
// Make newform 1600.3.b.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_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_1600_3_b_s();".
// 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_1600_3_b_s();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-2, 4, -2, 2], [0, 0, 4, 4], [2, 0, 4, -4]];
    Rf_basisdens := [1, 1, 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_1600_b();"
function MakeCharacter_1600_b()
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 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_1600_b_Hecke();"
function MakeCharacter_1600_b_Hecke(Kf)
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    char_values := [[-1, 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, 1, 0, 0], [0, 0, 0, 0], [0, 1, -1, 0], [0, 2, 2, 0], [-4, 0, 0, -1], [6, 0, 0, -4], [0, 0, -2, 0], [0, 3, -1, 0], [2, 0, 0, 2], [0, 10, -2, 0], [4, 0, 0, -5], [-28, 0, 0, 3], [0, -3, -2, 0], [0, 13, 3, 0], [-44, 0, 0, -5], [0, -12, 6, 0], [-32, 0, 0, -13], [0, 11, -10, 0], [0, 2, 10, 0], [-66, 0, 0, 16], [0, -20, 8, 0], [0, 13, 6, 0], [-22, 0, 0, -20], [66, 0, 0, 6], [-82, 0, 0, 4], [0, 23, 11, 0], [0, -17, 12, 0], [32, 0, 0, -37], [-126, 0, 0, -10], [0, -23, -19, 0], [0, -42, 14, 0], [26, 0, 0, -6], [0, -4, -30, 0], [-48, 0, 0, 19], [0, -62, -18, 0], [84, 0, 0, 27], [0, 45, -28, 0], [0, -15, 9, 0], [-204, 0, 0, -5], [0, 76, 6, 0], [-122, 0, 0, 8], [0, -58, 22, 0], [-146, 0, 0, 8], [-16, 0, 0, -35], [0, 64, 16, 0], [0, -42, -14, 0], [0, -13, 35, 0], [0, -1, 8, 0], [-118, 0, 0, -10], [-6, 0, 0, -70], [0, -36, -20, 0], [-68, 0, 0, 15], [0, 26, -26, 0], [266, 0, 0, 2], [0, 79, 29, 0], [192, 0, 0, 35], [0, 86, 6, 0], [-256, 0, 0, 45], [-28, 0, 0, -5], [0, -15, 24, 0], [76, 0, 0, 23], [0, 11, 34, 0], [0, -10, -34, 0], [174, 0, 0, 88], [264, 0, 0, -23], [0, 70, 22, 0], [126, 0, 0, 32], [0, 91, 38, 0], [242, 0, 0, -54], [14, 0, 0, 72], [0, -16, -8, 0], [0, 157, 17, 0], [256, 0, 0, 77], [0, 32, 30, 0], [0, 43, -57, 0], [-48, 0, 0, -77], [124, 0, 0, 47], [282, 0, 0, -36], [-272, 0, 0, -75], [0, 28, -70, 0], [328, 0, 0, -57], [0, 126, 22, 0], [234, 0, 0, -30], [0, 48, -64, 0], [0, -19, 44, 0], [528, 0, 0, -51], [-554, 0, 0, -60], [278, 0, 0, 40], [0, 167, 1, 0], [0, -13, -116, 0], [0, 172, 16, 0], [0, -67, 79, 0], [0, -38, -26, 0], [0, -24, 114, 0], [0, -73, 107, 0], [602, 0, 0, 70], [202, 0, 0, 180], [0, 161, 52, 0], [438, 0, 0, -40], [0, -1, 20, 0], [704, 0, 0, -43], [0, 89, -92, 0], [-112, 0, 0, -27], [0, -122, 10, 0], [186, 0, 0, -118], [0, -153, -34, 0], [94, 0, 0, -48], [0, 12, 108, 0], [172, 0, 0, -41], [0, -251, -69, 0], [656, 0, 0, -35], [-54, 0, 0, -110], [0, 200, -2, 0], [0, -198, 26, 0], [212, 0, 0, -141], [0, -271, 26, 0], [0, 117, 19, 0], [-464, 0, 0, 21], [0, -92, -14, 0], [48, 0, 0, 171], [114, 0, 0, -130], [-136, 0, 0, 153], [0, 61, -22, 0], [0, -86, 2, 0], [568, 0, 0, 31], [-518, 0, 0, 158], [0, 76, -44, 0], [0, -55, -77, 0], [236, 0, 0, 223], [0, 36, -10, 0], [0, -213, -67, 0], [0, 282, 58, 0], [-576, 0, 0, -67], [122, 0, 0, -172], [-102, 0, 0, 164], [-104, 0, 0, -71], [0, -169, -58, 0], [744, 0, 0, 105], [218, 0, 0, -100], [0, 122, 22, 0], [-232, 0, 0, -201], [0, 115, 67, 0], [0, 267, -110, 0], [352, 0, 0, 35], [0, -220, 104, 0], [-264, 0, 0, -295], [426, 0, 0, 130], [0, -52, 6, 0], [0, 147, -79, 0], [284, 0, 0, 71], [-348, 0, 0, 