
// Make newform 400.6.c.h in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_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_400_6_c_h();".
// 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_400_6_c_h();".
// 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];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_400_c();"
function MakeCharacter_400_c()
    N := 400;
    order := 2;
    char_gens := [351, 101, 177];
    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_400_c_Hecke();"
function MakeCharacter_400_c_Hecke(Kf)
    N := 400;
    order := 2;
    char_gens := [351, 101, 177];
    char_values := [[1, 0], [1, 0], [-1, 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 := 6;
    raw_aps := [[0, 0], [0, 4], [0, 0], [0, -54], [604, 0], [0, -153], [0, -465], [-1324, 0], [0, 426], [-5902, 0], [3320, 0], [0, -5387], [-17958, 0], [0, -4632], [0, -4898], [0, -15717], [33228, 0], [-40210, 0], [0, 29432], [55312, 0], [0, 13629], [31456, 0], [0, -12276], [90854, 0], [0, -77353], [-72714, 0], [0, 64698], [0, -103340], [70146, 0], [0, -75927], [0, -168298], [-275308, 0], [0, 114251], [224284, 0], [183802, 0], [-296032, 0], [0, -67383], [0, 30124], [0, -31006], [0, -295841], [-241404, 0], [187622, 0], [-37560, 0], [0, 82217], [0, -180259], [-654168, 0], [693156, 0], [0, -247378], [0, 453544], [-1089494, 0], [0, 249853], [1620384, 0], [1001218, 0], [-368980, 0], [0, -139505], [0, -405870], [-353214, 0], [1716216, 0], [0, 122941], [-1676182, 0], [0, 629408], [0, 359579], [0, 930064], [-278384, 0], [0, -237091], [0, 918689], [-2997428, 0], [0, -937657], [0, 90156], [87058, 0], [0, 1329505], [-2149368, 0], [0, -1541290], [0, 1143635], [-1301540, 0], [0, 1016382], [-94230, 0], [0, 2777753], [-784814, 0], [4594006, 0], [1413012, 0], [5945558, 0], [6481144, 0], [0, 2027985], [-1214504, 0], [0, 2765768], [-2201106, 0], [0, -1649173], [-3942658, 0], [0, -4410202], [0, -643544], [6511792, 0], [0, -2897614], [-990276, 0], [2914996, 0], [0, -1239358], [6758066, 0], [-6339030, 0], [0, -115960], [-9444402, 0], [0, 1550808], [0, 611649], [0, 7044796], [-1484682, 0], [2864700, 0], [0, -2108641], [0, -1005236], [0, 3668457], [1148840, 0], [11640938, 0], [0, -77770], [0, -5906845], [0, -3211261], [3852524, 0], [6751360, 0], [-7354926, 0], [0, -7984724], [0, 8633346], [0, -681257], [-8818076, 0], [-1520346, 0], [0, -157543], [0, 870461], [0, 9946744], [-20151916, 0], [8107662, 0], [13561258, 0], [4283136, 0], [0, 5604194], [0, 2352743], [-23109868, 0], [0, -2876470], [19242376, 0], [0, -2246051], [7335002, 0], [5855262, 0], [0, 6727875], [0, 5035304], [0, 5942209], [-10605258, 0], [1389436, 0], [-11266138, 0], [0, 13854666], [0, 6183084], [-12318190, 0], [-11727656, 0], [0, 7852939], [0, -12619501], [3662476, 0], [0, 20896886], [0, -5449487], [-30433630, 0], [0, -3045140], [0, 13895394], [0, 18575516], [7859592, 0], [-16269304, 0], [36936462, 0], [0, -2448525], [-6839426, 0], [0, 5189520], [0, 12979373], [0, -19605990], [10687596, 0], [0, -13863281], [0, 4746362], [20324312, 0], [0, -23525379]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_6_c_h();".
// 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_400_6_c_h(:prec:=2)
    chi := MakeCharacter_400_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_400_6_c_h();".
// 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_400_6_c_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![64, 0, 1]>],Snew);
    return Vf;
end function;