
// Make newform 1050.3.f.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_f_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_1050_3_f_a();".
// 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_1050_3_f_a();".
// 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, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [1, 0, 1, 0], [0, 4, 0, 1]];
    Rf_basisdens := [1, 2, 1, 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_1050_f();"
function MakeCharacter_1050_f()
    N := 1050;
    order := 2;
    char_gens := [701, 127, 451];
    v := [2, 2, 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_1050_f_Hecke();"
function MakeCharacter_1050_f_Hecke(Kf)
    N := 1050;
    order := 2;
    char_gens := [701, 127, 451];
    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, -1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [-2, 3, -1, 2], [-6, -3, 0, 0], [0, 0, -8, -2], [0, 0, 2, -11], [0, 0, -2, -8], [-6, 9, 0, 0], [30, 0, 0, 0], [0, 0, -12, 12], [20, -36, 0, 0], [0, 0, 14, 7], [32, 30, 0, 0], [0, 0, -28, 4], [54, 12, 0, 0], [0, 0, -28, -20], [0, 0, 4, 4], [-44, 12, 0, 0], [-30, 57, 0, 0], [0, 0, -4, 26], [32, 72, 0, 0], [0, 0, -32, 20], [0, 0, -54, 21], [0, 0, -44, 10], [0, 0, 42, -27], [0, 0, -28, -52], [42, -3, 0, 0], [-2, 24, 0, 0], [90, 12, 0, 0], [-56, 24, 0, 0], [0, 0, 24, -12], [-30, 96, 0, 0], [0, 0, -116, -8], [126, -60, 0, 0], [4, 54, 0, 0], [0, 0, -84, -36], [124, 36, 0, 0], [0, 0, 68, -32], [0, 0, -18, 3], [-102, -63, 0, 0], [0, 0, -16, 38], [6, 75, 0, 0], [-28, 36, 0, 0], [150, -24, 0, 0], [0, 0, 2, 68], [64, 30, 0, 0], [0, 0, 18, 36], [0, 0, 132, -24], [0, 0, -72, 6], [-210, -108, 0, 0], [150, -93, 0, 0], [0, 0, 100, -2], [0, 0, 16, 32], [0, 0, 138, 21], [246, 111, 0, 0], [0, 0, 194, -59], [0, 0, -52, -52], [-104, 132, 0, 0], [-66, -60, 0, 0], [0, 0, 198, 24], [0, 0, -74, 167], [0, 0, -54, -24], [0, 0, -56, -76], [0, 0, 136, -116], [18, -396, 0, 0], [256, -54, 0, 0], [-190, -72, 0, 0], [-66, -273, 0, 0], [0, 0, -164, -20], [0, 0, 182, 139], [-102, -3, 0, 0], [0, 0, 10, 124], [118, 216, 0, 0], [-128, 162, 0, 0], [0, 0, -116, -172], [-78, -216, 0, 0], [0, 0, -348, -48], [222, -336, 0, 0], [0, 0, 28, 22], [0, 0, 28, 140], [32, -228, 0, 0], [318, -189, 0, 0], [0, 0, -96, 228], [0, 0, -6, -156], [522, -207, 0, 0], [-210, -276, 0, 0], [-382, -72, 0, 0], [0, 0, 38, 139], [-152, 216, 0, 0], [0, 0, 168, 144], [0, 0, -112, 148], [164, 114, 0, 0], [198, 315, 0, 0], [136, 438, 0, 0], [0, 0, -164, 92], [0, 0, 242, -143], [0, 0, -70, 169], [0, 0, 100, -272], [-376, 468, 0, 0], [436, 216, 0, 0], [-750, -60, 0, 0], [0, 0, -360, -84], [690, -12, 0, 0], [-316, 36, 0, 0], [0, 0, 248, -16], [0, 0, 168, 168], [0, 0, -110, -307], [-6, 669, 0, 0], [0, 0, 48, 264], [0, 0, 202, 268], [-38, -168, 0, 0], [-54, -288, 0, 0], [0, 0, 92, -88], [-160, -684, 0, 0], [66, -672, 0, 0], [0, 0, 30, 144], [0, 0, 0, 384], [-138, 420, 0, 0], [654, -177, 0, 0], [0, 0, 236, -304], [-292, 636, 0, 0], [0, 0, -678, -15], [1026, -147, 0, 0], [0, 0, -84, 24], [-234, -60, 0, 0], [-14, -144, 0, 0], [0, 0, -352, 244], [0, 0, 156, 60], [0, 0, -480, -186], [1028, -216, 0, 0], [-246, -723, 0, 0], [244, 138, 0, 0], [974, -216, 0, 0], [0, 0, -634, -89], [0, 0, 304, -260], [0, 0, 362, 37], [0, 0, 76, 88], [0, 0, 186, -279], [-198, -60, 0, 0], [0, 0, -228, 216], [882, 276, 0, 0], [280, -468, 0, 0], [-810, 111, 0, 0], [0, 0, -280, -286], [0, 0, 580, 152], [0, 0, 