
// Make newform 2736.3.o.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_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_2736_3_o_m();".
// 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_2736_3_o_m();".
// 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 := [39, -93, 112, -39, 22, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-15, 16, 43, -32, 5, -2], [-7, -8, 9, -2, 1, 0], [-17, 12, -13, 2, -1, 0], [-168, 392, -172, 128, -20, 8], [-510, 1114, -286, 204, -20, 8]];
    Rf_basisdens := [1, 19, 2, 2, 19, 19];
    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_2736_o();"
function MakeCharacter_2736_o()
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    v := [2, 2, 2, 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_2736_o_Hecke();"
function MakeCharacter_2736_o_Hecke(Kf)
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 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, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [-4, 0, -1, 0, 0, 0], [0, -1, 0, 0, 1, 0], [8, 0, -1, 2, 0, 0], [1, 0, 0, 2, -1, 1], [6, 0, -3, -1, 0, 0], [0, 2, 0, 0, 2, 0], [0, 1, 0, 0, 1, 2], [0, 3, 0, 0, 1, 0], [0, 0, 0, 0, 3, -4], [36, 0, -1, -2, 0, 0], [4, 0, 4, -5, 0, 0], [0, -2, 0, 0, 1, -2], [0, -4, 0, 0, 5, 2], [32, 0, 7, 4, 0, 0], [0, 2, 0, 0, 0, 2], [0, 4, 0, 0, 1, 2], [12, 0, 1, 14, 0, 0], [0, -3, 0, 0, 8, -4], [-66, 0, 0, -4, 0, 0], [0, 0, 0, 0, 3, -2], [0, 0, 0, 0, 7, -2], [30, 0, -15, 7, 0, 0], [0, -1, 0, 0, 6, -4], [0, -10, 0, 0, -4, -4], [0, 3, 0, 0, 12, -2], [0, 2, 0, 0, -4, -6], [0, 5, 0, 0, -10, 4], [28, 0, -17, 16, 0, 0], [8, 0, 17, 8, 0, 0], [32, 0, 15, 10, 0, 0], [-8, 0, 16, 9, 0, 0], [0, 3, 0, 0, 11, -2], [-110, 0, -18, 26, 0, 0], [14, 0, 12, 28, 0, 0], [0, -2, 0, 0, 34, -8], [0, 0, 0, 0, -24, 8], [0, -6, 0, 0, -6, -6], [0, -1, 0, 0, -6, -2], [-4, 0, 14, 33, 0, 0], [0, 0, 0, 0, -26, -8], [102, 0, 27, -11, 0, 0], [120, 0, -13, -26, 0, 0], [0, -8, 0, 0, -27, -8], [0, 9, 0, 0, -9, -14], [0, 0, 0, 0, 3, -16], [168, 0, -25, -12, 0, 0], [-56, 0, -23, 24, 0, 0], [-4, 0, 32, 39, 0, 0], [0, 22, 0, 0, -3, 10], [-12, 0, -15, 14, 0, 0], [0, 2, 0, 0, 32, 0], [84, 0, 0, 23, 0, 0], [0, 6, 0, 0, 27, 6], [-122, 0, -24, -48, 0, 0], [-48, 0, -1, -36, 0, 0], [0, -2, 0, 0, -38, -10], [104, 0, -17, 26, 0, 0], [0, 12, 0, 0, 6, 16], [0, 18, 0, 0, 42, 10], [-156, 0, 42, 5, 0, 0], [-74, 0, -24, 20, 0, 0], [0, 10, 0, 0, -23, -14], [0, -2, 0, 0, 68, -2], [0, -4, 0, 0, 43, -22], [132, 0, -57, 8, 0, 0], [120, 0, -11, -34, 0, 0], [-354, 0, 54, 10, 0, 0], [444, 0, 12, -13, 0, 0], [-50, 0, -12, 60, 0, 0], [0, 13, 0, 0, -58, 2], [0, -4, 0, 0, -14, 22], [0, 0, 0, 0, -75, 2], [-264, 0, 18, 31, 0, 0], [56, 0, -63, -6, 0, 0], [0, 2, 0, 0, -4, -18], [0, -36, 0, 0, -5, -22], [-402, 0, -42, -58, 0, 0], [0, -3, 0, 0, -25, 24], [0, -26, 0, 0, 13, 10], [0, 2, 0, 0, 80, -12], [0, 11, 0, 0, -40, -16], [-564, 0, 3, -28, 0, 0], [0, 2, 0, 0, 98, -8], [8, 0, 21, -42, 0, 0], [-544, 