
// Make newform 1200.3.c.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_c_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_1200_3_c_g();".
// 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_1200_3_c_g();".
// 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;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 2], [0, 2, 0, -2]];
    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_1200_c();"
function MakeCharacter_1200_c()
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    v := [2, 2, 1, 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_1200_c_Hecke();"
function MakeCharacter_1200_c_Hecke(Kf)
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    char_values := [[1, 0, 0, 0], [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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 1, 0, -1], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, -3, 0], [0, 15, 0, 0], [0, 0, 0, 7], [-23, 0, 0, 0], [0, 0, 0, 1], [0, 0, -9, 0], [-33, 0, 0, 0], [0, -66, 0, 0], [0, 0, 13, 0], [0, 7, 0, 0], [0, 0, 0, 16], [0, 0, 0, -13], [0, 0, -36, 0], [39, 0, 0, 0], [0, 113, 0, 0], [0, 0, -9, 0], [0, 58, 0, 0], [70, 0, 0, 0], [0, 0, 0, 54], [0, 0, -32, 0], [0, 1, 0, 0], [0, 0, 67, 0], [0, 26, 0, 0], [0, 0, 0, 17], [145, 0, 0, 0], [0, 0, 0, 54], [0, -226, 0, 0], [0, 0, -37, 0], [0, 0, 0, -6], [102, 0, 0, 0], [0, 0, -102, 0], [207, 0, 0, 0], [0, 89, 0, 0], [0, -17, 0, 0], [0, 0, 0, 95], [0, 0, 0, -39], [0, 0, 26, 0], [-145, 0, 0, 0], [0, 0, 19, 0], [0, -57, 0, 0], [0, 0, 0, -80], [225, 0, 0, 0], [231, 0, 0, 0], [0, 327, 0, 0], [0, 0, 0, 27], [-183, 0, 0, 0], [0, 0, 0, -35], [0, 0, 3, 0], [215, 0, 0, 0], [0, 0, -58, 0], [0, 0, 0, 54], [0, 0, 0, -21], [0, 0, -3, 0], [-86, 0, 0, 0], [0, -319, 0, 0], [0, 0, -151, 0], [0, -1, 0, 0], [0, 0, 0, -172], [0, -215, 0, 0], [0, 0, -76, 0], [0, 535, 0, 0], [0, 0, 0, -166], [-94, 0, 0, 0], [0, -295, 0, 0], [0, 0, 0, -156], [374, 0, 0, 0], [0, 0, 0, 99], [0, 0, 131, 0], [0, 241, 0, 0], [0, 47, 0, 0], [73, 0, 0, 0], [0, 0, 0, -40], [0, 0, 189, 0], [0, -695, 0, 0], [0, 0, -255, 0], [425, 0, 0, 0], [0, 0, 233, 0], [306, 0, 0, 0], [0, 0, -37, 0], [0, -281, 0, 0], [-671, 0, 0, 0], [0, 0, 0, 150], [0, 0, 214, 0], [0, -202, 0, 0], [0, 0, 119, 0], [0, -166, 0, 0], [0, 0, 0, -43], [0, 0, -282, 0], [0, 369, 0, 0], [0, 0, 208, 0], [785, 0, 0, 0], [0, 0, 0, -99], [0, 0, 78, 0], [0, 0, -233, 0], [0, -129, 0, 0], [119, 0, 0, 0], [0, -650, 0, 0], [0, 0, 0, -38], [0, 0, 0, -279], [0, 0, 185, 0], [503, 0, 0, 0], [0, 353, 0, 0], [0, 0, 0, 381], [0, 0, 0, 257], [0, 0, 308, 0], [-1033, 0, 0, 0], [0, -330, 0, 0], [0, 770, 0, 0], [0, 0, 0, 223], [401, 0, 0, 0], [-889, 0, 0, 0], [0, 0, -240, 0], [0, 42, 0, 0], [0, 0, 0, -162], [0, 0, 0, 354], [0, 0, -133, 0], [-974, 0, 0, 0], [0, 26, 0, 0], [0, 0, 0, 33], [0, 0, 0, -6], [722, 0, 0, 0], [0, 0, -23, 0], [-103, 0, 0, 0], [0, 0, -237, 0], [0, -895, 0, 0], [0, -406, 0, 0], [-578, 0, 0, 0], [0, 0, 0, -174], [-30, 0, 0, 0], [0, 897, 0, 0], [0, 0, -58, 0], [465, 0, 0, 0], [0, 0, 0, -61], [0, 801, 0, 0], [0, 0, 0, 70], [0, 0, -237, 0], [879, 0, 0, 0], [0, 0, -298, 0], [0, -521, 