
// Make newform 800.3.b.i in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_800_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_800_b_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_800_3_b_i();".
// 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_800_3_b_i();".
// 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, 14, 0, 9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0], [-9, 0, 20, 0, 4, 0], [0, 76, 0, 40, 0, 4], [51, 0, 40, 0, 4, 0], [0, -194, 0, -140, 0, -16]];
    Rf_basisdens := [1, 1, 5, 5, 5, 5];
    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_800_b();"
function MakeCharacter_800_b()
    N := 800;
    order := 2;
    char_gens := [351, 101, 577];
    v := [1, 2, 2];
    // 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_800_b_Hecke();"
function MakeCharacter_800_b_Hecke(Kf)
    N := 800;
    order := 2;
    char_gens := [351, 101, 577];
    char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 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, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1], [0, -2, 0, 1, 0, 2], [1, 0, -2, 0, -1, 0], [12, 0, 2, 0, 2, 0], [0, -6, 0, -3, 0, -2], [0, -2, 0, 4, 0, 3], [-6, 0, -4, 0, 0, 0], [0, -4, 0, 0, 0, -4], [37, 0, -4, 0, -3, 0], [-10, 0, -3, 0, -1, 0], [0, -1, 0, 0, 0, -6], [0, 8, 0, 12, 0, -1], [11, 0, -6, 0, 5, 0], [0, 6, 0, -13, 0, 2], [-20, 0, 3, 0, 7, 0], [0, 13, 0, 16, 0, 2], [0, 0, 0, 10, 0, 8], [10, 0, 6, 0, 4, 0], [0, 12, 0, 20, 0, -4], [0, 3, 0, 16, 0, 14], [-22, 0, 12, 0, 16, 0], [32, 0, 2, 0, 6, 0], [62, 0, -4, 0, -16, 0], [0, -2, 0, -4, 0, 3], [0, -21, 0, 0, 0, -20], [-32, 0, -7, 0, -7, 0], [44, 0, -16, 0, -20, 0], [0, 26, 0, 0, 0, -3], [0, -42, 0, -1, 0, 2], [22, 0, 8, 0, 6, 0], [0, -30, 0, -33, 0, -2], [-4, 0, -3, 0, 17, 0], [0, -12, 0, 6, 0, -20], [-113, 0, -12, 0, -9, 0], [0, 21, 0, -40, 0, -20], [0, -34, 0, -12, 0, 1], [61, 0, 20, 0, -3, 0], [0, -50, 0, -27, 0, 2], [38, 0, -28, 0, 16, 0], [0, -56, 0, 0, 0, 8], [-120, 0, 2, 0, 14, 0], [-111, 0, 2, 0, -5, 0], [0, -16, 0, 54, 0, 40], [0, 30, 0, -13, 0, 26], [0, -54, 0, -16, 0, 31], [0, 35, 0, -24, 0, -4], [66, 0, -12, 0, 0, 0], [-14, 0, -10, 0, -36, 0], [0, 12, 0, -4, 0, -4], [150, 0, -11, 0, 7, 0], [0, -18, 0, 29, 0, 2], [-152, 0, 24, 0, -16, 0], [0, 20, 0, 8, 0, -3], [-140, 0, 29, 0, -7, 0], [0, -36, 0, 12, 0, -20], [-199, 0, -4, 0, 33, 0], [-190, 0, -23, 0, -33, 0], [0, 13, 0, -88, 0, 12], [91, 0, -40, 0, -1, 0], [0, 1, 0, -64, 0, -2], [0, 84, 0, -34, 0, 12], [-266, 0, 16, 0, -2, 0], [-37, 0, -10, 0, 9, 0], [0, -86, 0, 17, 0, 30], [-12, 0, 52, 0, 0, 0], [0, 41, 0, -24, 0, 14], [66, 0, 52, 0, 48, 0], [8, 0, 68, 0, 52, 0], [0, -92, 0, -58, 0, -4], [0, 34, 0, -8, 0, 21], [-177, 0, -36, 0, -49, 0], [0, -46, 0, 51, 0, 30], [0, -122, 0, -24, 0, -45], [300, 0, -19, 0, -47, 0], [-73, 0, -42, 0, -75, 0], [270, 0, -4, 