
// Make newform 1800.3.l.a in Magma, downloaded from the LMFDB on 12 August 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1800_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1800_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_1800_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_1800_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 := [2, 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_1800_l();"
function MakeCharacter_1800_l()
    N := 1800;
    order := 2;
    char_gens := [1351, 901, 1001, 577];
    v := [2, 2, 1, 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_1800_l_Hecke();"
function MakeCharacter_1800_l_Hecke( : Kf := false)
    N := 1800;
    order := 2;
    char_gens := [1351, 901, 1001, 577];
    char_values := [[1, 0], [1, 0], [-1, 0], [1, 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], [-12, 0], [0, 4], [8, 0], [0, 7], [-16, 0], [0, -28], [0, 21], [-4, 0], [-30, 0], [0, 15], [8, 0], [0, -12], [0, 35], [0, -56], [-14, 0], [88, 0], [0, 20], [80, 0], [100, 0], [0, -92], [0, -105], [112, 0], [0, -45], [44, 0], [0, 0], [136, 0], [0, -15], [20, 0], [0, -40], [0, 63], [120, 0], [0, -91], [-196, 0], [-238, 0], [168, 0], [0, 120], [0, -147], [0, 144], [-216, 0], [0, -264], [178, 0], [0, -147], [196, 0], [-56, 0], [-28, 0], [0, 60], [184, 0], [0, 215], [0, -160], [384, 0], [0, 140], [0, -65], [0, -280], [0, 59], [140, 0], [248, 0], [0, 119], [-368, 0], [0, 147], [552, 0], [0, -24], [222, 0], [0, -315], [-520, 0], [-48, 0], [0, 252], [210, 0], [0, 193], [0, -420], [-172, 0], [-2, 0], [-736, 0], [0, 0], [0, 325], [-338, 0], [0, 105], [144, 0], [0, 540], [680, 0], [0, -4], [482, 0], [-756, 0], [0, -140], [0, 55], [-464, 0], [0, -35], [-20, 0], [0, 492], [0, 476], [-388, 0], [0, 0], [96, 0], [0, -300], [0, -219], [0, -511], [-352, 0], [936, 0], [-56, 0], [0, 227], [0, 332], [0, 289], [816, 0], [-242, 0], [0, -280], [0, -145], [0, -420], [286, 0], [100, 0], [-770, 0], [0, -615], [256, 0], [196, 0], [0, -455], [-120, 0], [0, -500], [0, 93], [0, 840], [-130, 0], [-610, 0], [0, -427], [0, -700], [632, 0], [0, 21], [56, 0], [0, 436], [-372, 0], [520, 0], [1040, 0], [0, 440], [-180, 0], [-152, 0], [0, -385], [-526, 0], [0, -1005], [-1072, 0], [0, 1085], [0, 95], [544, 0], [0, -621], [868, 0], [0, 1000], [-1208, 0], [0, -756], [1282, 0], [0, -735], [-1032, 0], [0, -40], [462, 0], [0, -175], [-8, 0], [0, -632], [200, 0], [0, -56], [-884, 0], [0, 761], [-78, 0], [0, 189], [0, 408], [0, -945], [-20, 0], [0, -84], [0, 297], [0, -840], [-116, 0], [-130, 0], [1616, 0], [0, -427], [0, -336], [-114, 0], [0, 896], [32, 0], [116, 0], [0, -1225], [1856, 0], [0, -459], [308, 0], [1454, 0], [-212, 0], [0, -700], [-888, 0], [0, 1113], [0, 528], [0, -21], [-952, 0], [1872, 0], [1696, 0], [0, 984], [1472, 0], [0, -720], [304, 0], [0, -1379], [0, 168], [0, -1337], [850, 0], [-280, 0], [0, -1583], [0, 1008], [0, -445], [188, 0], [1624, 0], [-560, 0], [0, 540], [0, -843], [644, 0], [0, -828], [0, -1015], [-1736, 0], [-978, 0], [0, -1029], [1412, 0], [0, 1120], [0, -1204], [110, 0], [-1988, 0], [0, -329], [0, 0], [0, -1203], [-856, 0], [-1476, 0], [0, 201], [2380, 0], [0, 540], [1694, 0], [0, -777], [0, -1116], [-492, 0], [0, -1304], [2328, 0], [280, 0], [-316, 0], [0, 409], [1560, 0], [0, -1460], [-1360, 0], [0, 635], [0, 1316], [0, 820], [0, -228], [1496, 0], [-52, 0], [-626, 0], [0, 63], [0, -1036], [1772, 0], [0, 696], [664, 0], [0, -392], [1560, 0], [0, 609], [0, 1180], [-1184, 0], [0, 693], [0, -840], [-1816, 0], [2720, 0], [0, 195], [2450, 0], [-2692, 0], [0, -1008], [2440, 0], [-1518, 0], [0, 137], [-1048, 0], [0, 1331], [0, -1735], [-560, 0], [0, -53], [1234, 0], [1456, 0], [-640, 0], [-2084, 0], [1568, 0], [-1628, 0], [0, 2028], [1586, 0], [-446, 0], [0, -560], [0, 140], [1180, 0], [0, 92], [-1534, 0], [112, 0], [0, 1024], [2272, 0], [0, -405], [1340, 0], [0, 2015], [0, 259], [0, 2520], [0, 905], [0, -1680], [1938, 0], [0, 1165], [3404, 0], [0, -1155], [0, -684], [400, 0], [-3710, 0], [0, 2515], [-2764, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1800_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1800_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_1800_3_l_a(:prec:=2)
    chi := MakeCharacter_1800_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(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_1800_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_1800_3_l_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1800_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![12, 1]>],Snew);
    return Vf;
end function;