0, 0], [0, 0, 0, -61], [-818, 0, 0, 0], [0, 0, -463, 0], [0, -233, 0, 0], [0, 0, 0, -286], [1230, 0, 0, 0], [0, 0, 0, 454], [0, 113, 0, 0], [0, 0, 458, 0], [0, -793, 0, 0], [0, 0, 0, 24], [0, 1678, 0, 0], [0, 0, 602, 0], [-1079, 0, 0, 0], [0, 0, 51, 0], [0, 73, 0, 0], [0, 0, 484, 0], [0, 0, 0, -211], [0, 0, 0, 270], [0, -10, 0, 0], [0, 0, -172, 0], [0, 0, 0, -290], [0, 0, 0, -48], [79, 0, 0, 0], [0, -378, 0, 0], [1753, 0, 0, 0], [0, 0, 0, 306], [0, 0, -605, 0], [1391, 0, 0, 0], [0, 0, -422, 0], [0, 391, 0, 0], [-1639, 0, 0, 0], [0, 0, -312, 0], [-393, 0, 0, 0], [0, 0, -51, 0], [0, 1311, 0, 0], [481, 0, 0, 0], [0, 1153, 0, 0], [0, 0, -530, 0], [0, -593, 0, 0], [0, 0, 0, 282], [0, 0, 0, 318], [0, 0, 622, 0], [0, -47, 0, 0], [0, -1145, 0, 0], [-919, 0, 0, 0], [0, 0, -207, 0], [0, 1495, 0, 0], [0, 0, 0, 447], [-1129, 0, 0, 0], [0, 0, -273, 0], [0, 0, 0, -407], [0, 0, 0, -254], [-1958, 0, 0, 0], [0, 682, 0, 0], [0, 0, 0, -786], [0, 0, 0, -713], [0, 0, -431, 0], [-2294, 0, 0, 0], [0, 1742, 0, 0], [-1226, 0, 0, 0], [0, 0, -16, 0], [0, 0, 0, -843], [1001, 0, 0, 0], [0, 0, 0, -593], [0, 0, 522, 0], [-841, 0, 0, 0], [0, -407, 0, 0], [0, 0, -514, 0], [0, 2135, 0, 0], [0, 0, 0, 796], [0, 0, 668, 0], [-422, 0, 0, 0], [0, 145, 0, 0], [0, 0, -47, 0], [0, 0, 0, -163], [0, 0, 0, -273], [1295, 0, 0, 0], [-367, 0, 0, 0], [0, 0, -498, 0], [0, 1450, 0, 0], [0, 0, 0, -481], [-2479, 0, 0, 0], [0, 0, 0, -795], [0, 0, -120, 0], [0, -1175, 0, 0], [0, 0, 369, 0], [0, 599, 0, 0], [-1090, 0, 0, 0], [127, 0, 0, 0], [0, 0, 304, 0], [0, 738, 0, 0], [0, 0, 0, 246], [806, 0, 0, 0], [0, 0, 0, 741], [0, 0, 226, 0], [0, 0, -649, 0], [0, 0, 0, -183], [-665, 0, 0, 0], [0, -681, 0, 0], [-2951, 0, 0, 0], [0, 0, 0, 326], [0, 0, 798, 0], [0, -1319, 0, 0], [0, 0, -381, 0], [3033, 0, 0, 0], [0, 0, 0, -1072], [0, -82, 0, 0], [0, 0, 105, 0], [0, 0, 0, -826], [1777, 0, 0, 0], [0, 0, 0, -43], [0, 0, 846, 0], [-1865, 0, 0, 0], [0, -1586, 0, 0], [0, 0, 0, 1023], [0, -722, 0, 0], [0, -433, 0, 0], [0, 0, 0, 265], [2270, 0, 0, 0], [0, -481, 0, 0], [0, 0, 0, 200], [2062, 0, 0, 0], [0, 0, 46, 0], [0, 0, 328, 0], [0, -2014, 0, 0], [0, 0, 0, 388], [2735, 0, 0, 0], [0, 606, 0, 0], [0, -1526, 0, 0], [1057, 0, 0, 0], [0, -1671, 0, 0], [0, -681, 0, 0], [0, 0, 0, 708], [-351, 0, 0, 0], [-1377, 0, 0, 0], [0, 0, 606, 0], [0, 0, 0, -75], [-502, 0, 0, 0], [0, 0, 0, -583], [2479, 0, 0, 0], [0, 1537, 0, 0], [0, 0, 780, 0], [0, 3135, 0, 0], [0, 0, 0, -691], [-434, 0, 0, 0], [0, 0, -171, 0], [0, 0, -884, 0], [0, 0, 0, -72], [0, 0, 0, 180], [0, 0, -82, 0], [0, 2455, 0, 0], [0, 0, -305, 0], [215, 0, 0, 0], [0, 0, 0, -9], [0, 0, -243, 0], [0, -610, 0, 0], [0, 1618, 0, 0], [0, 0, 0, 758], [1033, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_c_g();".
// 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_1200_3_c_g(:prec:=4)
    chi := MakeCharacter_1200_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_1200_3_c_g();".
// 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_1200_3_c_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![1, 0, 1]>,<11,R![72, 0, 1]>],Snew);
    return Vf;
end function;