
// Make newform 1200.3.e.n in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_e_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_e_n();".
// 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_e_n();".
// 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, 1, 2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -2, 2, -1], [2, 12, -4, 4], [-12, 0, 0, -6]];
    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_e();"
function MakeCharacter_1200_e()
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    v := [1, 2, 2, 2];
    // 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_e_Hecke();"
function MakeCharacter_1200_e_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, 0], [0, 0, 0, 0], [0, 2, 1, 0], [0, 2, -1, 0], [8, 0, 0, 1], [-12, 0, 0, -1], [0, -4, -2, 0], [0, 4, 4, 0], [-12, 0, 0, 1], [0, 20, -2, 0], [-20, 0, 0, 3], [-30, 0, 0, 0], [0, 4, -4, 0], [0, 32, 2, 0], [-24, 0, 0, -3], [0, 2, 11, 0], [14, 0, 0, -6], [0, 48, -6, 0], [0, 68, 2, 0], [2, 0, 0, 2], [0, -44, 2, 0], [0, 24, -6, 0], [6, 0, 0, 2], [14, 0, 0, 0], [-12, 0, 0, 9], [0, -14, 5, 0], [0, 56, -10, 0], [70, 0, 0, -2], [-144, 0, 0, 5], [0, 66, 9, 0], [0, 62, -7, 0], [120, 0, 0, 1], [0, 32, 4, 0], [24, 0, 0, -5], [0, 56, -20, 0], [-8, 0, 0, 1], [0, 72, 18, 0], [0, 44, 20, 0], [-240, 0, 0, -7], [0, 26, -25, 0], [142, 0, 0, 2], [0, 84, 18, 0], [10, 0, 0, -26], [48, 0, 0, -7], [0, -72, 0, 0], [0, -48, -24, 0], [0, -62, -19, 0], [0, -44, -8, 0], [-206, 0, 0, -12], [-300, 0, 0, 7], [0, -92, 22, 0], [58, 0, 0, 20], [0, -178, -7, 0], [-72, 0, 0, 25], [0, -116, -8, 0], [12, 0, 0, 5], [0, 16, 56, 0], [-364, 0, 0, -1], [-162, 0, 0, 22], [0, 80, -26, 0], [-240, 0, 0, 25], [0, -28, 40, 0], [0, -148, 14, 0], [346, 0, 0, 6], [312, 0, 0, -11], [0, -292, 10, 0], [-118, 0, 0, -18], [0, -228, 12, 0], [14, 0, 0, 2], [-276, 0, 0, -21], [0, -8, 4, 0], [0, -126, -39, 0], [64, 0, 0, 13], [0, 168, -48, 0], [0, -40, 2, 0], [144, 0, 0, 39], [268, 0, 0, 27], [-210, 0, 0, -34], [370, 0, 0, -16], [0, 198, -39, 0], [-14, 0, 0, -36], [0, -56, -32, 0], [106, 0, 0, 6], [0, -64, -20, 0], [0, 124, 40, 0], [-114, 0, 0, 6], [86, 0, 0, 28], [-600, 0, 0, 3], [0, -194, 23, 0], [0, 112, 34, 0], [0, -156, 30, 0], [0, 54, 87, 0], [0, -210, 33, 0], [0, 220, 38, 0], [0, -336, -18, 0], [660, 0, 0, -15], [-90, 0, 0, 18], [0, 256, -58, 0], [-2, 0, 0, 18], [0, 4, -16, 0], [624, 0, 0, -7], [0, -52, -112, 0], [-186, 0, 0, 58], [0, -196, -38, 0], [286, 0, 0, 0], [0, 120, -66, 0], [696, 0, 0, 25], [0, -288, -72, 0], [410, 0, 0, -12], [0, 166, -49, 0], [-92, 0, 0, 15], [-324, 0, 0, 43], [0, 152, 112, 0], [0, 228, 30, 0], [-78, 0, 0, 32], [0, 72, -30, 0], [0, -184, -58, 0], [-288, 0, 0, 57], [0, -282, -63, 0], [-658, 0, 0, -30], [130, 0, 0, -30], [0, 0, 0, -15], [0, 152, -46, 0], [0, 348, -30, 0], [612, 0, 0, 17], [-806, 0, 0, 24], [0, -152, 28, 0], [0, -98, 59, 0], [332, 0, 0, -29], [0, -180, 6, 0], [0, -176, 70, 0], [0, 532, -34, 0], [20, 0, 0, -33], [-66, 0, 0, -34], [386, 0, 0, 72], [168, 0, 0, -43], [0, 760, -10, 0], [144, 0, 0, 41], [306, 0, 0, 32], [0, 88, -4, 0], [492, 0, 0, -19], [0, 198, 51, 0], [0, -212, 124, 0], [-74, 0, 0, 54], [0, 208, -32, 0], [4, 0, 0, -49], [-420, 0, 0, -53], [0, -88, -152, 0], [0, 228, 12, 0], [352, 0, 