
// Make newform 8100.2.d.m in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8100_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8100_d_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_8100_2_d_m();".
// 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_8100_2_d_m();".
// 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, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [0, 0, 0, 2], [-1, 0, 2, -1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_8100_d();"
function MakeCharacter_8100_d()
    N := 8100;
    order := 2;
    char_gens := [4051, 6401, 7777];
    v := [2, 2, 1];
    // 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_8100_d_Hecke();"
function MakeCharacter_8100_d_Hecke(Kf)
    N := 8100;
    order := 2;
    char_gens := [4051, 6401, 7777];
    char_values := [[1, 0, 0, 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, 0, 0, 0], [0, 0, 0, -1], [0, -1, 0, 0], [0, 0, 0, -2], [0, 0, 1, -1], [1, -2, 0, 0], [0, 0, -1, -2], [6, -1, 0, 0], [-1, -4, 0, 0], [0, 0, -1, -3], [-6, 3, 0, 0], [0, 0, 3, 1], [0, 0, 2, 1], [0, 0, -5, -1], [6, -1, 0, 0], [-4, 0, 0, 0], [0, 0, 5, 6], [-6, 3, 0, 0], [0, 0, 4, 3], [4, -6, 0, 0], [0, 0, 2, 7], [0, 3, 0, 0], [0, 0, 1, 1], [-6, -7, 0, 0], [0, 0, 1, -6], [0, 0, 1, 2], [1, -4, 0, 0], [0, 0, -7, -5], [0, 0, 2, 3], [12, 1, 0, 0], [0, 0, 6, 6], [-5, 6, 0, 0], [0, 0, 0, 0], [-7, 6, 0, 0], [0, 0, -3, -1], [0, 0, 8, -1], [0, 0, 1, -7], [0, 0, 7, 14], [0, -7, 0, 0], [-13, 2, 0, 0], [0, -11, 0, 0], [0, 0, 7, 9], [0, 0, -4, -8], [-2, 8, 0, 0], [-13, 4, 0, 0], [0, 0, 8, 8], [0, 0, -9, 3], [4, -8, 0, 0], [0, 0, 0, -9], [-12, -2, 0, 0], [-7, -10, 0, 0], [24, 2, 0, 0], [0, 0, -5, 2], [0, 0, 7, 14], [18, 1, 0, 0], [-4, -12, 0, 0], [0, 0, 7, 7], [12, -10, 0, 0], [0, 0, 11, 2], [0, 0, 0, -9], [0, 0, 5, 3], [6, 13, 0, 0], [0, 0, 2, -4], [0, 0, 8, 7], [-19, 6, 0, 0], [0, 0, -2, -14], [0, 0, -15, -3], [25, 0, 0, 0], [0, 0, 6, 15], [-18, -5, 0, 0], [0, 0, 5, -6], [0, 0, 8, 11], [-8, -2, 0, 0], [0, 0, -10, -14], [-24, 2, 0, 0], [0, 0, -5, 10], [6, 12, 0, 0], [4, -8, 0, 0], [-18, 2, 0, 0], [-7, -12, 0, 0], [-18, 9, 0, 0], [0, 0, 3, -2], [-17, -6, 0, 0], [0, 0, 20, 1], [-12, 7, 0, 0], [0, 0, -7, 3], [-6, -9, 0, 0], [0, 0, 7, 6], [0, 0, -9, 3], [12, 5, 0, 0], [0, 0, 18, 2], [-6, -1, 0, 0], [-17, 6, 0, 0], [0, 0, -4, 10], [6, -12, 0, 0], [0, 6, 0, 0], [0, 0, 10, 6], [-1, -2, 0, 0], [0, 0, 14, 12], [0, 0, -6, -6], [0, 0, 10, -1], [24, -11, 0, 0], [-13, -8, 0, 0], [0, 0, -10, -3], [0, 0, -6, -3], [0, 0, -17, -4], [30, 9, 0, 0], [-7, 10, 0, 0], [0, 0, 11, 9], [0, 0, 4, -6], [0, 0, 6, 0], [10, 0, 0, 0], [-7, -4, 0, 0], [-18, -3, 0, 0], [0, 0, -5, -9], [0, 0, 19, 14], [0, 0, -4, -2], [6, 2, 0, 0], [5, -6, 0, 0], [0, 0, -11, 6], [0, 0, 7, 20], [0, 0, 8, -2], [-10, 0, 0, 0], [-30, -7, 0, 0], [-14, -2, 0, 0], [12, -3, 0, 0], [0, 0, -10, -18], [0, 0, -11, -24], [-5, -18, 0, 0], [0, 0, -15, -9], [20, -12, 0, 0], [0, 0, 2, -6], [-18, 7, 0, 