Properties

Label 1800.1.c.a
Level 18001800
Weight 11
Character orbit 1800.c
Analytic conductor 0.8980.898
Analytic rank 00
Dimension 44
Projective image S4S_{4}
CM/RM no
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,1,Mod(449,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.449");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 1800=233252 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1800.c (of order 22, degree 11, 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: 0.8983170227390.898317022739
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ8)\Q(\zeta_{8})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2 2
Twist minimal: yes
Projective image: S4S_{4}
Projective field: Galois closure of 4.2.10800.2

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == q+ζ82q7+(ζ83ζ8)q11+ζ82q13+(ζ83+ζ8)q17q19+(ζ83+ζ8)q23+(ζ83+ζ8)q29+ζ82q97+O(q100) q + \zeta_{8}^{2} q^{7} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + \zeta_{8}^{2} q^{13} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{17} - q^{19} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{23} + (\zeta_{8}^{3} + \zeta_{8}) q^{29}+ \cdots - \zeta_{8}^{2} q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q19+4q31+4q614q91+O(q100) 4 q - 4 q^{19} + 4 q^{31} + 4 q^{61} - 4 q^{91}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1800Z)×\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times.

nn 577577 901901 10011001 13511351
χ(n)\chi(n) 1-1 11 1-1 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
449.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 0 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.c.a 4
3.b odd 2 1 inner 1800.1.c.a 4
4.b odd 2 1 3600.1.c.a 4
5.b even 2 1 inner 1800.1.c.a 4
5.c odd 4 1 1800.1.l.a 2
5.c odd 4 1 1800.1.l.b yes 2
12.b even 2 1 3600.1.c.a 4
15.d odd 2 1 inner 1800.1.c.a 4
15.e even 4 1 1800.1.l.a 2
15.e even 4 1 1800.1.l.b yes 2
20.d odd 2 1 3600.1.c.a 4
20.e even 4 1 3600.1.l.a 2
20.e even 4 1 3600.1.l.b 2
60.h even 2 1 3600.1.c.a 4
60.l odd 4 1 3600.1.l.a 2
60.l odd 4 1 3600.1.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.c.a 4 1.a even 1 1 trivial
1800.1.c.a 4 3.b odd 2 1 inner
1800.1.c.a 4 5.b even 2 1 inner
1800.1.c.a 4 15.d odd 2 1 inner
1800.1.l.a 2 5.c odd 4 1
1800.1.l.a 2 15.e even 4 1
1800.1.l.b yes 2 5.c odd 4 1
1800.1.l.b yes 2 15.e even 4 1
3600.1.c.a 4 4.b odd 2 1
3600.1.c.a 4 12.b even 2 1
3600.1.c.a 4 20.d odd 2 1
3600.1.c.a 4 60.h even 2 1
3600.1.l.a 2 20.e even 4 1
3600.1.l.a 2 60.l odd 4 1
3600.1.l.b 2 20.e even 4 1
3600.1.l.b 2 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace S1new(1800,[χ])S_{1}^{\mathrm{new}}(1800, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
1111 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
1313 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
1717 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
1919 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
2323 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
2929 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
3131 (T1)4 (T - 1)^{4} Copy content Toggle raw display
3737 T4 T^{4} Copy content Toggle raw display
4141 T4 T^{4} Copy content Toggle raw display
4343 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
4747 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
5353 T4 T^{4} Copy content Toggle raw display
5959 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
6161 (T1)4 (T - 1)^{4} Copy content Toggle raw display
6767 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 T4 T^{4} Copy content Toggle raw display
7979 T4 T^{4} Copy content Toggle raw display
8383 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
8989 T4 T^{4} Copy content Toggle raw display
9797 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
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