
// Make newform 1200.3.bg.e in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_bg();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_bg_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_1200_3_bg_e();".
// 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_1200_3_bg_e();".
// 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 := [9, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 3, 3];
    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_1200_bg();"
function MakeCharacter_1200_bg()
    N := 1200;
    order := 4;
    char_gens := [751, 901, 401, 577];
    v := [4, 4, 4, 3];
    // 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_1200_bg_Hecke();"
function MakeCharacter_1200_bg_Hecke(Kf)
    N := 1200;
    order := 4;
    char_gens := [751, 901, 401, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 0, -1, 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], [-3, 0, 3, -2], [4, -1, 0, 1], [-12, 2, -12, 0], [6, 0, -6, 14], [0, -6, 2, -6], [-14, -6, -14, 0], [0, -9, -22, -9], [20, 10, 0, -10], [-28, 0, 28, -6], [-14, -14, 0, 14], [-2, -28, -2, 0], [-4, 0, 4, -38], [30, 20, 30, 0], [0, -21, 4, -21], [-6, -22, 0, 22], [-2, 0, 2, -68], [-68, -8, 0, 8], [-27, -40, -27, 0], [0, 6, -112, 6], [68, -22, 68, 0], [0, -4, 62, -4], [37, 0, -37, 0], [38, -23, 0, 23], [31, -34, 31, 0], [-40, 0, 40, 22], [0, -34, -14, -34], [36, -54, 36, 0], [93, 0, -93, 22], [-104, 7, 0, -7], [-24, 0, 24, -78], [0, -24, 82, -24], [0, -39, -42, -39], [262, 12, 0, -12], [-38, 0, 38, -74], [0, 96, 0, 0], [-90, 0, 90, 46], [-16, 10, -16, 0], [0, 25, 16, 25], [-50, 86, 0, -86], [-100, -62, 0, 62], [-95, -64, -95, 0], [-42, 0, 42, -36], [0, -68, -138, -68], [-38, -96, 0, 96], [231, -46, 231, 0], [-156, 0, 156, -64], [0, -60, 138, -60], [146, -78, 146, 0], [0, 124, -24, 124], [-64, -136, 0, 136], [-120, 27, 0, -27], [-224, 0, 224, 70], [-146, 14, -146, 0], [0, 45, 2, 45], [-246, 32, 0, -32], [232, 0, -232, -54], [-18, -130, 0, 130], [-166, 152, -166, 0], [178, 56, 178, 0], [238, 0, -238, -104], [76, 32, 0, -32], [63, 160, 63, 0], [20, 0, -20, 14], [238, 106, 0, -106], [-69, 0, 69, 168], [-216, 0, 216, 204], [0, -34, 158, -34], [-46, 66, -46, 0], [0, -22, -76, -22], [-187, 0, 187, -22], [-90, 98, -90, 0], [0, 112, 150, 112], [372, -38, 372, 0], [0, 67, 534, 67], [58, 0, -58, 258], [-122, -96, 0, 96], [0, 32, -12, 32], [0, -153, -40, -153], [410, 144, 0, -144], [-232, 86, 0, -86], [-137, -124, -137, 0], [0, 36, 322, 36], [8, 208, 8, 0], [0, 114, 562, 114], [161, 0, -161, 112], [-110, 63, 0, -63], [-29, -134, -29, 0], [468, 0, -468, 74], [0, 116, -172, 116], [-15, 0, 15, 22], [52, 357, 0, -357], [0, -50, 190, -50], [-276, 78, -276, 0], [0, -153, -546, -153], [-270, -26, 0, 26], [-258, -100, -258, 0], [230, -30, 0, 30], [-42, 0, 42, -24], [-366, 0, 366, -316], [532, -140, 532, 0], [0, 188, 26, 188], [-158, -118, 0, 118], [-21, 0, 21, -220], [364, 0, -364, -150], [52, 22, 52, 0], [0, -182, 64, -182], [-26, -84, 0, 84], [65, 0, -65, -26], [78, -150, 78, 0], [402, 0, -402, 294], [0, -72, -258, -72], [-400, -170, 0, 170], [-742, -80, 0, 80], [-40, 84, -40, 0], [-368, 0, 368, -230], [-72, -182, -72, 0], [0, 291, 444, 291], [610, 122, 0, -122], [-143, -564, -143, 0], [-248, 0, 248, -366], [36, 98, 36, 0], [-966, 78, 0, -78], [562, -115, 0, 115], [0, -224, -378, -224], [0, -68, -772, -68], [177, 0, -177, -186], [64, 474, 64, 0], [0, -374, -266, -374], [-240, -270, -240, 0], [60, -98, 0, 98], [-168, 0, 168, -290], [94, 334, 0, -334], [0, -44, -94, -44], [356, -170, 356, 0], [-330, 0, 330, 96], [72, 0, -72, -46], [0, -24, 758, -24], [-370, 188, 0, -188], [410, -65, 0, 65], [235, -214, 235, 0], [732, 0, -732, 116], [0, -638, -14, -638], [0, -346, -116, -346], [150, -330, 150, 0], [6, 0, -6, -566], [0, 212, -766, 212], [-882, -138, -882, 