0], [-11, 14, 0, 0], [0, 0, 2, -5], [0, 0, 14, -3], [0, 0, 2, 10], [24, 9, 0, 0], [-1, 14, 0, 0], [12, 5, 0, 0], [0, 0, -3, 4], [0, 0, 20, -2], [31, -12, 0, 0], [24, 9, 0, 0], [0, 0, 4, 3], [0, 0, 20, 4], [13, 12, 0, 0], [0, 0, -7, 13], [0, 0, 12, 14], [12, 3, 0, 0], [0, 0, 8, 23], [0, 0, -8, -1], [0, 0, 17, 12], [-12, -3, 0, 0], [-41, 6, 0, 0], [-24, -13, 0, 0], [0, 0, -15, -4], [-12, -2, 0, 0], [0, 0, -1, -20], [0, 0, -4, -2], [0, 0, -16, -9], [24, 1, 0, 0], [0, 0, 6, 6], [0, 0, -25, 1], [35, -12, 0, 0], [0, 0, -22, 3], [13, 6, 0, 0], [0, 0, 11, 7], [-18, 17, 0, 0], [-19, -6, 0, 0], [-48, -1, 0, 0], [0, 0, 3, 4], [-50, -4, 0, 0], [-30, 17, 0, 0], [2, -28, 0, 0], [6, -5, 0, 0], [0, 0, -18, -8], [19, -22, 0, 0], [0, 0, 13, -11], [24, 10, 0, 0], [0, 0, -2, 15], [0, 0, -23, -22], [0, 0, -4, -11], [-24, -1, 0, 0], [0, 0, -13, -6], [0, 0, -4, -13], [19, 22, 0, 0], [-18, -22, 0, 0], [0, 0, -19, -10], [0, 0, -22, -23], [-4, 10, 0, 0], [-18, 27, 0, 0], [0, 0, 6, 21], [0, 0, 1, 17], [11, 12, 0, 0], [0, 0, -11, 15], [0, 0, -17, -1], [0, 0, 16, 8], [-12, 11, 0, 0], [11, 0, 0, 0], [0, 0, -28, -9], [1, 12, 0, 0], [-6, -6, 0, 0], [0, 0, -2, -4], [31, 4, 0, 0], [0, 0, 22, 20], [36, -18, 0, 0], [20, 8, 0, 0], [0, 0, 13, 22], [48, -1, 0, 0], [0, 0, -20, -24], [0, 0, -6, -12], [12, -5, 0, 0], [-1, 30, 0, 0], [0, 0, -3, -22], [36, 13, 0, 0], [0, 0, 19, 26], [0, 0, 3, 18], [5, 14, 0, 0], [25, -24, 0, 0], [-42, -11, 0, 0], [0, 0, -26, -12], [0, 0, -3, 3], [25, 26, 0, 0], [0, 0, 3, -12], [6, -6, 0, 0], [0, 0, -5, 13], [18, 10, 0, 0], [0, 0, -18, -5], [-35, -12, 0, 0], [26, 28, 0, 0], [6, -32, 0, 0], [0, 0, -14, -12], [0, 0, 1, -16], [13, -18, 0, 0], [0, 0, 9, 21], [12, 24, 0, 0], [18, -23, 0, 0], [0, 0, -11, -31], [-4, -6, 0, 0], [0, 0, 7, 21], [40, 0, 0, 0], [0, 0, 28, 20], [-24, -3, 0, 0], [0, 0, 25, 19], [0, -40, 0, 0], [-23, -20, 0, 0], [0, 0, -2, 23], [0, 0, -25, 9], [6, -25, 0, 0], [0, 0, -10, 1], [1, -30, 0, 0], [0, 0, -16, -23], [-54, -10, 0, 0], [17, -12, 0, 0], [0, 0, 29, 18], [0, 0, -23, -25], [0, 0, -16, 9], [0, 0, -13, -13], [0, 0, -6, -15], [19, -12, 0, 0], [0, 0, -6, -2], [0, 0, -2, 17], [67, -6, 0, 0], [12, 20, 0, 0], [30, 9, 0, 0], [0, 0, -2, 12], [0, 0, 21, 6], [23, -26, 0, 0], [0, 0, -4, -30], [0, 0, -20, -6], [-47, 6, 0, 0], [0, 0, -35, -14], [0, 0, -11, -27], [0, 0, -5, 14], [-23, -20, 0, 0], [38, -16, 0, 0], [18, 14, 0, 0], [0, 0, -22, -11], [2, 12, 0, 0], [0, 0, 11, -8], [38, 6, 0, 0], [0, 0, 14, 9], [6, -5, 0, 0], [0, 0, -21, 1], [0, 0, -4, 7], [7, 0, 0, 0], [-18, 28, 0, 0], [-12, 9, 0, 0], [0, 0, 10, 20], [0, 0, -17, 11], [-18, -35, 0, 0], [0, 0, -20, -33], [-30, 11, 0, 0], [59, 2, 0, 0], [0, 0, 3, -15], [-18, -3, 0, 0], [0, 0, 5, -24], [0, 0, -23, -18], [0, 0, 11, 1], [-29, -4, 0, 0], [0, 0, 9, 30], [11, -12, 0, 0], [0, 0, -24, -7], [0, 0, -27, -21], [25, 24, 0, 0], [-24, -9, 0, 0], [0, 0, -8, 6], [0, 0, -17, -10], [-18, 7, 0, 0], [24, 17, 0, 0], [0, 0, 5, -7], [0, 0, 39, -3], [7, -28, 