
// Make newform 1620.3.o.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1620_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1620_o_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_1620_3_o_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_1620_3_o_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 := [25, 0, -5, 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, 5, 5];
    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_1620_o();"
function MakeCharacter_1620_o()
    N := 1620;
    order := 6;
    char_gens := [811, 1541, 1297];
    v := [6, 5, 6];
    // 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_1620_o_Hecke();"
function MakeCharacter_1620_o_Hecke(Kf)
    N := 1620;
    order := 6;
    char_gens := [811, 1541, 1297];
    char_values := [[1, 0, 0, 0], [0, 0, 1, 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, 0, 0, 0], [0, 0, 0, 0], [0, -1, 0, 1], [-2, 0, 2, 0], [0, -6, 0, 0], [0, 0, -8, 0], [0, 0, 0, 6], [-34, 0, 0, 0], [0, 18, 0, -18], [0, 18, 0, 0], [0, 0, -14, 0], [56, 0, 0, 0], [0, -12, 0, 12], [-8, 0, 8, 0], [0, 18, 0, 0], [0, 0, 0, -18], [0, 6, 0, -6], [46, 0, -46, 0], [0, 0, -32, 0], [0, 0, 0, 24], [-106, 0, 0, 0], [22, 0, -22, 0], [0, 54, 0, 0], [0, 0, 0, 48], [-122, 0, 122, 0], [0, 78, 0, 0], [0, 0, 46, 0], [0, 0, 0, 6], [86, 0, 0, 0], [0, 42, 0, -42], [-34, 0, 0, 0], [0, 66, 0, -66], [0, 18, 0, 0], [0, 0, 82, 0], [0, 66, 0, -66], [46, 0, -46, 0], [0, 0, -92, 0], [68, 0, 0, 0], [0, 30, 0, -30], [0, 54, 0, 0], [0, 0, 0, -126], [194, 0, 0, 0], [0, 36, 0, 0], [0, 0, -218, 0], [0, 0, 0, 42], [-34, 0, 0, 0], [0, 0, 46, 0], [-398, 0, 398, 0], [0, 150, 0, 0], [0, 0, -86, 0], [0, 0, 0, 138], [0, 0, 0, 0], [-134, 0, 134, 0], [0, 0, 0, -138], [0, 78, 0, -78], [0, -114, 0, 0], [0, 0, 0, 150], [2, 0, 0, 0], [448, 0, -448, 0], [0, -84, 0, 0], [0, 0, -248, 0], [0, -66, 0, 66], [272, 0, 0, 0], [0, 24, 0, -24], [-278, 0, 278, 0], [0, -54, 0, 0], [598, 0, -598, 0], [0, 0, 358, 0], [0, 114, 0, -114], [-518, 0, 518, 0], [0, 126, 0, 0], [0, 0, 0, 156], [178, 0, -178, 0], [0, 0, 532, 0], [86, 0, 0, 0], [0, -54, 0, 54], [0, -186, 0, 0], [-28, 0, 0, 0], [0, -120, 0, 120], [0, 0, 214, 0], [0, -78, 0, 78], [238, 0, -238, 0], [0, 0, 0, -108], [-382, 0, 0, 0], [274, 0, -274, 0], [0, -342, 0, 0], [0, 0, 0, -348], [-446, 0, 446, 0], [0, 42, 0, 0], [0, 0, -854, 0], [0, 0, 0, 6], [0, -24, 0, 0], [2, 0, 0, 0], [0, -210, 0, 210], [0, 0, 514, 0], [0, 0, 0, -294], [0, -138, 0, 138], [0, 0, 0, 300], [-832, 0, 0, 0], [314, 0, 0, 0], [-536, 0, 536, 0], [0, 0, 0, 402], [0, 366, 0, -366], [0, 0, 0, 0], [0, 0, 526, 0], [122, 0, 0, 0], [0, -102, 0, 0], [0, 0, 0, 330], [0, -36, 0, 36], [766, 0, -766, 0], [0, 0, -206, 0], [-556, 0, 0, 0], [0, 174, 0, -174], [514, 0, -514, 0], [1094, 0, 0, 0], [0, -72, 0, 0], [0, 0, -404, 0], [0, 0, 0, -522], [0, -342, 0, 342], [0, -354, 0, 0], [0, 0, 118, 0], [-194, 0, 194, 0], [0, 186, 0, 0], [0, 0, 0, 42], [-122, 0, 122, 0], [0, 0, 0, -258], [-1166, 0, 1166, 0], [0, 0, 0, -96], [1318, 0, -1318, 0], [0, 0, 136, 0], [-682, 0, 0, 0], [0, -390, 0, 390], [0, 0, 658, 0], [1112, 0, 0, 0], [0, -36, 0, 36], [0, 0, -206, 0], [0, 0, 0, -486], [0, 0, 1168, 0], [0, 534, 0, -534], [0, 0, 0, 408], [674, 0, 0, 0], [0, 606, 0, 0], [0, 0, -98, 0], [0, 0, 0, -366], [398, 0, 0, 0], [0, 444, 0, 0], [-668, 