
// Make newform 784.3.s.g in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_784_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_784_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_784_3_s_g();".
// 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_784_3_s_g();".
// 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 := [4, 0, -8, 0, 14, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, -1], [0, -41, 0, 0, 0, 0, 0, -1], [16, 0, -28, 0, 7, 0, -2, 0], [-20, 0, 0, 0, 0, 0, -1, 0], [0, 96, 0, -168, 0, 49, 0, -12], [24, 0, -42, 0, 14, 0, -3, 0]];
    Rf_basisdens := [1, 1, 14, 7, 14, 2, 14, 2];
    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_784_s();"
function MakeCharacter_784_s()
    N := 784;
    order := 6;
    char_gens := [687, 197, 689];
    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_784_s_Hecke();"
function MakeCharacter_784_s_Hecke(Kf)
    N := 784;
    order := 6;
    char_gens := [687, 197, 689];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, 1], [0, -3, 0, 4, 0, 0, 4, 0], [0, 0, 0, 3, 0, 0, 0, 0], [0, -5, 5, 0, 0, 0, 1, 0], [26, 0, 0, 0, -26, 0, 0, 0], [8, 0, 0, 0, 0, 2, 0, 0], [0, 0, 2, -8, 0, 0, 0, 0], [32, 0, 0, 0, -32, 0, 0, 0], [0, -3, 0, -5, 0, 0, -5, 0], [-10, 0, 0, 0, 0, 7, 0, 0], [0, 12, -12, 0, 0, 0, -2, 0], [0, 0, 0, 0, -6, 0, 0, 2], [0, 0, 11, 3, 0, 0, 0, 0], [0, -9, 9, 0, 0, 0, -8, 0], [0, 0, 0, 0, 28, 0, 0, -6], [-56, 0, 0, 0, 0, 4, 0, 0], [0, 0, 11, -3, 0, 0, 0, 0], [60, 0, 0, 0, -60, -2, 0, -2], [0, -13, 0, -33, 0, 0, -33, 0], [0, -11, 11, 0, 0, 0, 25, 0], [0, -22, 0, -33, 0, 0, -33, 0], [0, 0, -17, -22, 0, 0, 0, 0], [0, -8, 8, 0, 0, 0, -4, 0], [-10, 0, 0, 0, 10, 4, 0, 4], [0, 0, 0, 0, -68, 0, 0, -8], [104, 0, 0, 0, 0, 3, 0, 0], [-22, 0, 0, 0, 0, -10, 0, 0], [0, -22, 22, 0, 0, 0, -37, 0], [0, 0, 0, 0, 140, 0, 0, -1], [0, 0, 0, 45, 0, 0, 45, 0], [-110, 0, 0, 0, 110, 6, 0, 6], [0, 0, 0, 0, -22, 0, 0, 2], [0, 0, 21, -34, 0, 0, 0, 0], [-54, 0, 0, 0, 54, 17, 0, 17], [0, 44, 0, -42, 0, 0, -42, 0], [0, -37, 37, 0, 0, 0, -66, 0], [0, 0, 0, 0, 182, 0, 0, 16], [0, 27, 0, 44, 0, 0, 44, 0], [-8, 0, 0, 0, 8, -24, 0, -24], [0, 0, 0, 0, -8, 0, 0, 5], [214, 0, 0, 0, 0, -4, 0, 0], [0, 0, 28, 28, 0, 0, 0, 0], [-136, 0, 0, 0, 0, 22, 0, 0], [0, -16, 0, -30, 0, 0, -30, 0], [0, 0, -56, 45, 0, 0, 0, 0], [0, -1, 1, 0, 0, 0, -58, 0], [-28, 0, 0, 0, 28, 3, 0, 3], [-142, 0, 0, 0, 0, -6, 0, 0], [0, 0, -17, -31, 0, 0, 0, 0], [0, 14, 0, 35, 0, 0, 35, 0], [0, -58, 58, 0, 0, 0, -7, 0], [0, 0, 0, 0, -272, 0, 0, 14], [0, 0, -49, -4, 0, 0, 0, 0], [0, 32, -32, 0, 0, 0, 74, 0], [0, 0, 0, 0, 82, 0, 0, 18], [-264, 0, 0, 0, 0, 11, 0, 0], [0, 0, -49, 13, 0, 0, 0, 0], [0, 31, 0, -22, 0, 0, -22, 0], [0, 62, 0, 39, 0, 0, 39, 0], [0, 0, 28, -92, 0, 0, 0, 0], [0, 73, -73, 0, 0, 0, 27, 0], [-174, 0, 0, 0, 174, 26, 0, 26], [-94, 0, 0, 0, 94, 25, 0, 25], [64, 0, 0, 0, 0, 46, 0, 0], [0, 0, 0, 0, -262, 0, 0, -35], [0, 31, 0, 24, 0, 0, 24, 0], [0, 0, 17, -19, 0, 0, 0, 0], [186, 0, 0, 0, -186, 48, 0, 48], [0, 0, -8, 118, 0, 0, 0, 0], [-446, 0, 0, 0, 446, 2, 0, 2], [314, 0, 0, 0, 0, 41, 0, 0], [0, -42, 42, 0, 0, 0, -74, 0], [0, 0, 0, 0, 404, 0, 0, 14], [0, -27, 27, 0, 0, 0, 34, 0], [282, 0, 0, 0, -282, 14, 0, 14], [0, 0, -20, 63, 0, 0, 0, 0], [0, -87, 0, -85, 0, 