
// Make newform 900.3.l.b in Magma, downloaded from the LMFDB on 12 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_l_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_900_3_l_b();".
// 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_900_3_l_b();".
// 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, 0, 1, 0], [0, 3, 0, 1], [0, 3, 0, -1]];
    Rf_basisdens := [1, 3, 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_900_l();"
function MakeCharacter_900_l()
    N := 900;
    order := 4;
    char_gens := [451, 101, 577];
    v := [4, 4, 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_900_l_Hecke();"
function MakeCharacter_900_l_Hecke( : Kf := false)
    N := 900;
    order := 4;
    char_gens := [451, 101, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, -1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 0], [-5, 5, 1, -1], [4, 0, 0, 5], [0, 0, -5, -5], [-10, 10, -1, 1], [0, -10, 10, 0], [-10, -10, 3, 3], [0, -22, -5, 0], [4, 0, 0, -10], [0, 0, 3, -3], [-50, 0, 0, 10], [-30, -30, 2, 2], [0, 0, -9, 9], [-50, -50, 6, 6], [0, -52, -15, 0], [-78, 0, 0, -10], [10, -10, -6, 6], [20, 0, 0, -40], [5, 5, 16, 16], [0, 24, 10, 0], [-60, -60, -17, -17], [0, 10, 60, 0], [-75, 75, -8, 8], [-58, 0, 0, 5], [-55, -55, 13, 13], [-40, 40, -23, 23], [0, -86, -30, 0], [40, 40, -11, -11], [-5, 5, 1, -1], [-32, 0, 0, 5], [-60, 60, 17, -17], [0, -122, 40, 0], [0, -114, -35, 0], [-110, 0, 0, -20], [30, -30, -3, 3], [40, 40, -20, -20], [130, -130, 17, -17], [80, 80, 37, 37], [0, 88, -5, 0], [-10, 0, 0, -110], [-4, 0, 0, 30], [-15, -15, 28, 28], [30, -30, -94, 94], [0, 114, -20, 0], [110, 0, 0, -40], [105, 105, 55, 55], [100, -100, 40, -40], [0, 138, 60, 0], [90, 90, 25, 25], [0, 96, -60, 0], [-32, 0, 0, 0], [312, 0, 0, 25], [-140, 140, 27, -27], [130, 130, 33, 33], [0, 242, -15, 0], [-66, 0, 0, 0], [-180, 180, -13, 13], [-6, 0, 0, -130], [-50, -50, -104, -104], [90, 90, -96, -96], [-150, 150, -12, 12], [116, 0, 0, -40], [255, 255, 28, 28], [60, -60, -1, 1], [202, 0, 0, 170], [75, -75, 8, -8], [320, -320, -38, 38], [0, 38, 90, 0], [90, 90, -71, -71], [0, -212, 30, 0], [-5, 5, -169, 169], [-70, -70, 87, 87], [0, 194, 0, 0], [-240, -240, 119, 119], [0, 222, -105, 0], [-10, 10, 163, -163], [114, 0, 0, -200], [0, -12, -80, 0], [0, 504, 125, 0], [-310, 0, 0, -80], [-128, 0, 0, 90], [-65, -65, -2, -2], [0, 214, -140, 0], [-120, -120, -88, -88], [0, -154, -150, 0], [-255, 255, 120, -120], [-326, 0, 0, 75], [-515, -515, -9, -9], [140, -140, 207, -207], [0, -52, 60, 0], [-105, 105, 57, -57], [-4, 0, 0, -65], [0, -278, 110, 0], [-240, -240, 117, 117], [0, -226, -45, 0], [-522, 0, 0, 110], [-30, -30, 26, 26], [-482, 0, 0, -50], [-390, 390, 92, -92], [-150, 150, 118, -118], [-420, -420, 150, 150], [0, -338, -140, 0], [230, 0, 0, 10], [-205, 205, -198, 198], [-180, 180, -201, 201], [-40, -40, -157, -157], [0, -568, 70, 0], [-202, 0, 0, 180], [135, -135, 93, -93], [210, 210, -209, -209], [-150, 150, 91, -91], [0, 522, 120, 0], [232, 0, 0, 130], [-634, 0, 0, -120], [-480, -480, 118, 118], [-620, 620, 7, -7], [-240, -240, -67, -67], [0, 196, 105, 0], [938, 0, 0, -90], [-295, -295, 170, 170], [-160, 160, 209, -209], [540, 540, 107, 107], [-1122, 0, 0, -10], [514, 0, 0, 65], [0, 662, 120, 0], [0, 236, -60, 0], [-585, 585, -103, 103], [-100, -100, -53, -53], [0, 130, -430, 0], [-260, -260, -133, -133], [252, 0, 0, 450], [420, -420, -231, 231], [-182, 0, 0, -10], [0, 706, 60, 0], [-420, -420, -29, -29], [890, -890, -28, 28], [-240, 240, -167, 167], [0, -518, 160, 0], [-310, 0, 0, 140], [-86, 0, 0, -365], [85, 85, 171, 171], [780, -780, -122, 122], [0, 586, -290, 0], [0, -1052, -30, 0], [-670, -670, -151, -151], [750, -750, 53, -53], [0, -410, -220, 0], [-430, -430, -259, -259], [450, -450, 49, -49], [550, 0, 0, -280], [380, 380, -136, -136], [150, -150, 79, -79], [140, -140, 318, -318], [-568, 0, 0, 90], [0, -836, 330, 0], [0, 1046, 10, 0], [155, -155, 88, -88], [-630, 0, 0, 365], [-380, 380, -108, 108], [-420, -420, 17, 17], [-205, 205, -105, 105], [872, 0, 0, 275], [-150, 150, 283, -283], [460, 460, 319, 319], [-662, 0, 0, 360], [-700, 700, -41, 41]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_l_b();".
// 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_900_3_l_b(:prec:=4)
    chi := MakeCharacter_900_l();
    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_900_3_l_b();".
// 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_900_3_l_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![1444, 760, 200, 20, 1]>,<11,R![-134, -8, 1]>],Snew);
    return Vf;
end function;