
// Make newform 3600.3.e.x in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3600_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3600_e_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_3600_3_e_x();".
// 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_3600_3_e_x();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-4, 8]];
    Rf_basisdens := [1, 1];
    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_3600_e();"
function MakeCharacter_3600_e()
    N := 3600;
    order := 2;
    char_gens := [3151, 901, 2801, 577];
    v := [1, 2, 2, 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_3600_e_Hecke();"
function MakeCharacter_3600_e_Hecke( : Kf := false)
    N := 3600;
    order := 2;
    char_gens := [3151, 901, 2801, 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], [0, -1], [0, -2], [24, 0], [24, 0], [0, 4], [0, -5], [10, 0], [0, -2], [24, 0], [34, 0], [0, 3], [0, 1], [-48, 0], [0, 2], [70, 0], [0, 13], [0, -8], [48, 0], [0, 6], [0, 13], [-14, 0], [96, 0], [38, 0], [0, 21], [0, -1], [-166, 0], [-168, 0], [0, -5], [0, -10], [-24, 0], [0, -28], [86, 0], [0, 10], [-120, 0], [0, -9], [0, -35], [-192, 0], [0, -46], [122, 0], [0, 28], [144, 0], [-48, 0], [0, -42], [0, 28], [0, -19], [0, 39], [166, 0], [264, 0], [0, -12], [-242, 0], [0, -30], [-72, 0], [0, 39], [230, 0], [0, 30], [264, 0], [-254, 0], [0, 31], [432, 0], [0, -43], [0, -76], [-240, 0], [240, 0], [0, 16], [-48, 0], [0, 99], [-458, 0], [72, 0], [0, -92], [0, -21], [-24, 0], [0, -48], [0, 27], [182, 0], [-264, 0], [350, 0], [-190, 0], [0, 114], [502, 0], [0, -28], [-528, 0], [0, -26], [0, -103], [-82, 0], [-384, 0], [278, 0], [0, 33], [0, 63], [0, -56], [0, 39], [0, -82], [0, 112], [0, -37], [-374, 0], [-82, 0], [0, -73], [842, 0], [0, 65], [-864, 0], [0, 89], [830, 0], [0, 8], [96, 0], [0, -109], [-456, 0], [0, 44], [-1154, 0], [0, -17], [-456, 0], [-600, 0], [0, 36], [0, -2], [-1070, 0], [0, -133], [0, 49], [-1056, 0], [0, -138], [-554, 0], [-624, 0], [-864, 0], [0, 21], [0, 108], [634, 0], [358, 0], [0, 68], [0, 103], [840, 0], [0, -88], [0, -113], [0, 126], [-504, 0], [-754, 0], [-578, 0], [-1296, 0], [0, -195], [-288, 0], [-494, 0], [0, -104], [298, 0], [0, 125], [0, -53], [-454, 0], [0, 28], [1560, 0], [1272, 0], [0, -152], [0, 71], [936, 0], [562, 0], [0, -1], [0, 81], [0, 241], [0, 248], [0, 86], [-1042, 0], [864, 0], [-490, 0], [0, 163], [360, 0], [0, -61], [0, 178], [-1128, 0], [0, -41], [0, -158], [-24, 0], [-1058, 0], [288, 0], [0, 58], [650, 0], [0, -56], [-1488, 0], [0, 78], [-1742, 0], [0, 108], [-154, 0], [0, 37], [602, 0], [0, -141], [0, -78], [-1848, 0], [-1224, 0], [0, 215], [442, 0], [-168, 0], [0, -5], [-286, 0], [0, -12], [1776, 0], [0, 233], [0, -252], [2170, 0], [0, 259], [-1416, 0], [142, 0], [1080, 0], [-936, 0], [0, -29], [-1238, 0], [0, 98], [-1512, 0], [-962, 0], [0, 150], [1104, 0], [0, -206], [0, 205], [-110, 0], [0, 36], [-720, 0], [-86, 0], [0, 137], [0, -93], [0, -176], [-1826, 0], [0, 179], [-1186, 0], [0, 109], [2304, 0], [1078, 0], [0, 158], [898, 0], [0, -371], [0, -77], [-214, 0], [1176, 0], [0, -180], [0, -269], [0, -174], [-360, 0], [0, -232], [0, -282], [-2462, 0], [0, -301], [0, 293], [-718, 0], [384, 0], [0, 174], [0, -16], [0, 373], [0, 24], [0, 185], [-1322, 0], [1416, 0], [0, 224], [0, -277], [0, 426], [0, -104], [0, 247], [1224, 0], [-1858, 0], [0, 297], [674, 0], [2448, 0], [0, 174], [314, 0], [0, 229], [1392, 0], [3120, 0], [0, -295], [0, -209], [1894, 0], [1944, 0], [2760, 0], [0, -232], [106, 0], [-2674, 0], [0, -73], [1152, 0], [2858, 0], [0, 113], [2640, 0], [0, -10], [1968, 0], [0, 449], [0, 271], [-650, 0], [1630, 0], [0, 294], [0, 255], [0, 174], [0, -291], [-1510, 0], [0, 437], [0, 4], [1152, 0], [-480, 0], [0, -62], [1646, 0], [2842, 0], [0, 123], [-1944, 0], [0, 10], [-264, 0], [2630, 0], [0, 562], [-1680, 0], [0, -418], [0, 81], [1680, 0], [-2400, 0], [0, -118], [0, -75], [0, -236], [-1632, 0], [0, -25], [-362, 0], [0, -280], [1416, 0], [0, -381], [-518, 0], [254, 0], [0, -329], [0, 261], [1390, 0], [0, 154], [0, 304], [3120, 0], [2318, 0], [0, 120], [-432, 0], [-3590, 0], [0, -39], [-456, 0], [1102, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3600_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3600_3_e_x();".
// 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_3600_3_e_x(:prec:=2)
    chi := MakeCharacter_3600_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2161) - 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_3600_3_e_x();".
// 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_3600_3_e_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3600_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![48, 0, 1]>,<11,R![192, 0, 1]>,<13,R![-24, 1]>,<17,R![-24, 1]>],Snew);
    return Vf;
end function;