.a.o 1 180.p odd 6 1
400.4.c.j 2 180.x even 12 2
720.4.a.k 1 36.h even 6 1
900.4.a.m 1 45.h odd 6 1
900.4.d.k 2 45.l even 12 2
980.4.a.c 1 63.l odd 6 1
980.4.i.e 2 63.g even 3 1
980.4.i.e 2 63.h even 3 1
980.4.i.n 2 63.k odd 6 1
980.4.i.n 2 63.t odd 6 1
1280.4.d.c 2 144.v odd 12 2
1280.4.d.n 2 144.x even 12 2
1600.4.a.p 1 360.z odd 6 1
1600.4.a.bl 1 360.bk even 6 1
1620.4.i.d 2 1.a even 1 1 trivial
1620.4.i.d 2 9.c even 3 1 inner
1620.4.i.j 2 3.b odd 2 1
1620.4.i.j 2 9.d odd 6 1
2420.4.a.d 1 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{2} - 16T_{7} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} - 60T_{11} + 3600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$11$ \( T^{2} - 60T + 3600 \) Copy content Toggle raw display
$13$ \( T^{2} + 86T + 7396 \) Copy content Toggle raw display
$17$ \( (T - 18)^{2} \) Copy content Toggle raw display
$19$ \( (T - 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 48T + 2304 \) Copy content Toggle raw display
$29$ \( T^{2} - 186T + 34596 \) Copy content Toggle raw display
$31$ \( T^{2} + 176T + 30976 \) Copy content Toggle raw display
$37$ \( (T - 254)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 186T + 34596 \) Copy content Toggle raw display
$43$ \( T^{2} - 100T + 10000 \) Copy content Toggle raw display
$47$ \( T^{2} + 168T + 28224 \) Copy content Toggle raw display
$53$ \( (T + 498)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 252T + 63504 \) Copy content Toggle raw display
$61$ \( T^{2} - 58T + 3364 \) Copy content Toggle raw display
$67$ \( T^{2} - 1036 T + 1073296 \) Copy content Toggle raw display
$71$ \( (T - 168)^{2} \) Copy content Toggle raw display
$73$ \( (T - 506)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 272T + 73984 \) Copy content Toggle raw display
$83$ \( T^{2} + 948T + 898704 \) Copy content Toggle raw display
$89$ \( (T + 1014)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 766T + 586756 \) Copy content Toggle raw display
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