
// Make newform 3600.3.e.ba in Magma, downloaded from the LMFDB on 19 December 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_ba();".
// 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_ba();".
// 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, 2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-2, -12, 4, -4], [0, -20, 20, -10], [-12, 0, 0, -6]];
    Rf_basisdens := [1, 1, 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, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 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], [0, -1, 0, 0], [0, 0, 1, 0], [-4, 0, 0, 0], [0, 0, 0, -1], [0, -4, 0, 0], [0, 0, 2, 0], [0, 0, 0, 3], [0, 2, 0, 0], [16, 0, 0, 0], [0, 0, 0, 4], [0, -8, 0, 0], [0, 0, -2, 0], [0, 0, 0, -7], [0, 0, 1, 0], [-58, 0, 0, 0], [0, 6, 0, 0], [0, 0, -2, 0], [-94, 0, 0, 0], [0, -2, 0, 0], [0, 0, -6, 0], [0, 0, 0, 2], [14, 0, 0, 0], [0, 0, 0, 7], [0, 7, 0, 0], [0, 0, 10, 0], [22, 0, 0, 0], [0, 0, 0, 1], [0, -21, 0, 0], [0, 0, -5, 0], [0, 0, 0, 13], [0, 8, 0, 0], [0, 0, 0, 9], [0, -28, 0, 0], [196, 0, 0, 0], [0, 30, 0, 0], [0, 0, -14, 0], [0, 0, 0, 9], [0, 0, 13, 0], [-242, 0, 0, 0], [0, 0, 6, 0], [-326, 0, 0, 0], [0, 0, 0, 3], [0, 48, 0, 0], [0, 36, 0, 0], [0, -17, 0, 0], [0, 0, -4, 0], [82, 0, 0, 0], [0, 0, 0, 27], [0, 0, -10, 0], [-362, 0, 0, 0], [0, 0, 19, 0], [0, 0, 0, 17], [0, 0, 14, 0], [0, 0, 0, -9], [0, -44, 0, 0], [224, 0, 0, 0], [0, 0, 0, -34], [0, -10, 0, 0], [0, 0, 0, -29], [0, -52, 0, 0], [0, 0, -2, 0], [-446, 0, 0, 0], [0, 0, 0, 15], [0, 44, 0, 0], [-46, 0, 0, 0], [0, 0, -24, 0], [-82, 0, 0, 0], [0, 0, 0, 35], [0, 0, 32, 0], [0, -33, 0, 0], [-596, 0, 0, 0], [0, -48, 0, 0], [0, 0, -38, 0], [0, 0, 0, 21], [736, 0, 0, 0], [0, 0, 0, 26], [-158, 0, 0, 0], [0, 0, -45, 0], [-302, 0, 0, 0], [0, 0, -28, 0], [-326, 0, 0, 0], [0, 44, 0, 0], [0, 0, 8, 0], [0, 0, 0, 2], [-466, 0, 0, 0], [0, 0, 0, -7], [0, 13, 0, 0], [0, 0, 2, 0], [0, 0, 30, 0], [0, 45, 0, 0], [0, 0, 39, 0], [0, 100, 0, 0], [0, 0, 30, 0], [0, 0, 0, 47], [0, 0, 0, 54], [0, -74, 0, 0], [-2, 0, 0, 0], [0, -8, 0, 0], [0, 0, 0, 21], [0, 0, 28, 0], [0, 0, 0, -2], [0, -52, 0, 0], [-74, 0, 0, 0], [0, 0, -30, 0], [0, 0, 0, -11], [0, 0, 0, 0], [-418, 0, 0, 0], [0, 1, 0, 0], [-56, 0, 0, 0], [0, 0, 0, -73], [0, -16, 0, 0], [0, -54, 0, 0], [0, 0, 0, -60], [0, 66, 0, 0], [0, 0, -38, 0], [0, 0, 0, 39], [0, 0, 39, 0], [62, 0, 0, 0], [994, 0, 0, 0], [0, 0, 0, 15], [0, 0, 22, 0], [0, -48, 0, 0], [0, 0, 0, -101], [562, 0, 0, 0], [0, 0, -76, 0], [0, 61, 0, 0], [-184, 0, 0, 0], [0, 96, 0, 0], [0, 0, -34, 0], [0, 94, 0, 0], [704, 0, 0, 0], [0, 0, 0, 74], [-802, 0, 0, 0], [0, 0, 0, 47], [0, 130, 0, 0], [0, 0, 0, 97], [0, 0, 0, 28], [0, 40, 0, 0], [0, 0, 0, -85], [0, -183, 0, 0], [0, 0, 20, 0], [1582, 0, 0, 0], [0, 0, -76, 0], [1384, 0, 0, 0], [0, 0, 0, -5], [0, -88, 0, 0], [0, 0, -30, 0], [1036, 0, 0, 0], [0, 0, 0, -30], [0, 98, 0, 0], [0, 0, 38, 0], [0, -148, 0, 0], [0, 0, -4, 0], [0, -62, 0, 0], [0, 0, 0, -78], [-434, 0, 0, 0], [0, 0, 0, -43], [0, 0, 72, 0], [0, 0, 0, -79], [0, -241, 0, 0], [0, 