0, 0], [6, -17, 0, 0], [0, 29, 0, 0], [0, 0, 23, 23], [66, 7, 0, 0], [-52, -6, 0, 0], [0, 0, 18, 29], [0, -9, 0, 0], [0, 0, -16, 17], [0, 0, -36, -6], [35, 0, 0, 0], [1, 12, 0, 0], [0, 0, 19, 24], [0, 0, -26, -10], [0, 0, -11, -22], [-13, 6, 0, 0], [0, 0, 36, -3], [-11, -2, 0, 0], [0, 0, -21, -21], [-1, 18, 0, 0], [0, 0, -36, 3], [-29, -18, 0, 0], [0, 0, 39, -6], [11, -44, 0, 0], [0, 0, -34, -12], [0, 0, -31, -1], [0, 0, 33, 12], [24, -15, 0, 0], [-4, 22, 0, 0], [0, 0, -11, -19], [36, 20, 0, 0], [-13, 6, 0, 0], [0, 0, -15, -13], [24, -13, 0, 0], [0, 0, -8, -28], [-13, -32, 0, 0], [0, 0, 14, -18], [-30, -28, 0, 0], [0, 0, -5, 24], [-65, 0, 0, 0], [0, 0, 5, -2], [6, 19, 0, 0], [-36, 7, 0, 0], [0, 0, -1, 22], [0, 0, 6, -6], [0, 0, 34, 34], [-72, 9, 0, 0], [0, 0, 21, 27], [30, 1, 0, 0], [0, 0, -1, 3], [0, 0, 12, 10], [0, 0, 12, 0], [0, 0, -11, 15], [5, 28, 0, 0], [72, -5, 0, 0], [-32, -4, 0, 0], [0, 0, -23, -4], [24, -19, 0, 0], [41, 6, 0, 0], [0, 0, -20, -23], [42, 13, 0, 0], [6, -19, 0, 0], [0, 0, 3, 7], [6, 17, 0, 0], [0, 0, -28, 9], [-30, -4, 0, 0], [0, 0, -29, 2], [0, 0, 35, 3], [0, 0, 7, -22], [-23, 6, 0, 0], [0, 0, 5, 25], [-4, -6, 0, 0], [0, 0, 11, -24], [0, 0, 16, 15], [0, 0, -27, -18], [4, 28, 0, 0], [0, 0, 20, 28], [-12, -28, 0, 0], [0, 0, -11, -41], [-30, -27, 0, 0], [0, 0, 7, -21], [19, -2, 0, 0], [-36, -24, 0, 0], [-76, 16, 0, 0], [-24, 7, 0, 0], [79, 12, 0, 0], [0, 0, 34, 8], [0, 0, 19, -8], [0, 0, 25, 17], [30, 17, 0, 0], [-34, 24, 0, 0], [0, 0, 10, -5], [-36, -8, 0, 0], [0, 0, 7, 15], [0, -9, 0, 0], [0, 0, -16, -43], [0, 0, -13, 10], [0, 0, -29, -37], [29, 6, 0, 0], [0, 0, -24, -34], [18, -16, 0, 0], [-42, -17, 0, 0], [0, 0, -11, 16], [0, 0, 19, -7], [0, 0, 7, -28], [60, 26, 0, 0], [0, 0, 15, 17], [0, 0, 42, 0], [-42, -19, 0, 0], [0, 0, 1, 15], [0, 0, 24, 21], [0, 0, 24, 9], [12, 28, 0, 0], [-67, -18, 0, 0], [-48, -16, 0, 0], [41, -18, 0, 0], [-48, 17, 0, 0], [37, -6, 0, 0], [0, 0, 15, 9], [0, 0, 23, 18], [30, 23, 0, 0], [55, -6, 0, 0], [-7, -30, 0, 0], [0, 0, 20, 21], [10, -12, 0, 0], [0, 0, 32, -2], [24, 33, 0, 0], [-26, -36, 0, 0], [18, -22, 0, 0], [-73, -16, 0, 0], [0, 0, -15, 9], [0, 0, 45, 16], [0, 0, 0, 3], [31, 38, 0, 0], [-34, -6, 0, 0], [0, 0, 44, 18], [0, 44, 0, 0], [0, 0, -4, -29], [66, 0, 0, 0], [0, 0, -25, -54], [-72, 18, 0, 0], [13, 12, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8100_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8100_2_d_m();".
// 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_8100_2_d_m(:prec:=4)
    chi := MakeCharacter_8100_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3229) - 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_8100_2_d_m();".
// 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_8100_2_d_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8100_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 0, 8, 0, 1]>,<11,R![-3, 0, 1]>,<29,R![33, -12, 1]>],Snew);
    return Vf;
end function;