
// Make newform 684.3.y.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_684_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_684_y_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_684_3_y_g();".
// 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_684_3_y_g();".
// 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 := [55225, 0, -7990, 0, 921, 0, -34, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -1514105, 0, 105301, 0, -5219, 0, 36], [55225, 0, -31314, 0, 921, 0, -34, 0], [7667, 0, 0, 0, 0, 0, 1, 0], [0, -888535, 0, 58637, 0, -2149, 0, 48], [0, -3995, 0, 686, 0, -34, 0, 1], [2740570, 0, -315903, 0, 15657, 0, -343, 0], [0, 250980, 0, 58637, 0, -2149, 0, 95]];
    Rf_basisdens := [1, 649305, 216435, 2763, 129861, 2115, 649305, 129861];
    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_684_y();"
function MakeCharacter_684_y()
    N := 684;
    order := 6;
    char_gens := [343, 533, 325];
    v := [6, 6, 5];
    // 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_684_y_Hecke();"
function MakeCharacter_684_y_Hecke(Kf)
    N := 684;
    order := 6;
    char_gens := [343, 533, 325];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 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, 0], [0, 1, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [-5, 0, 5, -4, 0, 0, 2, 0], [0, -1, 0, 0, 0, 1, 0, 1], [5, 0, 2, -7, 0, 0, 8, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 2, 0, 0, 1, -1, 0, -1], [-9, 0, -18, -5, 0, 0, 10, 0], [-1, 0, -2, -6, 0, 0, 12, 0], [0, -2, 0, 0, 4, -2, 0, 2], [0, 0, 5, -11, 0, 0, 11, 0], [0, 0, 0, 0, -4, -4, 0, -4], [0, -4, 0, 0, 1, 2, 0, -1], [0, 5, 0, 0, 4, 5, 0, 2], [31, 0, 31, 0, 0, 0, 34, 0], [-19, 0, 19, -38, 0, 0, 19, 0], [0, -3, 0, 0, -6, -3, 0, -3], [0, 0, -25, -2, 0, 0, 2, 0], [-66, 0, -33, 1, 0, 0, 1, 0], [0, 17, 0, 0, -3, 0, 0, 0], [0, -4, 0, 0, -5, 2, 0, 5], [36, 0, 18, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 1, 0, -9], [-41, 0, -82, -7, 0, 0, 14, 0], [0, 1, 0, 0, -1, -2, 0, -2], [-60, 0, -30, 4, 0, 0, 4, 0], [0, 18, 0, 0, -1, -36, 0, -2], [-12, 0, 12, -64, 0, 0, 32, 0], [0, -25, 0, 0, 0, 25, 0, 7], [0, 0, 0, 0, -1, 17, 0, -1], [-11, 0, -11, 0, 0, 0, 3, 0], [0, -27, 0, 0, 0, 27, 0, -4], [44, 0, 88, 20, 0, 0, -40, 0], [0, 0, 71, 78, 0, 0, -78, 0], [29, 0, 0, -33, 0, 0, 0, 0], [0, -38, 0, 0, 0, 19, 0, 0], [0, 19, 0, 0, -6, 19, 0, -3], [0, 1, 0, 0, -2, -2, 0, -4], [82, 0, -82, 52, 0, 0, -26, 0], [0, -24, 0, 0, -7, 0, 0, 0], [94, 0, 47, -6, 0, 0, -6, 0], [0, 32, 0, 0, -3, 0, 0, 0], [49, 0, 49, 0, 0, 0, -53, 0], [110, 0, 55, -35, 0, 0, -35, 0], [-106, 0, -53, -25, 0, 0, -25, 0], [0, 20, 0, 0, 1, -40, 0, 2], [125, 0, 0, 28, 0, 0, 0, 0], [0, -9, 0, 0, 0, 9, 0, 17], [0, 48, 0, 0, -7, 0, 0, 0], [37, 0, -37, 164, 0, 0, -82, 0], [0, 0, 0, 0, 0, 40, 0, 0], [0, 4, 0, 0, -17, -2, 0, 17], [0, -17, 0, 0, 0, 17, 0, 3], [0, -32, 0, 0, -18, -32, 0, -9], [0, 0, 38, -36, 0, 0, 36, 0], [68, 0, 0, 160, 0, 0, 0, 0], [0, -6, 0, 0, -2, 3, 0, 2], [0, 0, -382, -10, 0, 0, 10, 0], [0, -30, 0, 0, 0, 60, 0, 0], [-92, 0, -46, -4, 0, 0, -4, 0], [0, 53, 0, 0, 12, 0, 0, 0], [328, 0, 328, 0, 0, 0, 0, 0], [0, 2, 0, 0, 20, -1, 0, -20], [97, 0, 194, 7, 0, 0, -14, 0], [-326, 0, -163, 12, 0, 0, 12, 0], [0, -2, 0, 0, 0, 2, 0, -39], [-109, 0, 0, 110, 0, 0, 0, 0], [0, -25, 0, 0, 18, 0, 0, 0], [0, -44, 0, 0, 0, 44, 0, -30], [217, 0, 217, 0, 0, 0, 99, 0], [-114, 0, -228, -58, 0, 0, 116, 0], [109, 0, 218, -79, 0, 0, 158, 0], [0, 14, 0, 0, -38, 14, 0, -19], [0, 0, 0, 0, 27, 28, 0, 