
// Make newform 900.3.l.a 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_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_900_3_l_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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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], [1, 0], [0, 1]];
    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], [7, 7], [-10, 0], [-9, 9], [1, 1], [0, -8], [-23, 23], [0, 8], [-14, 0], [-33, -33], [14, 0], [15, -15], [-39, -39], [-7, 7], [0, 56], [42, 0], [7, 7], [-98, 0], [-49, 49], [0, 96], [-63, 63], [0, 112], [-33, -33], [-26, 0], [-73, 73], [121, 121], [0, -136], [-127, 127], [7, 7], [230, 0], [-63, -63], [0, -88], [0, -168], [130, 0], [63, 63], [-65, 65], [-103, -103], [73, -73], [0, -56], [-70, 0], [142, 0], [63, -63], [-63, -63], [0, 336], [314, 0], [135, -135], [281, 281], [0, 168], [273, -273], [0, 288], [-446, 0], [150, 0], [161, 161], [-151, 151], [0, -376], [210, 0], [-129, -129], [174, 0], [-113, 113], [345, -345], [327, 327], [-2, 0], [-81, 81], [-159, -159], [-182, 0], [447, 447], [25, 25], [0, -200], [321, -321], [0, 112], [-377, -377], [-217, 217], [0, -56], [57, -57], [0, 312], [-193, -193], [30, 0], [0, 432], [0, 168], [-454, 0], [494, 0], [511, -511], [0, 176], [177, -177], [0, 608], [-609, -609], [-490, 0], [7, -7], [25, 25], [0, 128], [-249, -249], [-650, 0], [0, 632], [-471, 471], [0, 8], [366, 0], [-273, 273], [394, 0], [231, 231], [-735, -735], [609, -609], [0, -560], [938, 0], [-97, -97], [-39, -39], [-479, 479], [0, 1040], [-430, 0], [423, 423], [-249, 249], [321, 321], [0, -600], [-638, 0], [-482, 0], [-33, 33], [-7, -7], [-471, 471], [0, 328], [-742, 0], [287, -287], [577, 577], [-399, 399], [378, 0], [470, 0], [0, 1192], [0, 608], [-441, -441], [-361, 361], [0, 920], [-343, 343], [786, 0], [63, 63], [398, 0], [0, 704], [825, -825], [71, 71], [33, 33], [0, -368], [-886, 0], [614, 0], [-1001, 1001], [-615, -615], [0, -616], [0, -1424], [935, -935], [-63, -63], [0, 392], [217, -217], [831, 831], [-322, 0], [287, -287], [-903, -903], [-1081, -1081], [238, 0], [0, -1552], [0, 224], [383, 383], [-42, 0], [1337, 1337], [273, -273], [743, 743], [-266, 0], [417, 417], [-631, 631], [-14, 0], [671, 671]];
    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_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_900_3_l_a(:prec:=2)
    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_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_900_3_l_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![98, -14, 1]>,<11,R![10, 1]>],Snew);
    return Vf;
end function;