Properties

Label 3800.1.o.c
Level 38003800
Weight 11
Character orbit 3800.o
Self dual yes
Analytic conductor 1.8961.896
Analytic rank 00
Dimension 33
Projective image D9D_{9}
CM discriminant -152
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(1101,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.1101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 3800=235219 3800 = 2^{3} \cdot 5^{2} \cdot 19
Weight: k k == 1 1
Character orbit: [χ][\chi] == 3800.o (of order 22, degree 11, 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: yes
Analytic conductor: 1.896447048011.89644704801
Analytic rank: 00
Dimension: 33
Coefficient field: Q(ζ18)+\Q(\zeta_{18})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x33x1 x^{3} - 3x - 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D9D_{9}
Projective field: Galois closure of 9.1.8340544000000.1

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β21,\beta_1,\beta_2 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qq2+(β2β1)q3+q4+(β2+β1)q6β1q7q8+(β1+1)q9+(β2β1)q12β2q13+β1q14+q16++(β21)q98+O(q100) q - q^{2} + (\beta_{2} - \beta_1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{6} - \beta_1 q^{7} - q^{8} + ( - \beta_1 + 1) q^{9} + (\beta_{2} - \beta_1) q^{12} - \beta_{2} q^{13} + \beta_1 q^{14} + q^{16}+ \cdots + ( - \beta_{2} - 1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q3q2+3q43q8+3q9+3q163q183q19+3q21+3q273q32+3q36+3q37+3q383q393q423q47+3q49+3q513q54+3q98+O(q100) 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 3 q^{9} + 3 q^{16} - 3 q^{18} - 3 q^{19} + 3 q^{21} + 3 q^{27} - 3 q^{32} + 3 q^{36} + 3 q^{37} + 3 q^{38} - 3 q^{39} - 3 q^{42} - 3 q^{47} + 3 q^{49} + 3 q^{51} - 3 q^{54}+ \cdots - 3 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of ν=ζ18+ζ181\nu = \zeta_{18} + \zeta_{18}^{-1}:

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν22 \nu^{2} - 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+2 \beta_{2} + 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3800Z)×\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times.

nn 401401 951951 19011901 19771977
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}
1101.1
−0.347296
1.87939
−1.53209
−1.00000 −1.53209 1.00000 0 1.53209 0.347296 −1.00000 1.34730 0
1101.2 −1.00000 −0.347296 1.00000 0 0.347296 −1.87939 −1.00000 −0.879385 0
1101.3 −1.00000 1.87939 1.00000 0 −1.87939 1.53209 −1.00000 2.53209 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
152.g odd 2 1 CM by Q(38)\Q(\sqrt{-38})

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.o.c 3
5.b even 2 1 3800.1.o.e yes 3
5.c odd 4 2 3800.1.b.d 6
8.b even 2 1 3800.1.o.f yes 3
19.b odd 2 1 3800.1.o.f yes 3
40.f even 2 1 3800.1.o.d yes 3
40.i odd 4 2 3800.1.b.c 6
95.d odd 2 1 3800.1.o.d yes 3
95.g even 4 2 3800.1.b.c 6
152.g odd 2 1 CM 3800.1.o.c 3
760.b odd 2 1 3800.1.o.e yes 3
760.t even 4 2 3800.1.b.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.b.c 6 40.i odd 4 2
3800.1.b.c 6 95.g even 4 2
3800.1.b.d 6 5.c odd 4 2
3800.1.b.d 6 760.t even 4 2
3800.1.o.c 3 1.a even 1 1 trivial
3800.1.o.c 3 152.g odd 2 1 CM
3800.1.o.d yes 3 40.f even 2 1
3800.1.o.d yes 3 95.d odd 2 1
3800.1.o.e yes 3 5.b even 2 1
3800.1.o.e yes 3 760.b odd 2 1
3800.1.o.f yes 3 8.b even 2 1
3800.1.o.f yes 3 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(3800,[χ])S_{1}^{\mathrm{new}}(3800, [\chi]):

T333T31 T_{3}^{3} - 3T_{3} - 1 Copy content Toggle raw display
T733T7+1 T_{7}^{3} - 3T_{7} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+1)3 (T + 1)^{3} Copy content Toggle raw display
33 T33T1 T^{3} - 3T - 1 Copy content Toggle raw display
55 T3 T^{3} Copy content Toggle raw display
77 T33T+1 T^{3} - 3T + 1 Copy content Toggle raw display
1111 T3 T^{3} Copy content Toggle raw display
1313 T33T1 T^{3} - 3T - 1 Copy content Toggle raw display
1717 T33T+1 T^{3} - 3T + 1 Copy content Toggle raw display
1919 (T+1)3 (T + 1)^{3} Copy content Toggle raw display
2323 T33T+1 T^{3} - 3T + 1 Copy content Toggle raw display
2929 T33T1 T^{3} - 3T - 1 Copy content Toggle raw display
3131 T3 T^{3} Copy content Toggle raw display
3737 (T1)3 (T - 1)^{3} Copy content Toggle raw display
4141 T3 T^{3} Copy content Toggle raw display
4343 T3 T^{3} Copy content Toggle raw display
4747 (T+1)3 (T + 1)^{3} Copy content Toggle raw display
5353 T33T1 T^{3} - 3T - 1 Copy content Toggle raw display
5959 T33T1 T^{3} - 3T - 1 Copy content Toggle raw display
6161 T3 T^{3} Copy content Toggle raw display
6767 T33T1 T^{3} - 3T - 1 Copy content Toggle raw display
7171 T3 T^{3} Copy content Toggle raw display
7373 T33T+1 T^{3} - 3T + 1 Copy content Toggle raw display
7979 T3 T^{3} Copy content Toggle raw display
8383 T3 T^{3} Copy content Toggle raw display
8989 T3 T^{3} Copy content Toggle raw display
9797 T3 T^{3} Copy content Toggle raw display
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