27], [0, 0, -37, -66, 0, 0, 66, 0], [0, 30, 0, 0, -2, 30, 0, -1], [286, 0, -286, 24, 0, 0, -12, 0], [0, 14, 0, 0, -39, 0, 0, 0], [-140, 0, -70, 90, 0, 0, 90, 0], [0, -8, 0, 0, -8, 4, 0, 8], [-149, 0, 149, -268, 0, 0, 134, 0], [214, 0, 107, 97, 0, 0, 97, 0], [0, 0, 0, 0, 27, 25, 0, 27], [0, 17, 0, 0, 3, -34, 0, 6], [161, 0, 0, 24, 0, 0, 0, 0], [0, 14, 0, 0, 0, -14, 0, 1], [-475, 0, 0, -83, 0, 0, 0, 0], [0, 16, 0, 0, -50, 0, 0, 0], [0, 0, 0, 0, 15, 37, 0, 15], [-156, 0, -312, -106, 0, 0, 212, 0], [0, 43, 0, 0, 0, -43, 0, -21], [0, 0, 281, -179, 0, 0, 179, 0], [0, 0, 0, 0, -21, -71, 0, -21], [0, 108, 0, 0, -12, -54, 0, 12], [0, -15, 0, 0, -14, 30, 0, -28], [99, 0, -99, -302, 0, 0, 151, 0], [511, 0, 511, 0, 0, 0, 168, 0], [-131, 0, 131, 194, 0, 0, -97, 0], [0, 0, 0, 0, 37, 51, 0, 37], [0, -61, 0, 0, 9, 122, 0, 18], [0, -36, 0, 0, -20, 72, 0, -40], [383, 0, 0, 141, 0, 0, 0, 0], [260, 0, 0, -188, 0, 0, 0, 0], [0, 55, 0, 0, 0, -55, 0, -18], [0, 0, 0, 0, -33, -68, 0, -33], [0, -100, 0, 0, 21, 50, 0, -21], [-101, 0, -202, 156, 0, 0, -312, 0], [319, 0, 638, 63, 0, 0, -126, 0], [0, 0, 68, -242, 0, 0, 242, 0], [0, 0, 0, 0, 10, 135, 0, 10], [-163, 0, 0, 105, 0, 0, 0, 0], [757, 0, 757, 0, 0, 0, -77, 0], [0, 6, 0, 0, -24, 6, 0, -12], [0, 0, -1153, 1, 0, 0, -1, 0], [0, 33, 0, 0, 59, 0, 0, 0], [0, -147, 0, 0, -31, 0, 0, 0], [0, 116, 0, 0, -13, -58, 0, 13], [-516, 0, 516, 8, 0, 0, -4, 0], [-185, 0, -370, 122, 0, 0, -244, 0], [0, 20, 0, 0, -16, -40, 0, -32], [0, -70, 0, 0, -13, 140, 0, -26], [62, 0, 0, 420, 0, 0, 0, 0], [0, 85, 0, 0, 0, -85, 0, 78], [337, 0, 337, 0, 0, 0, -12, 0], [0, 115, 0, 0, 0, -115, 0, 54], [0, 0, 299, 363, 0, 0, -363, 0], [-364, 0, 0, -168, 0, 0, 0, 0], [0, 0, -499, 183, 0, 0, -183, 0], [0, -26, 0, 0, 58, -26, 0, 29], [299, 0, -299, 226, 0, 0, -113, 0], [0, 0, -751, 180, 0, 0, -180, 0], [0, 128, 0, 0, -3, 0, 0, 0], [607, 0, 607, 0, 0, 0, -364, 0], [0, 8, 0, 0, -36, -4, 0, 36], [279, 0, 558, 161, 0, 0, -322, 0], [0, -13, 0, 0, 53, 26, 0, 106], [0, -157, 0, 0, 30, 0, 0, 0], [-160, 0, 160, 352, 0, 0, -176, 0], [0, 0, 0, 0, 55, -21, 0, 55], [-470, 0, -470, 0, 0, 0, -228, 0], [0, -138, 0, 0, -7, 69, 0, 7], [-421, 0, -842, -8, 0, 0, 16, 0], [0, 110, 0, 0, 2, 110, 0, 1], [0, 0, -139, -132, 0, 0, 132, 0], [0, -90, 0, 0, -8, -90, 0, -4], [943, 0, 943, 0, 0, 0, 127, 0], [0, 68, 0, 0, -22, -136, 0, -44], [-354, 0, -177, 68, 0, 0, 68, 0], [0, -179, 0, 0, 28, 0, 0, 0], [1141, 0, 1141, 0, 0, 0, -43, 0], [0, -38, 0, 0, 22, 19, 0, -22], [-1880, 0, -940, -10, 0, 0, -10, 0], [0, -136, 0, 0, 2, 272, 0, 4], [-1021, 0, 0, 15, 0, 0, 0, 0], [0, 43, 0, 0, 0, -43, 0, 42], [223, 0, 223, 0, 0, 0, -216, 0], [0, -148, 0, 0, 54, 74, 0, -54], [0, 85, 0, 0, 0, -85, 0, -51], [0, -1, 0, 0, 60, -1, 0, 30], [0, 0, -883, 201, 0, 0, -201, 0], [0, -108, 0, 0, -68, -108, 0, -34], [0, -70, 0, 0, 48, 140, 0, 96], [0, 36, 0, 0, 22, 36, 0, 11], [22, 0, 11, 291, 0, 0, 291, 0], [637, 0, 637, 0, 0, 0, 150, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_684_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_684_3_y_g();".
// 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_684_3_y_g(:prec:=8)
    chi := MakeCharacter_684_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_684_3_y_g();".
// 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_684_3_y_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_684_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![883600, 0, 63920, 0, 3684, 0, 68, 0, 1]>,<7,R![-5, 2, 1]>],Snew);
    return Vf;
end function;