
// Make newform 2880.2.t.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2880_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2880_t_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2880_2_t_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2880_2_t_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2880_t();"
function MakeCharacter_2880_t()
    N := 2880;
    order := 4;
    char_gens := [2431, 901, 641, 577];
    v := [4, 3, 4, 4];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2880_t_Hecke();"
function MakeCharacter_2880_t_Hecke(Kf)
    N := 2880;
    order := 4;
    char_gens := [2431, 901, 641, 577];
    char_values := [[1, 0, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, -1, 0, 0], [0, 2, 2, 2], [0, 2, 0, 0], [-2, 0, 2, -2], [-4, -1, 0, 1], [-1, 0, 1, 4], [0, -1, -4, -1], [2, 0, -2, 2], [2, -4, 0, 4], [6, -2, 6, 0], [0, -2, -2, -2], [2, -8, 2, 0], [0, -5, 0, 5], [4, 0, 4, 0], [-6, -2, -6, 0], [1, 0, -1, -12], [2, 0, -2, -8], [0, 0, -8, 0], [0, -2, 6, -2], [8, 4, 0, -4], [4, 0, -4, 10], [0, -4, 2, -4], [-2, -8, 0, 8], [2, -2, 2, 0], [0, 0, -10, 0], [8, 6, 8, 0], [3, 0, -3, 12], [-16, -1, 0, 1], [-14, -4, 0, 4], [-12, 0, 12, 2], [0, 3, 0, 3], [-1, -8, -1, 0], [0, -10, 0, 0], [0, -4, 2, -4], [0, 0, 0, 10], [-8, 0, 8, -8], [0, -5, 4, -5], [0, 0, 0, 16], [-12, 0, 12, -2], [11, -4, 11, 0], [4, -6, 0, 6], [-2, 2, 0, -2], [12, -6, 12, 0], [0, 4, -12, 4], [3, 0, -3, 8], [18, -2, 0, 2], [-16, 0, 16, -2], [-13, 0, -13, 0], [0, 11, -12, 11], [4, 12, 0, -12], [-16, 4, 0, -4], [6, -14, 6, 0], [8, -13, 0, 13], [0, 13, 4, 13], [6, 0, -6, -18], [-12, -8, 0, 8], [18, -2, 18, 0], [0, 10, 2, 10], [6, 8, 6, 0], [8, -8, 8, 0], [-2, 0, 2, -8], [0, 12, -16, 12], [0, 2, -14, 2], [8, 0, -8, 2], [-13, 4, -13, 0], [-6, 6, 0, -6], [12, -4, 12, 0], [1, 0, -1, 12], [-20, -1, 0, 1], [0, -2, -16, -2], [18, 12, 0, -12], [-4, -10, -4, 0], [-11, 8, -11, 0], [-16, -5, 0, 5], [-8, 26, -8, 0], [0, 0, 0, 14], [-10, -12, 0, 12], [0, -16, -8, -16], [6, 0, -6, -6], [3, 0, 3, 0], [-4, -6, 0, 6], [34, 2, 0, -2], [0, -4, 12, -4], [-12, 2, -12, 0], [-6, -2, 0, 2], [0, 4, 18, 4], [-4, 0, 4, -18], [-6, -4, 0, 4], [-4, 0, 4, -16], [-4, 0, 0, 0], [0, 0, 26, 0], [-2, 6, -2, 0], [13, 0, -13, 12], [0, 3, 16, 3], [10, 0, -10, 2], [0, -6, -10, -6], [-6, 32, -6, 0], [-9, 0, 9, 12], [6, 0, -6, -12], [-16, 0, 16, -4], [0, 0, 0, -12], [0, 8, -22, 8], [21, -4, 21, 0], [-22, -4, 0, 4], [4, -10, 4, 0], [-24, -7, 0, 7], [0, 10, -16, 10], [0, 12, -12, 12], [10, 6, 0, -6], [0, -2, 0, 0], [0, 9, 28, 9], [5, 16, 5, 0], [0, 8, 0, 8], [2, -8, 0, 8], [-16, 0, 16, -24], [0, -5, 0, -5], [-4, 0, 4, 8], [22, 0, -22, 18], [9, 4, 9, 0], [6, 14, 0, -14], [20, 0, 20, 0], [-20, 8, -20, 0], [7, 0, -7, 4], [-22, 0, 22, 10], [1, 40, 1, 0], [-8, 12, 0, -12], [0, -8, -6, -8], [-6, 0, 6, 2], [5, 0, -5, -4], [0, -27, -8, -27], [6, 4, 0, -4], [-6, 2, -6, 0], [0, 2, 42, 2], [-24, -12, 0, 12], [16, -18, 16, 0], [-2, 0, 2, 20], [0, 0, 0, -14], [0, -12, -6, -12], [-21, 0, -21, 0], [-10, 10, -10, 0], [0, 34, 6, 34], [12, 20, 12, 0], [-21, 0, 21, -12], [0, 14, -4, 14], [-16, -34, -16, 0], [0, 11, 36, 11], [-1, -40, -1, 0], [-28, -1, 0, 