
// Make newform 400.6.c.d in Magma, downloaded from the LMFDB on 24 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_d();".
// 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_d();".
// 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, 10], [0, 0], [0, 12], [-124, 0], [0, -239], [0, -599], [3044, 0], [0, 92], [3282, 0], [5728, 0], [0, 5163], [-8886, 0], [0, -4594], [0, -11832], [0, -5843], [16876, 0], [-18482, 0], [0, 7766], [31960, 0], [0, 2443], [44560, 0], [0, 33682], [-71994, 0], [0, 24433], [51606, 0], [0, 90212], [0, 32850], [112706, 0], [0, 11751], [0, 47296], [-70292, 0], [0, 138645], [-130308, 0], [401530, 0], [75976, 0], [0, -197161], [0, -5862], [0, 275964], [0, -216447], [559620, 0], [604710, 0], [409152, 0], [0, -270433], [0, -314949], [283048, 0], [-142756, 0], [0, 444848], [0, -571578], [695786, 0], [0, 173563], [-1642960, 0], [-1167438, 0], [790612, 0], [0, -64895], [0, 35444], [-1790174, 0], [1773616, 0], [0, -137725], [594170, 0], [0, 546214], [0, -166827], [0, -529986], [1336488, 0], [0, -822093], [0, -861849], [-2749628, 0], [0, -1707447], [0, -365382], [2297490, 0], [0, 585359], [3886536, 0], [0, -466520], [0, 196109], [-4729300, 0], [0, 948672], [3722954, 0], [0, 1669039], [4274898, 0], [2573190, 0], [5268276, 0], [-973354, 0], [-3557360, 0], [0, 977479], [-3296808, 0], [0, -2529098], [-2127298, 0], [0, 144565], [2668702, 0], [0, 3793096], [0, 709806], [-1884064, 0], [0, 3006940], [-4292316, 0], [1345092, 0], [0, 101004], [-9783438, 0], [-10483030, 0], [0, 3105086], [5080878, 0], [0, -1673434], [0, 3501887], [0, -6490942], [-1899418, 0], [1662996, 0], [0, 4386721], [0, -2591966], [0, -4249289], [11247096, 0], [-3464390, 0], [0, 499856], [0, -4906699], [0, -2673723], [-6827684, 0], [3609704, 0], [-13385342, 0], [0, -4955574], [0, 8917964], [0, 2161617], [19785828, 0], [15777158, 0], [0, -3393809], [0, -7497109], [0, 5779022], [220156, 0], [4789326, 0], [-4268918, 0], [-16195952, 0], [0, -3267132], [0, -6580855], [-14234764, 0], [0, -10791772], [-18659376, 0], [0, -1283405], [-25958566, 0], [-5532674, 0], [0, -4164699], [0, 6826126], [0, -4273129], [-7584842, 0], [-6184732, 0], [-2781018, 0], [0, 8194732], [0, -11475574], [3501362, 0], [5296680, 0], [0, 10147421], [0, -2413923], [-13021044, 0], [0, -19619344], [0, 6731103], [-917710, 0], [0, 12274402], [0, 8073140], [0, 10168066], [-10772560, 0], [41856568, 0], [-29984546, 0], [0, 7120101], [-41454594, 0], [0, 7703934], [0, 10316403], [0, -5936212], [-1532220, 0], [0, 8716041], [0, 1116348], [-22250080, 0], [0, 26633099]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![400, 0, 1]>],Snew);
    return Vf;
end function;