0, -22, 23, 0, 0], [-352, 0, 19, -34, 0, 0], [100, 0, 1, -26, 0, 0], [-234, 0, -21, 17, 0, 0], [0, 1, 0, 0, -27, 38], [102, 0, 78, -58, 0, 0], [-68, 0, -13, 26, 0, 0], [222, 0, -51, 23, 0, 0], [0, -32, 0, 0, -35, -6], [0, -34, 0, 0, 32, -32], [0, 0, 0, 0, -32, 26], [-232, 0, 129, -6, 0, 0], [0, 34, 0, 0, -37, -28], [280, 0, 10, 15, 0, 0], [0, -32, 0, 0, 49, -14], [0, -8, 0, 0, -41, -30], [154, 0, -48, -84, 0, 0], [264, 0, 17, -6, 0, 0], [468, 0, -3, -94, 0, 0], [366, 0, -138, 10, 0, 0], [0, -46, 0, 0, 32, -4], [0, -20, 0, 0, -59, 6], [0, 23, 0, 0, -5, -18], [-472, 0, 31, -44, 0, 0], [-704, 0, 7, 18, 0, 0], [26, 0, 12, 64, 0, 0], [-584, 0, -7, -40, 0, 0], [0, 16, 0, 0, -5, -18], [-216, 0, 119, 34, 0, 0], [-364, 0, -46, -51, 0, 0], [400, 0, 22, -165, 0, 0], [0, 50, 0, 0, -4, -34], [0, 33, 0, 0, -22, 2], [0, 16, 0, 0, 9, 38], [0, 32, 0, 0, 26, 20], [0, -44, 0, 0, -5, 0], [-332, 0, -57, 30, 0, 0], [-594, 0, -57, -47, 0, 0], [98, 0, -12, -12, 0, 0], [-468, 0, 60, 11, 0, 0], [64, 0, 51, 150, 0, 0], [170, 0, 36, 108, 0, 0], [416, 0, 87, -182, 0, 0], [0, -4, 0, 0, 68, -32], [0, 41, 0, 0, -86, 0], [-56, 0, 27, -112, 0, 0], [24, 0, 9, 88, 0, 0], [-132, 0, -67, -102, 0, 0], [0, -8, 0, 0, -5, 66], [0, 44, 0, 0, 30, 6], [0, 34, 0, 0, -74, 76], [384, 0, 99, 94, 0, 0], [0, -42, 0, 0, -69, 20], [520, 0, -2, -141, 0, 0], [224, 0, -63, 136, 0, 0], [0, 22, 0, 0, 34, -10], [0, -23, 0, 0, -100, 30], [0, -24, 0, 0, 6, -24], [-574, 0, -36, -84, 0, 0], [0, -52, 0, 0, 65, -26], [-380, 0, -97, 134, 0, 0], [0, -14, 0, 0, -29, -42], [0, 45, 0, 0, 86, -2], [1400, 0, 5, -4, 0, 0], [-300, 0, -89, 54, 0, 0], [0, -36, 0, 0, 33, 54], [0, -18, 0, 0, 55, 24], [0, -18, 0, 0, -108, 28], [110, 0, 6, -134, 0, 0], [-138, 0, -54, 94, 0, 0], [404, 0, -63, 30, 0, 0], [0, -70, 0, 0, -4, -32], [318, 0, 30, 14, 0, 0], [0, 30, 0, 0, -162, 38], [134, 0, -90, -158, 0, 0], [0, -2, 0, 0, -152, 34], [0, 56, 0, 0, -103, -52], [0, 28, 0, 0, -38, -40], [0, 9, 0, 0, -55, 42], [-704, 0, -21, -112, 0, 0], [0, 14, 0, 0, -132, 76], [1000, 0, 16, -87, 0, 0], [0, 72, 0, 0, 27, 72], [0, 47, 0, 0, 47, 24], [54, 0, -3, 143, 0, 0], [886, 0, -72, -28, 0, 0], [0, 1, 0, 0, 15, -34], [304, 0, 139, 150, 0, 0], [-418, 0, 24, -80, 0, 0], [-88, 0, 146, -1, 0, 0], [0, 85, 0, 0, 104, 28], [-272, 0, 5, -2, 0, 0], [-896, 0, 51, -90, 0, 0], [0, -20, 0, 0, -35, 2], [0, 25, 0, 0, 16, 66], [0, 24, 0, 0, 123, 12], [172, 0, 46, -239, 0, 0], [952, 0, 70, 123, 0, 0], [0, -23, 0, 0, -166, 18], [0, 76, 0, 0, -15, -40], [0, 32, 0, 0, 132, -40], [-210, 0, 51, -175, 0, 0], [0, -4, 0, 0, -98, 0], [-924, 0, -45, 164, 0, 0], [0, -16, 0, 0, -219, 4], [0, -40, 0, 0, -115, -46], [342, 0, 156, -136, 0, 0], [0, 80, 0, 0, 35, 12], [76, 0, -151, 190, 0, 0], [-142, 0, -72, 0, 0, 0], [510, 0, 84, -116, 0, 0], [-996, 0, -126, 5, 0, 0], [0, 20, 0, 0, 236, 28], [0, 27, 0, 0, 67, 38], [0, 9, 0, 0, -64, -62], [0, 30, 0, 0, -100, -52], [150, 0, -30, -94, 0, 0], [-56, 0, 22, 3, 0, 0], [-1512, 