
// Make newform 224.3.h.d in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_h_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_224_3_h_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_224_3_h_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 := [526, 300, 86, -300, 123, 0, -10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [4767050, -816140, -425950, 320592, 91850, -26304, -9000, -868], [982876622, -1121009602, -624881484, 411714584, -10053780, -31814108, 208186, 2679764], [1714881404, -1116611684, -633800883, 394683447, -55216902, -34456727, 6558661, 3872412], [556407, -5682368, 4853780, -1377571, -3353, 167753, 18417, -12992], [-98379406, 851928604, -1065326618, 684612832, -59878610, -52238584, 4313372, 5735372], [1079942072, 2981272772, -2992899254, 845538352, 109451638, -46895176, -164500, 8071492], [-15107251, -25469724, 23225460, -3401197, -1048471, 372167, 143219, -19240]];
    Rf_basisdens := [1, 1582309, 685139797, 685139797, 1582309, 685139797, 685139797, 1582309];
    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_224_h();"
function MakeCharacter_224_h()
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    v := [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_224_h_Hecke();"
function MakeCharacter_224_h_Hecke(Kf)
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0, 0, 0], [2, 1, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0], [0, 0, -3, -1, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 1], [0, 0, -4, -3, 0, 0, 0, 0], [4, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, -2, 0], [0, 1, 0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, -2, 2, 0], [0, -2, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, -9, -5, 0], [0, -1, 0, 0, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 6, 0], [0, 0, 12, 1, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9, 5, 0], [-68, -13, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 11, 0, 0, 1], [4, -9, 0, 0, 0, 0, 0, 0], [0, 0, 24, -17, 0, 0, 0, 0], [0, -7, 0, 0, 13, 0, 0, -1], [0, 9, 0, 0, -17, 0, 0, 1], [0, 0, -7, 19, 0, 0, 0, 0], [0, 4, 0, 0, -6, 0, 0, 2], [0, 0, 0, 0, 0, -11, 17, 0], [0, 0, 0, 0, 0, -24, -6, 0], [-52, 18, 0, 0, 0, 0, 0, 0], [-20, 38, 0, 0, 0, 0, 0, 0], [0, 0, -8, -13, 0, 0, 0, 0], [38, 32, 0, 0, 0, 0, 0, 0], [0, 0, -44, 17, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 22, 0], [52, -18, 0, 0, 0, 0, 0, 0], [0, 0, -31, -53, 0, 0, 0, 0], [0, 0, 0, 0, 0, -11, 5, 0], [0, 6, 0, 0, -14, 0, 0, -2], [0, 0, -3, 71, 0, 0, 0, 0], [0, 0, 0, 0, 0, -17, -1, 0], [0, 0, -53, -3, 0, 0, 0, 0], [-68, -15, 0, 0, 0, 0, 0, 0], [32, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -34, -14, 0], [0, 14, 0, 0, -26, 0, 0, 2], [0, 0, 0, 0, 0, -7, -27, 0], [0, -10, 0, 0, 12, 0, 0, -8], [0, 0, 4, 41, 0, 0, 0, 0], [0, 0, 5, -41, 0, 0, 0, 0], [134, -8, 0, 0, 0, 0, 0, 0], [268, 30, 0, 0, 0, 0, 0, 0], [0, 13, 0, 0, -21, 0, 0, 5], [0, 0, -16, 63, 0, 0, 0, 0], [0, -8, 0, 0, 4, 0, 0, -12], [-116, 15, 0, 0, 0, 0, 0, 0], [0, 0, 49, -117, 0, 0, 0, 0], [0, 12, 0, 0, -24, 0, 0, 0], [0, 0, 0, 0, 0, 16, 10, 0], [-214, -24, 0, 0, 0, 0, 0, 0], [0, 0, 64, -13, 0, 0, 0, 0], [0, 0, 65, 43, 0, 0, 0, 0], [0, 0, 8, -93, 0, 0, 0, 0], [0, -13, 0, 0, 38, 0, 0, 12], [0, -4, 0, 0, 2, 0, 0, -6], [0, 0, 0, 0, 0, 14, -18, 0], [0, 0, 0, 0, 0, 1, -63, 0], [188, -94, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -33, -37, 0], [0, 0, -9, -31, 0, 0, 0, 0], [0, -18, 0, 0, 46, 0, 0, 