0, 16, 0], [-142, 0, -37, 0, 17, 0], [0, -2, 0, 137, 0, 34], [396, 0, 43, 0, 15, 0], [0, -36, 0, -124, 0, -20], [-372, 0, 26, 0, 10, 0], [0, -100, 0, -62, 0, 36], [0, 17, 0, 96, 0, 8], [250, 0, 43, 0, 9, 0], [-202, 0, 52, 0, 106, 0], [-466, 0, -4, 0, -16, 0], [0, 112, 0, -36, 0, 25], [0, -25, 0, 112, 0, 56], [0, -112, 0, -72, 0, 16], [0, 124, 0, 24, 0, -45], [0, 34, 0, 121, 0, -10], [0, 86, 0, -11, 0, -86], [0, -18, 0, 76, 0, 35], [26, 0, -68, 0, 0, 0], [-414, 0, 64, 0, 48, 0], [0, 89, 0, -72, 0, -48], [-170, 0, -60, 0, -80, 0], [0, -161, 0, 8, 0, -16], [139, 0, -44, 0, 3, 0], [0, -59, 0, 8, 0, -4], [-222, 0, 91, 0, 65, 0], [0, 70, 0, -3, 0, 58], [352, 0, -44, 0, -100, 0], [0, 89, 0, 136, 0, -34], [620, 0, 48, 0, 12, 0], [0, 136, 0, -46, 0, 80], [-246, 0, 17, 0, -33, 0], [0, -184, 0, -68, 0, -17], [-113, 0, 118, 0, 45, 0], [150, 0, -26, 0, 0, 0], [0, 190, 0, 71, 0, 10], [0, 148, 0, -2, 0, -20], [334, 0, 13, 0, 39, 0], [0, -73, 0, 24, 0, 58], [0, 208, 0, -8, 0, 63], [1, 0, 18, 0, -69, 0], [0, 22, 0, 93, 0, 42], [344, 0, 119, 0, 31, 0], [-208, 0, -74, 0, 50, 0], [609, 0, -18, 0, -81, 0], [0, 99, 0, -56, 0, -106], [0, 22, 0, -69, 0, -78], [-64, 0, 183, 0, 55, 0], [186, 0, -68, 0, -16, 0], [0, 92, 0, -44, 0, 76], [0, -12, 0, 88, 0, 75], [-991, 0, -8, 0, 61, 0], [0, 38, 0, 73, 0, 42], [0, 216, 0, 136, 0, -7], [0, 32, 0, 172, 0, 112], [-711, 0, -96, 0, -83, 0], [-402, 0, -36, 0, -112, 0], [-34, 0, 32, 0, 48, 0], [-49, 0, -30, 0, -47, 0], [0, 37, 0, -192, 0, -118], [-765, 0, 90, 0, -3, 0], [-386, 0, -32, 0, 80, 0], [0, -10, 0, -183, 0, -134], [464, 0, -65, 0, -177, 0], [0, 58, 0, -52, 0, -41], [0, -59, 0, 8, 0, -126], [804, 0, -35, 0, -87, 0], [0, -44, 0, -162, 0, -148], [-709, 0, -102, 0, -75, 0], [-318, 0, 132, 0, -42, 0], [0, -138, 0, -133, 0, 50], [0, -268, 0, 28, 0, 133], [19, 0, 184, 0, 119, 0], [538, 0, 9, 0, 55, 0], [0, -183, 0, -128, 0, 56], [0, 40, 0, -136, 0, -41], [0, 67, 0, -248, 0, 68], [0, 292, 0, 12, 0, 20], [0, -56, 0, 66, 0, 16], [90, 0, -165, 0, -7, 0], [-250, 0, 34, 0, -28, 0], [-474, 0, 84, 0, 48, 0], [0, -243, 0, -168, 0, -14], [-154, 0, 36, 0, 42, 0], [0, 22, 0, -4, 0, 25], [0, -94, 0, 145, 0, 214], [-84, 0, -26, 0, 14, 0], [0, -44, 0, -224, 0, -3], [0, 224, 0, -136, 0, -192], [-903, 0, -2, 0, -73, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_800_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_800_3_b_i();".
// 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_800_3_b_i(:prec:=6)
    chi := MakeCharacter_800_b();
    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_800_3_b_i();".
// 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_800_3_b_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_800_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![64, 0, 224, 0, 36, 0, 1]>,<13,R![-832, -208, 0, 1]>],Snew);
    return Vf;
end function;