131], [0, -103, 32, 0], [0, -199, -9, 0], [0, 27, -52, 0], [0, 442, 54, 0], [0, -204, -44, 0], [968, 0, 0, -127], [-294, 0, 0, 10], [-1082, 0, 0, 80], [0, 259, 70, 0], [-866, 0, 0, -96], [0, 125, -91, 0], [0, 246, -198, 0], [1226, 0, 0, 34], [0, -65, 61, 0], [0, -14, -122, 0], [-316, 0, 0, -173], [-502, 0, 0, 252], [-1424, 0, 0, 109], [0, 96, 22, 0], [238, 0, 0, 220], [0, 158, -270, 0], [-426, 0, 0, 164], [0, -160, 204, 0], [578, 0, 0, 264], [0, 278, 62, 0], [158, 0, 0, -100], [0, 75, 103, 0], [-768, 0, 0, -53], [0, -307, -11, 0], [0, -66, 202, 0], [156, 0, 0, -377], [1166, 0, 0, -24], [0, -101, -55, 0], [642, 0, 0, -6], [-576, 0, 0, 229], [0, 433, -30, 0], [-72, 0, 0, -47], [0, -214, 234, 0], [-326, 0, 0, -118], [0, 37, 260, 0], [0, -62, 70, 0], [528, 0, 0, -205], [0, -329, -18, 0], [-1086, 0, 0, -42], [-668, 0, 0, -205], [776, 0, 0, -71], [1006, 0, 0, 120], [0, 431, -49, 0], [1082, 0, 0, -90], [0, 246, -78, 0], [4, 0, 0, -77], [-502, 0, 0, 276], [0, -396, 90, 0], [324, 0, 0, -445], [0, -72, -60, 0], [0, -455, -26, 0], [418, 0, 0, -248], [0, -34, -170, 0], [-674, 0, 0, 240], [-632, 0, 0, -25], [0, -1, -115, 0], [0, -265, -130, 0], [0, 332, -60, 0], [412, 0, 0, -185], [0, 113, 265, 0], [1042, 0, 0, -272], [0, -255, 209, 0], [1476, 0, 0, 75], [368, 0, 0, -277], [0, 4, 256, 0], [498, 0, 0, -224], [0, -457, -81, 0], [0, -273, 70, 0], [672, 0, 0, -93], [-546, 0, 0, 0], [0, -152, -24, 0], [0, 237, 69, 0], [0, 374, 86, 0], [-1284, 0, 0, 111], [0, 688, 86, 0], [0, 238, -206, 0], [-108, 0, 0, -333], [0, -319, -202, 0], [0, -183, -275, 0], [-1432, 0, 0, -175], [-1244, 0, 0, -13], [0, -824, -66, 0], [0, -266, 322, 0], [0, 629, -6, 0], [0, -22, 134, 0], [0, 67, 273, 0], [-838, 0, 0, -106], [734, 0, 0, 0], [0, 168, 128, 0], [0, 353, -339, 0], [0, -222, 170, 0], [0, -348, 110, 0], [0, -313, 229, 0], [-2196, 0, 0, -41], [762, 0, 0, 236], [0, -3, -217, 0], [1008, 0, 0, 77], [-484, 0, 0, 87], [0, -712, 30, 0], [1078, 0, 0, 48], [0, 547, 10, 0], [1744, 0, 0, 5], [126, 0, 0, 48], [0, 403, -135, 0], [0, -169, 180, 0], [-198, 0, 0, 190], [2856, 0, 0, 97], [-354, 0, 0, -304], [0, 440, 50, 0], [-318, 0, 0, 658], [922, 0, 0, 252], [0, 61, 12, 0], [476, 0, 0, -385], [-2002, 0, 0, -228], [0, 483, -244, 0], [1314, 0, 0, -330], [0, -864, 0, 0], [-1374, 0, 0, 150], [0, -117, -67, 0], [0, -37, 220, 0], [-1558, 0, 0, 30], [1372, 0, 0, -129], [0, 62, -158, 0], [0, 283, -211, 0], [0, -194, 242, 0], [0, 589, -3, 0], [2838, 0, 0, 32], [0, 287, 354, 0], [0, 170, 62, 0], [1014, 0, 0, 356], [1784, 0, 0, -295], [0, -44, -408, 0], [-1032, 0, 0, 337], [-312, 0, 0, -481], [0, 95, -510, 0], [-186, 0, 0, -260], [0, 226, 46, 0], [-1124, 0, 0, 295], [912, 0, 0, -421], [0, 170, 190, 0], [-1764, 0, 0, -65], [0, 808, 234, 0], [0, -389, -148, 0], [314, 0, 0, 618], [-676, 0, 0, -441], [0, 340, -80, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_3_b_s();".
// 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_1600_3_b_s(:prec:=4)
    chi := MakeCharacter_1600_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(1999) - 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_1600_3_b_s();".
// 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_1600_3_b_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![80, 0, 20, 0, 1]>,<7,R![2000, 0, 100, 0, 1]>,<13,R![-4, 8, 1]>],Snew);
    return Vf;
end function;