0], [222, 0, -222, 206], [1202, -184, 0, 184], [-324, -280, -324, 0], [-326, 0, 326, -46], [652, 0, -652, 60], [32, -50, 0, 50], [0, -434, -404, -434], [0, -110, -638, -110], [331, 0, -331, 400], [546, -495, 0, 495], [460, 0, -460, -472], [240, 18, 240, 0], [477, 0, -477, -654], [-1192, -183, 0, 183], [906, 0, -906, 118], [144, 58, 144, 0], [398, 344, 0, -344], [-456, 0, 456, 362], [0, -208, 308, -208], [-48, -202, -48, 0], [0, -91, 716, -91], [-1742, 104, 0, -104], [92, 28, 0, -28], [767, 460, 767, 0], [0, 210, 264, 210], [0, 120, -1190, 120], [-990, 288, 0, -288], [-202, 137, 0, -137], [255, -746, 255, 0], [0, 556, 598, 556], [85, 0, -85, -374], [1244, 33, 0, -33], [-246, -494, -246, 0], [-832, 0, 832, 74], [-440, -674, -440, 0], [0, -309, -446, -309], [70, 0, -70, 142], [624, 224, 624, 0], [0, -184, -1018, -184], [728, 534, 0, -534], [-763, 596, -763, 0], [-28, -576, -28, 0], [546, 304, 0, -304], [518, 123, 0, -123], [896, 0, -896, 26], [-48, 898, -48, 0], [76, 260, 0, -260], [532, -726, 532, 0], [408, 0, -408, 670], [-1332, -130, -1332, 0], [0, 69, -466, 69], [1268, 322, 0, -322], [-252, 0, 252, -190], [0, 440, 340, 440], [0, 629, 312, 629], [-18, 0, 18, 632], [0, 590, -872, 590], [-288, 828, -288, 0], [0, -488, 1206, -488], [246, 346, 0, -346], [-1489, 0, 1489, -240], [-806, 121, 0, -121], [741, 154, 741, 0], [-1188, 0, 1188, -396], [0, -166, -352, -166], [-80, 852, 0, -852], [11, 0, -11, 1118], [1046, 424, 0, -424], [412, 0, -412, 150], [-466, -432, -466, 0], [374, 32, 0, -32], [0, 244, -458, 244], [0, -478, -554, -478], [189, -282, 189, 0], [1212, 0, -1212, 270], [0, 488, -50, 488], [352, 14, 352, 0], [0, 208, -1416, 208], [653, 0, -653, 610], [-480, 545, 0, -545], [-600, 674, -600, 0], [0, 128, 674, 128], [-246, -28, 0, 28], [-486, 110, 0, -110], [354, 796, 354, 0], [14, 0, -14, 1282], [0, 360, 448, 360], [-414, 560, -414, 0], [0, 691, 192, 691], [-760, -424, 0, 424], [-12, -1676, -12, 0], [506, -366, 0, 366], [863, 738, 863, 0], [0, -34, -1074, -34], [-366, -738, -366, 0], [0, -670, -220, -670], [-553, 0, 553, 1030], [996, -615, 0, 615], [0, 618, 1442, 618], [-10, -574, -10, 0], [-526, 0, 526, 394], [-2, 648, 0, -648], [-550, 0, 550, -754], [0, -948, 140, -948], [146, -1104, 146, 0], [0, -831, -380, -831], [826, 78, 0, -78], [-200, 0, 200, -984], [-1338, 0, 1338, -56], [33, 0, -33, -1068], [1121, -842, 1121, 0], [-1364, 0, 1364, 352], [0, -20, -754, -20], [996, 118, 996, 0], [1364, 0, -1364, -78], [0, -866, -238, -866], [0, -137, 1674, -137], [-470, 850, 0, -850], [-906, 912, -906, 0], [-716, 1118, -716, 0], [-1810, -272, 0, 272], [1362, 0, -1362, -584], [-867, 772, -867, 0], [0, -274, -1088, -274], [835, 0, -835, 604], [-353, 382, -353, 0], [608, 0, -608, -700], [0, -350, 442, -350], [-576, 732, 0, -732], [1944, 57, 0, -57], [-1426, 246, -1426, 0], [-1134, -116, 0, 116], [-416, 0, 416, -550], [1582, -420, 0, 420], [768, 0, -768, 924], [-32, -578, 0, 578], [-947, -116, -947, 0], [-320, 0, 320, -14], [0, -496, 494, -496], [0, -498, 1790, -498], [742, -27, 0, 27], [240, 0, -240, -1018], [-570, 374, -570, 0], [2624, 59, 0, -59], [2208, 50, 2208, 0], [0, -477, 290, -477], [964, 998, 0, -998], [432, -550, 432, 0], [0, -1295, -256, -1295], [-890, 0, 890, -996], [-227, 1072, -227, 0], [-996, 0, 996, 730], [0, -876, 754, -876]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_bg_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_bg_e();".
// 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_1200_3_bg_e(:prec:=4)
    chi := MakeCharacter_1200_bg();
    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_1200_3_bg_e();".
// 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_1200_3_bg_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![36, 72, 72, 12, 1]>],Snew);
    return Vf;
end function;