1], [12, 0, -12, -10], [6, 24, 0, -24], [28, 0, -28, 12], [0, -15, 36, -15], [-28, 4, -28, 0], [4, 2, 0, -2], [0, -16, 6, -16], [22, -18, 0, 18], [0, 16, 14, 16], [0, 0, 0, -30], [-4, 0, 4, 10], [0, -41, 0, -41], [0, 18, -6, 18], [18, 6, 18, 0], [-4, 11, 0, -11], [0, 9, 8, 9], [4, 0, 0, 0], [14, -6, 14, 0], [6, -24, 0, 24], [-16, 10, -16, 0], [-22, 2, -22, 0], [7, 0, -7, 24], [0, 4, 48, 4], [0, -10, -38, -10], [-24, 20, 0, -20], [0, -8, 18, -8], [-7, 8, -7, 0], [0, -26, 0, 0], [0, 22, 6, 22], [-13, 0, 13, 8], [-6, -42, 0, 42], [24, 0, -24, -2], [-12, 42, -12, 0], [0, -9, 8, -9], [-8, -15, 0, 15], [-8, 34, -8, 0], [-4, 0, 4, -30], [-4, 0, 4, -8], [0, -24, 14, -24], [-16, -2, 0, 2], [-2, -30, 0, 30], [-4, 46, -4, 0], [-41, 0, 41, -8], [12, 0, -12, 6], [0, 0, 0, 0], [0, 7, -24, 7], [8, -32, 0, 32], [-14, 0, 14, 30], [-24, -13, 0, 13], [0, 15, 24, 15], [16, 0, -16, -10], [30, 0, 0, 0], [-10, -22, -10, 0], [-10, 20, 0, -20], [-14, 10, -14, 0], [-8, 0, 8, -40], [0, -8, 0, 8], [-4, 0, 4, -60], [0, 8, -2, 8], [19, 36, 19, 0], [26, 18, 0, -18], [16, 6, 16, 0], [0, -8, 10, -8], [28, -20, 28, 0], [0, 22, -24, 22], [0, -4, -58, -4], [-26, -20, 0, 20], [-6, -44, 0, 44], [0, -27, -8, -27], [-32, 0, 32, 22], [1, -24, 1, 0], [0, -20, 12, -20], [-10, -42, 0, 42], [30, -8, 0, 8], [20, 0, -20, 32], [-27, 24, -27, 0], [0, 9, 16, 9], [-4, -4, 0, 4], [0, 0, -30, 0], [-16, 14, -16, 0], [14, 0, -14, -10], [3, 0, -3, 56], [-44, -12, 0, 12], [0, 6, -62, 6], [-6, 12, -6, 0], [-12, 29, 0, -29], [26, 16, 0, -16], [-28, -12, -28, 0], [-22, -2, -22, 0], [0, 0, 28, 0], [-16, 0, 16, 4], [-3, -12, -3, 0], [0, -10, 30, -10], [13, 0, -13, 28], [52, 13, 0, -13], [0, -6, 44, -6], [14, 4, 0, -4], [14, 0, -14, -42], [27, 4, 27, 0], [28, 21, 0, -21], [20, 0, -20, 10], [38, 12, 0, -12], [0, -21, 28, -21], [0, 12, -28, 12], [-28, 0, 28, -10], [42, 0, -42, -10], [-27, -28, -27, 0], [-12, 24, -12, 0], [12, 62, 12, 0], [0, -18, -38, -18], [14, -26, 0, 26], [36, 0, -36, -10], [5, -24, 5, 0], [2, 0, -2, 26], [0, -3, 0, 3], [-25, 0, 25, 28], [4, 0, -4, -70], [0, 2, 42, 2], [2, 20, 2, 0], [-24, 6, -24, 0], [37, 0, -37, 0], [-10, 0, 10, 16], [0, 30, -10, 30], [-24, -12, 0, 12], [10, -2, 0, 2], [0, -20, -2, -20], [-28, -36, -28, 0], [-21, 0, 21, 4], [0, 0, 42, 0], [-8, 0, 8, 30], [44, -17, 0, 17], [0, 4, -42, 4], [0, 19, -32, 19], [-3, -8, -3, 0], [-16, 28, -16, 0], [-4, 10, 0, -10], [-10, 44, 0, -44], [12, 32, 12, 0], [0, -32, -28, -32], [6, -6, 0, 6], [20, 0, -20, 26], [-32, 0, 32, 24], [0, 29, 36, 29], [12, -38, 12, 0], [-22, 0, 22, -42], [-48, 0, 48, -14], [34, -24, 0, 24], [8, -14, 8, 0], [50, 2, 50, 0], [34, 0, -34, -36], [0, -10, -18, -10], [24, 0, -24, -30], [18, 28, 0, -28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2880_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2880_2_t_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_2880_2_t_a(:prec:=4)
    chi := MakeCharacter_2880_t();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2880_2_t_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_2880_2_t_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2880_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;