0, -85, 0], [370, 0, 0, 0, 0, 38, 0, 0], [0, 0, 0, 0, 110, 0, 0, -14], [0, 76, 0, -47, 0, 0, -47, 0], [0, 110, -110, 0, 0, 0, 122, 0], [20, 0, 0, 0, -20, -6, 0, -6], [52, 0, 0, 0, 0, -76, 0, 0], [-92, 0, 0, 0, 92, 45, 0, 45], [0, 55, 0, -112, 0, 0, -112, 0], [408, 0, 0, 0, 0, -10, 0, 0], [0, -71, 71, 0, 0, 0, 39, 0], [0, 0, -8, -112, 0, 0, 0, 0], [0, 0, 0, 0, 18, 0, 0, 46], [480, 0, 0, 0, 0, 34, 0, 0], [-348, 0, 0, 0, 348, 10, 0, 10], [0, -60, 0, 142, 0, 0, 142, 0], [0, -129, 129, 0, 0, 0, -122, 0], [0, 0, 80, -15, 0, 0, 0, 0], [0, 90, -90, 0, 0, 0, 125, 0], [490, 0, 0, 0, -490, 4, 0, 4], [-106, 0, 0, 0, 0, 9, 0, 0], [0, 0, 0, 0, -102, 0, 0, 80], [0, 0, 6, -187, 0, 0, 0, 0], [-384, 0, 0, 0, 384, -38, 0, -38], [0, 0, 0, 0, 170, 0, 0, -17], [0, 0, -36, 45, 0, 0, 0, 0], [0, 64, 0, -5, 0, 0, -5, 0], [0, -31, 31, 0, 0, 0, -149, 0], [0, 0, 0, 0, 184, 0, 0, -22], [0, -120, 0, 81, 0, 0, 81, 0], [0, -16, 16, 0, 0, 0, 96, 0], [0, 0, 0, 0, 288, 0, 0, -72], [6, 0, 0, 0, 0, 74, 0, 0], [0, 0, 160, 175, 0, 0, 0, 0], [268, 0, 0, 0, 0, 48, 0, 0], [0, 0, 0, 0, -576, 0, 0, -34], [0, 122, 0, -135, 0, 0, -135, 0], [0, 0, -114, -20, 0, 0, 0, 0], [40, 0, 0, 0, -40, -34, 0, -34], [786, 0, 0, 0, 0, -17, 0, 0], [0, 0, -65, -268, 0, 0, 0, 0], [-784, 0, 0, 0, 0, -1, 0, 0], [0, 61, -61, 0, 0, 0, 132, 0], [0, 0, 0, 0, -420, 0, 0, 78], [0, 38, -38, 0, 0, 0, -231, 0], [612, 0, 0, 0, 0, -20, 0, 0], [-332, 0, 0, 0, 332, -80, 0, -80], [0, -172, 172, 0, 0, 0, 2, 0], [0, 44, 0, 22, 0, 0, 22, 0], [0, 55, -55, 0, 0, 0, 122, 0], [0, 0, 0, 0, 718, 0, 0, 45], [146, 0, 0, 0, 0, 8, 0, 0], [-254, 0, 0, 0, 254, -26, 0, -26], [-740, 0, 0, 0, 0, -4, 0, 0], [0, 38, -38, 0, 0, 0, 141, 0], [0, -19, 0, -145, 0, 0, -145, 0], [0, 0, -35, -110, 0, 0, 0, 0], [0, 0, -97, 321, 0, 0, 0, 0], [0, 141, 0, -60, 0, 0, -60, 0], [0, 0, 0, 0, 528, 0, 0, 21], [0, -121, 0, 241, 0, 0, 241, 0], [662, 0, 0, 0, -662, -80, 0, -80], [0, 0, 0, 0, 108, 0, 0, -70], [582, 0, 0, 0, 0, -88, 0, 0], [0, 0, -55, -96, 0, 0, 0, 0], [0, 130, 0, 56, 0, 0, 56, 0], [0, -35, 0, 46, 0, 0, 46, 0], [0, 0, -215, 189, 0, 0, 0, 0], [0, 195, -195, 0, 0, 0, -189, 0], [400, 0, 0, 0, -400, -78, 0, -78], [640, 0, 0, 0, -640, 14, 0, 14], [0, 204, 0, 253, 0, 0, 253, 0], [-618, 0, 0, 0, 0, -52, 0, 0], [0, -110, 110, 0, 0, 0, 110, 0], [0, 0, 0, 0, -366, 0, 0, -36], [-1154, 0, 0, 0, 0, 14, 0, 0], [780, 0, 0, 0, -780, -30, 0, -30], [0, -136, 136, 0, 0, 0, 267, 0], [0, 247, 0, 217, 0, 0, 217, 0], [0, 0, -167, 142, 0, 0, 0, 0], [394, 0, 0, 0, -394, -9, 0, -9], [-368, 0, 0, 0, 0, -44, 0, 0], [450, 0, 0, 0, 0, -36, 0, 0], [0, 59, -59, 0, 0, 0, -149, 0], [0, 0, 0, 0, 660, 0, 0, -103], [0, 0, -38, 84, 0, 0, 0, 0], [0, 0, 0, 0, 1152, 0, 0, 26], [0, 0, 107, 114, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_784_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_784_3_s_g();".
// 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_784_3_s_g(:prec:=8)
    chi := MakeCharacter_784_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_784_3_s_g();".
// 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_784_3_s_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_784_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![4, 0, -40, 0, 398, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;