10], [52, -58, 0, 0, 0, 0, 0, 0], [0, 14, 0, 0, -12, 0, 0, 16], [0, 0, 0, 0, 0, 24, -38, 0], [0, 0, 0, 0, 0, -79, 1, 0], [0, -17, 0, 0, 24, 0, 0, -10], [0, 0, 0, 0, 0, -38, -62, 0], [0, 0, -73, 121, 0, 0, 0, 0], [200, 90, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, -10, 0, 0, 14], [0, 0, -8, -13, 0, 0, 0, 0], [0, 0, 0, 0, 0, 46, 22, 0], [-212, 42, 0, 0, 0, 0, 0, 0], [0, -23, 0, 0, 35, 0, 0, -11], [0, -21, 0, 0, 50, 0, 0, 8], [0, 0, 0, 0, 0, -107, -51, 0], [86, 40, 0, 0, 0, 0, 0, 0], [-292, -78, 0, 0, 0, 0, 0, 0], [0, 0, -73, -143, 0, 0, 0, 0], [172, -207, 0, 0, 0, 0, 0, 0], [0, 0, -68, -19, 0, 0, 0, 0], [0, -1, 0, 0, 12, 0, 0, 10], [244, -166, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 23, 0], [0, 0, 0, 0, 0, 69, -43, 0], [0, -45, 0, 0, 66, 0, 0, -24], [0, 0, 75, 77, 0, 0, 0, 0], [0, 54, 0, 0, -108, 0, 0, 0], [0, 0, 116, 117, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, -78, 0], [0, 0, 0, 0, 0, -59, 93, 0], [0, 0, 0, 0, 0, 4, -46, 0], [0, 0, 124, -43, 0, 0, 0, 0], [-76, 218, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 19, 79, 0], [0, -24, 0, 0, 36, 0, 0, -12], [0, 0, -20, -235, 0, 0, 0, 0], [0, -62, 0, 0, 134, 0, 0, 10], [148, 145, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, -29, 0, 0, -23], [0, -26, 0, 0, 36, 0, 0, -16], [0, 0, 0, 0, 0, 10, 134, 0], [56, -142, 0, 0, 0, 0, 0, 0], [0, 0, 40, 107, 0, 0, 0, 0], [-476, 41, 0, 0, 0, 0, 0, 0], [608, 18, 0, 0, 0, 0, 0, 0], [0, 0, 12, 109, 0, 0, 0, 0], [0, 66, 0, 0, -138, 0, 0, -6], [0, 0, 0, 0, 0, 146, 50, 0], [0, 0, 0, 0, 0, -21, -41, 0], [0, 0, 5, 127, 0, 0, 0, 0], [-118, 24, 0, 0, 0, 0, 0, 0], [0, 0, -141, -91, 0, 0, 0, 0], [0, 0, 0, 0, 0, -11, -55, 0], [0, 0, -104, 239, 0, 0, 0, 0], [0, 0, 0, 0, 0, -130, -110, 0], [0, 0, 0, 0, 0, -132, -106, 0], [0, 57, 0, 0, -84, 0, 0, 30], [0, -32, 0, 0, 94, 0, 0, 30], [0, 0, 131, -155, 0, 0, 0, 0], [0, 0, 0, 0, 0, -83, 53, 0], [-332, -2, 0, 0, 0, 0, 0, 0], [-548, 186, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -86, -10, 0], [0, 68, 0, 0, -114, 0, 0, 22], [0, 29, 0, 0, -57, 0, 0, 1], [0, 0, 55, -87, 0, 0, 0, 0], [0, 0, 12, -23, 0, 0, 0, 0], [0, 0, -43, 79, 0, 0, 0, 0], [-4, 162, 0, 0, 0, 0, 0, 0], [0, 0, 160, -229, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 18, 0], [652, 91, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 75, -45, 0], [0, 0, -133, 373, 0, 0, 0, 0], [0, 70, 0, 0, -146, 0, 0, -6], [0, 0, 89, -53, 0, 0, 0, 0], [0, 56, 0, 0, -82, 0, 0, 30], [0, 0, 96, -401, 0, 0, 0, 0], [-1124, 53, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 54, 0], [0, -34, 0, 0, 82, 0, 0, 14], [0, 0, 0, 0, 0, 195, 63, 0], [0, -40, 0, 0, 38, 0, 0, -42], [0, 0, 0, 0, 0, -67, 125, 0], [508, 246, 0, 0, 0, 0, 0, 0], [1180, -73, 0, 0, 0, 0, 0, 0], [0, 19, 0, 0, -3, 0, 0, 35], [0, -37, 0, 0, 103, 0, 0, 29], [0, 0, 329, 67, 0, 0, 0, 0], [0, 0, 0, 0, 0, -45, 151, 0], [-1318, 48, 0, 0, 0, 0, 0, 0], [-716, -210, 0, 0, 0, 0, 0, 0], [0, 0, -4, 393, 0, 0, 0, 0], [-202, 416, 0, 0, 0, 0, 0, 0], [0, -30, 0, 0, 98, 0, 0, 38], [-284, -345, 0, 0, 0, 0, 0, 0], [0, 0, 275, -347, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_h_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_224_3_h_d(:prec:=8)
    chi := MakeCharacter_224_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_224_3_h_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_224_3_h_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![52, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;