316, -152], [0, 0, 170, -95], [0, 0, 14, -88], [66, -303, 0, 0], [658, 144, 0, 0], [0, 0, 138, -39], [1208, 198, 0, 0], [0, 0, -508, -32], [680, 90, 0, 0], [-150, -207, 0, 0], [484, -426, 0, 0], [0, 0, -266, -445], [0, 0, -280, -112], [0, 0, 62, -29], [-258, -117, 0, 0], [954, -288, 0, 0], [-872, -276, 0, 0], [0, 0, -100, -512], [-222, -1116, 0, 0], [0, 0, -92, -472], [916, -90, 0, 0], [0, 0, 324, -564], [-664, 828, 0, 0], [0, 0, -202, -197], [-1506, -153, 0, 0], [0, 0, -108, -48], [-450, 63, 0, 0], [-752, 252, 0, 0], [0, 0, 170, -100], [0, 0, 562, 53], [716, 0, 0, 0], [-78, -132, 0, 0], [0, 0, -284, 76], [0, 0, 780, -12], [1408, -180, 0, 0], [0, 0, 344, 352], [-1160, -36, 0, 0], [0, 0, -278, 449], [-798, -111, 0, 0], [0, 0, -70, 85], [-286, -624, 0, 0], [0, 0, 236, 464], [8, -300, 0, 0], [0, 0, 328, -316], [0, 0, 1060, -74], [-210, 3, 0, 0], [-416, -306, 0, 0], [0, 0, -350, 185], [-294, 981, 0, 0], [0, 0, -298, 439], [100, 156, 0, 0], [472, 684, 0, 0], [0, 0, -150, 561], [0, 0, 160, 44], [-1206, 360, 0, 0], [0, 0, -254, -620], [0, 0, 104, -418], [0, 0, 20, -274], [0, 0, 520, 200], [0, 0, -146, -13], [0, 0, 514, 268], [-750, 741, 0, 0], [-1506, -120, 0, 0], [0, 0, 958, 328], [-1118, 0, 0, 0], [0, 0, -318, 477], [-508, 774, 0, 0], [0, 0, -44, -616], [0, 0, -544, -212], [0, 0, -680, 64], [1124, -42, 0, 0], [0, 0, 558, 207], [-546, 291, 0, 0], [-2130, 336, 0, 0], [514, 648, 0, 0], [0, 0, 420, 420], [-810, -36, 0, 0], [-700, -654, 0, 0], [0, 0, 40, -4], [508, 108, 0, 0], [0, 0, 902, 247], [438, -9, 0, 0], [0, 0, -1372, 116], [1446, 555, 0, 0], [-226, -408, 0, 0], [0, 0, -666, 384], [820, -390, 0, 0], [-654, -660, 0, 0], [0, 0, 682, -344], [0, 0, -276, -840], [0, 0, -188, -170], [930, 720, 0, 0], [1158, 327, 0, 0], [0, 0, 116, -692], [2034, 189, 0, 0], [0, 0, 92, 656], [0, 0, 1226, -76], [-596, 1020, 0, 0], [0, 0, -690, -393], [0, 0, 292, -736], [0, 0, 748, 124], [0, 0, -32, 728], [548, -348, 0, 0], [1590, -153, 0, 0], [-478, 1344, 0, 0], [0, 0, -582, 561], [-966, 729, 0, 0], [0, 0, -788, -470], [0, 0, 278, -173], [906, -423, 0, 0], [-304, 1188, 0, 0], [0, 0, -234, -336], [0, 0, -174, -579], [0, 0, -168, -24], [1264, -672, 0, 0], [-1638, -291, 0, 0], [0, 0, -208, 1034], [0, 0, 4, 232], [0, 0, 342, 291], [0, 0, -804, 96], [6, -120, 0, 0], [0, 0, 1074, 321], [-188, 1968, 0, 0], [-450, 1020, 0, 0], [0, 0, -284, 16], [-664, -750, 0, 0], [0, 0, -492, -390], [1468, -126, 0, 0], [0, 0, -564, -582], [0, 0, -1228, 236], [714, -1263, 0, 0], [-1292, -1092, 0, 0], [628, -1092, 0, 0], [0, 0, 856, 224], [0, 0, -424, 208], [-2440, -420, 0, 0], [0, 0, -200, 944], [0, 0, 884, -556], [0, 0, -364, -808], [42, -543, 0, 0], [-862, -1656, 0, 0], [1470, -480, 0, 0], [0, 0, -1220, -668], [0, 0, -578, -337], [2610, -384, 0, 0], [0, 0, 1456, -148], [1914, -408, 0, 0], [0, 0, -1364, -460], [2038, 456, 0, 0], [0, 0, -134, 41], [0, 0, 362, 356], [0, 0, -854, -343], [0, 0, -644, -592], [0, 0, -1772, 40], [0, 0, 88, 460], [2562, -360, 0, 0], [-2536, -396, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_3_f_a();".
// 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_1050_3_f_a(:prec:=4)
    chi := MakeCharacter_1050_f();
    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_1050_3_f_a();".
// 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_1050_3_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![18, 12, 1]>,<23,R![-126, 12, 1]>],Snew);
    return Vf;
end function;