0, -41, 0], [0, 0, 0, -57], [0, 0, 70, 0], [0, 64, 0, 0], [-164, 0, 0, 0], [502, 0, 0, 0], [0, 0, 0, -23], [0, 0, -39, 0], [722, 0, 0, 0], [0, 0, -28, 0], [1706, 0, 0, 0], [0, -138, 0, 0], [0, 0, 0, 98], [0, 16, 0, 0], [0, 0, 0, 37], [0, 249, 0, 0], [-382, 0, 0, 0], [0, 81, 0, 0], [0, 0, 71, 0], [1016, 0, 0, 0], [0, 0, 0, -139], [0, 0, 54, 0], [0, 0, 0, 145], [1244, 0, 0, 0], [0, -186, 0, 0], [-98, 0, 0, 0], [0, 0, 98, 0], [1586, 0, 0, 0], [0, 0, -32, 0], [0, 220, 0, 0], [0, 0, 0, 89], [0, 0, -82, 0], [0, 0, 0, -143], [962, 0, 0, 0], [716, 0, 0, 0], [0, 0, 0, 53], [0, 0, 18, 0], [0, 0, 0, 1], [0, -206, 0, 0], [404, 0, 0, 0], [-1778, 0, 0, 0], [0, 0, -79, 0], [0, 0, 0, -61], [0, 254, 0, 0], [0, 0, 28, 0], [0, 0, 0, 34], [0, -156, 0, 0], [-254, 0, 0, 0], [0, 0, 0, 25], [0, 123, 0, 0], [0, 0, 28, 0], [0, 0, -38, 0], [-118, 0, 0, 0], [0, -113, 0, 0], [0, 0, 0, 66], [0, 0, 130, 0], [0, 0, 0, 21], [2458, 0, 0, 0], [0, -20, 0, 0], [0, 0, 0, -200], [0, -3, 0, 0], [0, 0, -154, 0], [-502, 0, 0, 0], [0, 0, 0, -127], [0, 0, 28, 0], [0, -147, 0, 0], [0, 0, -123, 0], [-964, 0, 0, 0], [0, -216, 0, 0], [0, -116, 0, 0], [0, 0, 0, 2], [0, 34, 0, 0], [0, 0, -34, 0], [82, 0, 0, 0], [0, 0, 0, 23], [0, 0, -39, 0], [0, 0, -106, 0], [0, 0, 72, 0], [0, 168, 0, 0], [0, -215, 0, 0], [1358, 0, 0, 0], [0, 0, 0, 187], [0, 0, 106, 0], [0, -257, 0, 0], [0, 0, 17, 0], [0, -16, 0, 0], [0, 0, 46, 0], [1184, 0, 0, 0], [0, 0, 0, -32], [0, 0, -50, 0], [-1478, 0, 0, 0], [0, 0, 0, -135], [0, 0, 71, 0], [1678, 0, 0, 0], [0, -66, 0, 0], [0, 0, 0, 89], [1666, 0, 0, 0], [0, 107, 0, 0], [0, 0, -12, 0], [1402, 0, 0, 0], [-1496, 0, 0, 0], [0, 0, 0, 169], [0, -336, 0, 0], [0, 0, 0, -51], [0, 0, 0, -88], [0, -186, 0, 0], [0, 0, 0, 63], [-1198, 0, 0, 0], [0, 316, 0, 0], [-934, 0, 0, 0], [0, -6, 0, 0], [-1486, 0, 0, 0], [0, -253, 0, 0], [0, 0, -24, 0], [-1882, 0, 0, 0], [-2878, 0, 0, 0], [0, 0, 95, 0], [0, 0, 118, 0], [0, -72, 0, 0], [0, 0, -134, 0], [-2158, 0, 0, 0], [0, -72, 0, 0], [0, 0, 194, 0], [-1594, 0, 0, 0], [0, 0, 0, 159], [0, 200, 0, 0], [0, 0, 0, -196], [0, 0, 0, 193], [0, 0, 18, 0], [0, 0, 0, 279], [0, 0, 1, 0], [-1376, 0, 0, 0], [0, 0, 0, -59], [0, 10, 0, 0], [0, 0, 0, -159], [0, 0, 117, 0], [0, -412, 0, 0], [-2006, 0, 0, 0], [0, 0, 0, 199], [0, -184, 0, 0], [0, 0, -26, 0], [0, -436, 0, 0], [974, 0, 0, 0], [0, 0, 26, 0], [1762, 0, 0, 0], [0, 0, 2, 0], [-1076, 0, 0, 0], [0, 0, 114, 0], [0, 0, 0, -25], [0, 0, 0, -286], [0, 404, 0, 0], [0, 0, 114, 0], [2302, 0, 0, 0], [0, 0, -87, 0], [0, 0, -146, 0], [-334, 0, 0, 0], [0, 0, 0, 16], [0, 160, 0, 0], [2146, 0, 0, 0], [0, 0, 0, 111], [0, 277, 0, 0], [0, 0, 0, -61], [-1802, 0, 0, 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_ba();".
// 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_ba(:prec:=4)
    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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3600_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![60, 0, 1]>,<11,R![300, 0, 1]>,<13,R![4, 1]>,<17,R![-180, 0, 1]>],Snew);
    return Vf;
end function;