0, 668, 0], [0, 138, 0, 0], [0, 0, -278, 0], [0, 0, 0, 162], [0, 0, -572, 0], [0, 0, 0, -360], [-1132, 0, 0, 0], [0, -546, 0, 546], [-716, 0, 716, 0], [0, 564, 0, 0], [866, 0, 0, 0], [0, -132, 0, 0], [-1018, 0, 0, 0], [0, 534, 0, -534], [0, -330, 0, 0], [0, 0, 0, 690], [0, 0, 1414, 0], [0, 0, 0, 150], [0, -18, 0, 18], [0, -210, 0, 0], [914, 0, 0, 0], [1264, 0, -1264, 0], [1346, 0, 0, 0], [0, 858, 0, -858], [0, 546, 0, 0], [0, 0, 1846, 0], [0, -288, 0, 288], [-758, 0, 758, 0], [0, 0, -26, 0], [0, 288, 0, -288], [646, 0, -646, 0], [0, 0, 0, 930], [578, 0, 0, 0], [154, 0, -154, 0], [-422, 0, 422, 0], [0, -114, 0, 0], [0, 0, -908, 0], [0, 0, 0, -90], [0, 42, 0, -42], [0, 822, 0, 0], [-1468, 0, 0, 0], [172, 0, -172, 0], [0, 0, -206, 0], [0, 0, 0, -252], [-142, 0, 0, 0], [0, 474, 0, 0], [-1126, 0, 0, 0], [0, -6, 0, 0], [0, 0, 0, -186], [0, -486, 0, 486], [0, 0, 1018, 0], [-1184, 0, 1184, 0], [0, 810, 0, 0], [0, 0, 0, 450], [0, 126, 0, -126], [1858, 0, -1858, 0], [0, 0, 784, 0], [-1058, 0, 1058, 0], [0, 0, 0, 942], [0, 0, 0, -30], [-214, 0, 0, 0], [0, 702, 0, -702], [0, -192, 0, 0], [0, 0, 1006, 0], [842, 0, 0, 0], [0, -942, 0, 942], [1582, 0, -1582, 0], [0, -354, 0, 0], [0, 228, 0, -228], [-302, 0, 302, 0], [0, 0, 1258, 0], [0, -264, 0, 0], [0, 0, 0, 678], [0, 486, 0, -486], [0, 0, -674, 0], [0, 0, -1646, 0], [0, -228, 0, 228], [254, 0, 0, 0], [0, 54, 0, -54], [1354, 0, -1354, 0], [0, 318, 0, 0], [0, 0, 0, 1152], [118, 0, -118, 0], [0, 1002, 0, 0], [0, 0, -2048, 0], [-2122, 0, 0, 0], [0, 0, 1726, 0], [0, 1020, 0, -1020], [-824, 0, 824, 0], [0, -918, 0, 0], [0, 0, 2062, 0], [0, 0, 0, 222], [0, 462, 0, -462], [0, 0, 0, 432], [0, -654, 0, 0], [314, 0, 0, 0], [0, 0, -218, 0], [-1234, 0, 0, 0], [0, 714, 0, -714], [0, 732, 0, 0], [1982, 0, 0, 0], [0, 114, 0, -114], [0, 0, 1354, 0], [0, 0, 0, 498], [0, 0, -2276, 0], [0, 0, 0, -408], [0, -954, 0, 954], [-506, 0, 506, 0], [0, -870, 0, 0], [0, 0, 0, -822], [-886, 0, 0, 0], [-2516, 0, 2516, 0], [0, 0, 0, -774], [-1474, 0, 0, 0], [202, 0, -202, 0], [0, 1110, 0, 0], [0, 0, 1522, 0], [3248, 0, 0, 0], [0, -306, 0, 306], [922, 0, -922, 0], [0, 0, 0, -522], [0, -132, 0, 0], [0, 0, -1664, 0], [0, -870, 0, 870], [0, 0, -1274, 0], [-1384, 0, 0, 0], [-374, 0, 374, 0], [0, 0, -3146, 0], [0, 0, -2642, 0], [1418, 0, 0, 0], [0, 1218, 0, -1218], [2194, 0, -2194, 0], [-2386, 0, 0, 0], [0, -498, 0, 0], [0, -1110, 0, 1110], [0, 0, -1682, 0], [0, -174, 0, 0], [-794, 0, 794, 0], [0, 0, 2308, 0], [0, 0, 0, 876], [1658, 0, 0, 0], [0, 822, 0, -822], [1282, 0, -1282, 0], [0, 0, 0, 1044], [0, -846, 0, 0], [0, 0, 0, -1386], [0, -1470, 0, 1470], [0, -354, 0, 354], [2272, 0, -2272, 0], [0, 174, 0, -174], [706, 0, -706, 0], [0, -570, 0, 0], [0, 0, 0, 390], [-2336, 0, 2336, 0], [0, 0, 3466, 0], [0, 0, 0, 150], [1166, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1620_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1620_3_o_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_1620_3_o_b(:prec:=4)
    chi := MakeCharacter_1620_o();
    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_1620_3_o_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_1620_3_o_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1620_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![4, 2, 1]>],Snew);
    return Vf;
end function;