0, 47, 82, 0, 0], [0, 38, 0, 0, 62, -62], [-168, 0, 93, 28, 0, 0], [0, 16, 0, 0, 78, 30], [-254, 0, 216, 56, 0, 0], [-930, 0, 63, 217, 0, 0], [336, 0, 107, 70, 0, 0], [0, 18, 0, 0, -192, 96], [0, 32, 0, 0, -4, 4], [0, -24, 0, 0, -77, -82], [-530, 0, 180, -84, 0, 0], [0, 46, 0, 0, -86, -14], [0, 50, 0, 0, -151, 6], [-480, 0, -72, -287, 0, 0], [0, -33, 0, 0, -67, -84], [0, 77, 0, 0, -109, -58], [0, -58, 0, 0, -202, -48], [-56, 0, 113, 56, 0, 0], [0, 108, 0, 0, -15, -10], [72, 0, 215, 60, 0, 0], [0, -68, 0, 0, -155, -26], [0, -12, 0, 0, -198, 56], [0, -121, 0, 0, -5, 26], [-404, 0, 37, -206, 0, 0], [-880, 0, 7, 124, 0, 0], [0, -30, 0, 0, 75, -4], [0, 71, 0, 0, 77, -30], [0, 80, 0, 0, 17, -90], [-192, 0, -29, 30, 0, 0], [1268, 0, -70, -147, 0, 0], [588, 0, 193, 38, 0, 0], [-872, 0, -74, -213, 0, 0], [606, 0, -12, 200, 0, 0], [0, -62, 0, 0, 118, -92], [0, 70, 0, 0, 34, -12], [-324, 0, 263, 166, 0, 0], [-842, 0, 216, 0, 0, 0], [0, 59, 0, 0, -47, 44], [0, -110, 0, 0, -14, 24], [28, 0, 118, 337, 0, 0], [392, 0, -87, -296, 0, 0], [0, 2, 0, 0, -4, -88], [0, 34, 0, 0, 202, -46], [-604, 0, -64, 303, 0, 0], [490, 0, 258, -82, 0, 0], [-1704, 0, -39, 184, 0, 0], [414, 0, -165, 281, 0, 0], [0, 0, 0, 0, 112, -80], [-112, 0, -94, -145, 0, 0], [-1458, 0, 24, 260, 0, 0], [1072, 0, -31, 34, 0, 0], [0, -6, 0, 0, -156, 34], [0, -92, 0, 0, -203, -22], [-914, 0, 180, -112, 0, 0], [0, 13, 0, 0, -139, 78], [0, -76, 0, 0, -7, -30], [-200, 0, -103, 82, 0, 0], [0, -49, 0, 0, -92, 90], [368, 0, -91, -148, 0, 0], [0, 12, 0, 0, 81, 36], [0, 86, 0, 0, -163, -38], [1544, 0, 95, 194, 0, 0], [0, 42, 0, 0, 33, -32], [-568, 0, 242, 191, 0, 0], [0, 57, 0, 0, -103, -28], [0, -48, 0, 0, 125, 48], [332, 0, 229, 106, 0, 0], [728, 0, -315, 172, 0, 0], [0, -72, 0, 0, -156, -96], [734, 0, -222, 190, 0, 0], [-396, 0, -99, -310, 0, 0], [0, 33, 0, 0, 231, -16], [0, -24, 0, 0, -45, 78], [988, 0, -407, 46, 0, 0], [0, -24, 0, 0, 27, -82], [-154, 0, 294, -62, 0, 0], [-748, 0, 182, 201, 0, 0], [0, -157, 0, 0, -27, -128], [472, 0, 207, -318, 0, 0], [-1340, 0, -44, -293, 0, 0], [952, 0, 117, -418, 0, 0], [0, -88, 0, 0, 137, -50], [1160, 0, 21, 4, 0, 0], [0, -32, 0, 0, 25, 16], [1326, 0, 9, 31, 0, 0], [-980, 0, -53, -44, 0, 0], [0, -20, 0, 0, -167, -44], [0, -40, 0, 0, 215, 52], [0, 35, 0, 0, 221, -116], [-474, 0, 3, -323, 0, 0], [0, 31, 0, 0, -112, 60], [232, 0, -374, -105, 0, 0], [0, 20, 0, 0, -181, -2], [112, 0, -277, 206, 0, 0], [360, 0, 29, 78, 0, 0], [0, 62, 0, 0, -94, -16], [-232, 0, -233, -310, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_3_o_m();".
// 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_2736_3_o_m(:prec:=6)
    chi := MakeCharacter_2736_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_2736_3_o_m();".
// 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_2736_3_o_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-24, -44, -1, 1]>,<7,R![-164, -64, -1, 1]>],Snew);
    return Vf;
end function;