0, -75], [774, 0, 0, -30], [0, 128, 94, 0], [0, 292, -128, 0], [0, -28, -32, 0], [0, 496, -68, 0], [0, 52, 74, 0], [-210, 0, 0, 70], [1510, 0, 0, 12], [288, 0, 0, -65], [0, 36, -72, 0], [720, 0, 0, 13], [0, 2, -23, 0], [0, -226, -7, 0], [-900, 0, 0, 3], [0, 104, -190, 0], [0, 328, -52, 0], [-896, 0, 0, 29], [1402, 0, 0, 12], [1104, 0, 0, -15], [0, -6, -9, 0], [242, 0, 0, 68], [0, -416, 148, 0], [-742, 0, 0, 30], [0, -36, 150, 0], [-330, 0, 0, -6], [0, -152, -124, 0], [360, 0, 0, 11], [0, 342, -93, 0], [-1030, 0, 0, 0], [0, 750, -21, 0], [0, 214, 85, 0], [-244, 0, 0, 27], [-888, 0, 0, -39], [0, -432, 78, 0], [1368, 0, 0, -13], [920, 0, 0, 9], [0, 240, -90, 0], [298, 0, 0, -12], [0, 52, 154, 0], [-1078, 0, 0, 30], [0, -892, -4, 0], [0, -56, -76, 0], [2016, 0, 0, 7], [0, 232, -50, 0], [168, 0, 0, -87], [-622, 0, 0, -96], [536, 0, 0, -39], [-1056, 0, 0, 29], [0, 1008, -42, 0], [576, 0, 0, 95], [0, -308, -46, 0], [-208, 0, 0, 93], [-518, 0, 0, 12], [0, 58, -89, 0], [336, 0, 0, -15], [0, 884, -62, 0], [0, -124, -244, 0], [-1722, 0, 0, -30], [0, -372, 186, 0], [1330, 0, 0, -54], [1200, 0, 0, 55], [0, -1086, 21, 0], [0, -556, 176, 0], [0, 572, -118, 0], [698, 0, 0, -4], [0, -566, 173, 0], [510, 0, 0, 118], [0, -208, 218, 0], [960, 0, 0, 1], [-350, 0, 0, 36], [0, 952, 104, 0], [426, 0, 0, 4], [0, 774, -21, 0], [0, -920, -122, 0], [-1966, 0, 0, 4], [1344, 0, 0, 21], [0, 272, -112, 0], [0, -930, -93, 0], [0, -966, 3, 0], [-1432, 0, 0, -63], [0, -408, 264, 0], [0, 40, -4, 0], [-426, 0, 0, -46], [0, 520, 170, 0], [0, 724, -80, 0], [-206, 0, 0, -72], [-96, 0, 0, -15], [0, -582, -81, 0], [0, -932, 94, 0], [0, 612, 132, 0], [0, -1188, -78, 0], [0, -266, -61, 0], [1118, 0, 0, 58], [708, 0, 0, -105], [0, 788, 182, 0], [0, 250, -187, 0], [0, 538, 211, 0], [0, -652, -38, 0], [0, 452, 20, 0], [-460, 0, 0, -89], [642, 0, 0, -176], [0, -172, 152, 0], [-518, 0, 0, 164], [-2184, 0, 0, 77], [0, 70, 265, 0], [-506, 0, 0, -50], [0, -1464, 6, 0], [504, 0, 0, -107], [2122, 0, 0, 14], [0, 146, -131, 0], [0, 1236, 48, 0], [442, 0, 0, -104], [1060, 0, 0, -9], [1032, 0, 0, 81], [0, 12, 222, 0], [600, 0, 0, -101], [-318, 0, 0, -40], [0, -216, -150, 0], [-768, 0, 0, -95], [-622, 0, 0, -60], [0, -476, 32, 0], [1922, 0, 0, 34], [0, 612, -174, 0], [1466, 0, 0, -90], [0, 1154, 61, 0], [0, 564, 72, 0], [-1210, 0, 0, 62], [722, 0, 0, 168], [0, 406, 13, 0], [0, 1100, 128, 0], [0, -72, -84, 0], [0, -592, 170, 0], [338, 0, 0, 76], [0, 780, -36, 0], [0, 964, -62, 0], [-1906, 0, 0, -32], [720, 0, 0, -135], [0, -1024, -212, 0], [90, 0, 0, -140], [-2712, 0, 0, -69], [0, 1440, -30, 0], [-60, 0, 0, 191], [0, 434, 47, 0], [1804, 0, 0, 43], [-1608, 0, 0, 67], [0, 1132, -22, 0], [-384, 0, 0, 233], [0, 162, -309, 0], [0, 1652, 16, 0], [2890, 0, 0, -18], [2112, 0, 0, 57], [0, -568, 160, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_e_n();".
// 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_e_n(:prec:=4)
    chi := MakeCharacter_1200_e();
    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_e_n();".
// 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_e_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![2304, 0, 144, 0, 1]>,<11,R![2304, 0, 144, 0, 1]>,<13,R![-116, -16, 1]>],Snew);
    return Vf;
end function;