Properties

Label 2888.1.f.d
Level 28882888
Weight 11
Character orbit 2888.f
Self dual yes
Analytic conductor 1.4411.441
Analytic rank 00
Dimension 33
Projective image D9D_{9}
CM discriminant -8
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2888,1,Mod(723,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2888.723"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: N N == 2888=23192 2888 = 2^{3} \cdot 19^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 2888.f (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 1.441299756481.44129975648
Analytic rank: 00
Dimension: 33
Coefficient field: Q(ζ18)+\Q(\zeta_{18})^+
Copy content comment:defining polynomial
 
Copy content 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: no (minimal twist has level 152)
Projective image: D9D_{9}
Projective field: Galois closure of 9.1.69564674215936.1
Artin image: D9D_9
Artin field: Galois closure of 9.1.69564674215936.1

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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) == q+q2β1q3+q4β1q6+q8+(β2+1)q9+(β2+β1)q11β1q12+q16q17+(β2+1)q18+(β2+β1)q22++(β11)q99+O(q100) q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{11} - \beta_1 q^{12} + q^{16} - q^{17} + (\beta_{2} + 1) q^{18} + ( - \beta_{2} + \beta_1) q^{22}+ \cdots + (\beta_1 - 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+3q2+3q4+3q8+3q9+3q163q17+3q18+3q253q27+3q323q333q34+3q363q43+3q49+3q503q54+3q643q66+3q99+O(q100) 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 3 q^{25} - 3 q^{27} + 3 q^{32} - 3 q^{33} - 3 q^{34} + 3 q^{36} - 3 q^{43} + 3 q^{49} + 3 q^{50} - 3 q^{54} + 3 q^{64} - 3 q^{66}+ \cdots - 3 q^{99}+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/2888Z)×\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times.

nn 14451445 21672167 25292529
χ(n)\chi(n) 1-1 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
723.1
1.87939
−0.347296
−1.53209
1.00000 −1.87939 1.00000 0 −1.87939 0 1.00000 2.53209 0
723.2 1.00000 0.347296 1.00000 0 0.347296 0 1.00000 −0.879385 0
723.3 1.00000 1.53209 1.00000 0 1.53209 0 1.00000 1.34730 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
8.d odd 2 1 CM by Q(2)\Q(\sqrt{-2})

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.f.d 3
8.d odd 2 1 CM 2888.1.f.d 3
19.b odd 2 1 2888.1.f.c 3
19.c even 3 2 2888.1.k.b 6
19.d odd 6 2 2888.1.k.c 6
19.e even 9 2 152.1.u.a 6
19.e even 9 2 2888.1.u.b 6
19.e even 9 2 2888.1.u.g 6
19.f odd 18 2 2888.1.u.a 6
19.f odd 18 2 2888.1.u.e 6
19.f odd 18 2 2888.1.u.f 6
57.l odd 18 2 1368.1.eh.a 6
76.l odd 18 2 608.1.bg.a 6
95.p even 18 2 3800.1.cv.c 6
95.q odd 36 4 3800.1.cq.b 12
152.b even 2 1 2888.1.f.c 3
152.k odd 6 2 2888.1.k.b 6
152.o even 6 2 2888.1.k.c 6
152.t even 18 2 608.1.bg.a 6
152.u odd 18 2 152.1.u.a 6
152.u odd 18 2 2888.1.u.b 6
152.u odd 18 2 2888.1.u.g 6
152.v even 18 2 2888.1.u.a 6
152.v even 18 2 2888.1.u.e 6
152.v even 18 2 2888.1.u.f 6
456.bu even 18 2 1368.1.eh.a 6
760.bz odd 18 2 3800.1.cv.c 6
760.cp even 36 4 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 19.e even 9 2
152.1.u.a 6 152.u odd 18 2
608.1.bg.a 6 76.l odd 18 2
608.1.bg.a 6 152.t even 18 2
1368.1.eh.a 6 57.l odd 18 2
1368.1.eh.a 6 456.bu even 18 2
2888.1.f.c 3 19.b odd 2 1
2888.1.f.c 3 152.b even 2 1
2888.1.f.d 3 1.a even 1 1 trivial
2888.1.f.d 3 8.d odd 2 1 CM
2888.1.k.b 6 19.c even 3 2
2888.1.k.b 6 152.k odd 6 2
2888.1.k.c 6 19.d odd 6 2
2888.1.k.c 6 152.o even 6 2
2888.1.u.a 6 19.f odd 18 2
2888.1.u.a 6 152.v even 18 2
2888.1.u.b 6 19.e even 9 2
2888.1.u.b 6 152.u odd 18 2
2888.1.u.e 6 19.f odd 18 2
2888.1.u.e 6 152.v even 18 2
2888.1.u.f 6 19.f odd 18 2
2888.1.u.f 6 152.v even 18 2
2888.1.u.g 6 19.e even 9 2
2888.1.u.g 6 152.u odd 18 2
3800.1.cq.b 12 95.q odd 36 4
3800.1.cq.b 12 760.cp even 36 4
3800.1.cv.c 6 95.p even 18 2
3800.1.cv.c 6 760.bz odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T333T3+1 T_{3}^{3} - 3T_{3} + 1 acting on S1new(2888,[χ])S_{1}^{\mathrm{new}}(2888, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

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