
// Make newform 448.3.s.b in Magma, downloaded from the LMFDB on 30 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_448_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_448_s_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_448_3_s_b();".
// 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_448_3_s_b();".
// 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, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_448_s();"
function MakeCharacter_448_s()
    N := 448;
    order := 6;
    char_gens := [127, 197, 129];
    v := [6, 6, 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_448_s_Hecke();"
function MakeCharacter_448_s_Hecke( : Kf := false)
    N := 448;
    order := 6;
    char_gens := [127, 197, 129];
    char_values := [[1, 0], [1, 0], [0, 1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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], [1, 1], [-2, 1], [7, 0], [-15, 15], [-8, 16], [17, 17], [18, -9], [0, -9], [6, 0], [7, 7], [0, 31], [-32, 64], [10, 0], [-50, 25], [-57, 57], [-47, -47], [94, -47], [49, -49], [126, 0], [-15, -15], [0, -73], [-8, 16], [66, -33], [16, -32], [-57, -57], [-82, 41], [0, -39], [103, -103], [-78, 0], [-50, 0], [114, -57], [-63, 63], [136, -272], [0, -177], [55, -55], [-57, -57], [0, -23], [160, -320], [-162, 81], [-63, 63], [-72, 144], [0, -57], [209, -209], [150, 0], [119, 119], [346, 0], [192, -384], [81, 81], [-194, 97], [0, 273], [222, 0], [-127, -127], [56, -112], [-222, 111], [-153, 153], [263, 263], [-82, 41], [359, -359], [-222, 0], [1, 1], [-136, 272], [56, -112], [-281, -281], [258, -129], [0, 447], [0, -151], [274, 0], [-543, 543], [-104, 208], [-175, -175], [0, 39], [-201, -201], [0, 383], [-230, 0], [-370, 185], [-201, 201], [62, -31], [0, 177], [193, 193], [344, -688], [22, 0], [327, -327], [16, -32], [-690, 345], [0, 729], [-270, 0], [0, -799], [248, -496], [814, 0], [-462, 231], [311, 311], [-857, 857], [570, 0], [0, 425], [-112, 224], [446, -223], [-143, -143], [914, -457], [0, -721], [-118, 0], [-633, 633], [-159, -159], [0, -447], [-527, 527], [97, 97], [552, -1104], [-350, 175], [807, -807], [224, -448], [846, -423], [503, -503], [930, 0], [305, 305], [-194, 0], [33, -33], [-200, 400], [-441, -441], [0, -33], [-870, 0], [-585, -585], [-14, 0], [318, -159], [273, -273], [178, -89], [-906, 0], [0, 703], [686, -343], [-16, 32], [1118, -559], [577, -577], [-162, 0], [0, 775], [950, 0], [226, -113], [-528, 1056], [103, 103], [193, 193], [504, -1008], [465, -465], [72, -144], [0, -753], [-25, 25], [-246, 0], [855, 855], [208, -416], [-696, 1392], [113, 113], [-1646, 823], [0, 855], [0, -241], [-768, 1536], [-1286, 0], [1486, -743], [737, -737], [-1266, 0], [0, -473], [386, -193], [496, -992], [71, 71], [0, 201], [66, 0], [-194, 0], [1714, -857], [1521, -1521], [-649, -649], [823, -823], [-73, -73]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_448_s_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_448_3_s_b();".
// 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_448_3_s_b(:prec:=2)
    chi := MakeCharacter_448_s();
    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_448_3_s_b();".
// 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_448_3_s_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_448_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![3, -3, 1]>],Snew);
    return Vf;
end function;