
// Make newform 882.3.c.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_c_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_882_3_c_a();".
// 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_882_3_c_a();".
// 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 := [2, 0, 4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [2, 0, 1, 0], [0, 3, 0, 1]];
    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_882_c();"
function MakeCharacter_882_c()
    N := 882;
    order := 2;
    char_gens := [785, 199];
    v := [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_882_c_Hecke();"
function MakeCharacter_882_c_Hecke(Kf)
    N := 882;
    order := 2;
    char_gens := [785, 199];
    char_values := [[1, 0, 0, 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, 1, 0], [0, 0, 0, 0], [0, 1, 0, 4], [0, 0, 0, 0], [-12, 0, -4, 0], [0, -4, 0, 1], [0, 0, 0, 9], [0, 14, 0, -6], [-20, 0, 12, 0], [16, 0, -13, 0], [0, 8, 0, -4], [-32, 0, 3, 0], [0, 1, 0, -16], [-44, 0, -8, 0], [0, 28, 0, 20], [24, 0, -16, 0], [0, -18, 0, -46], [0, 52, 0, 23], [64, 0, 24, 0], [-8, 0, -40, 0], [0, -15, 0, 56], [-48, 0, 20, 0], [0, 6, 0, -26], [0, 32, 0, -9], [0, 81, 0, -8], [0, 124, 0, 55], [0, 88, 0, 28], [40, 0, 76, 0], [-112, 0, -13, 0], [-32, 0, 24, 0], [28, 0, -4, 0], [0, 62, 0, -6], [-16, 0, 65, 0], [0, 42, 0, 42], [184, 0, 0, 0], [188, 0, 20, 0], [0, 113, 0, 52], [-12, 0, -104, 0], [0, 96, 0, -80], [0, 63, 0, 76], [-56, 0, 188, 0], [0, -113, 0, -124], [-208, 0, -84, 0], [-16, 0, 88, 0], [50, 0, -64, 0], [0, 56, 0, -156], [-8, 0, -20, 0], [0, -48, 0, 68], [0, -66, 0, -146], [0, -199, 0, 84], [-64, 0, 57, 0], [92, 0, 216, 0], [0, -88, 0, -127], [0, -38, 0, 6], [0, -113, 0, 288], [-40, 0, 52, 0], [0, -20, 0, -63], [0, -44, 0, 264], [-82, 0, 96, 0], [240, 0, -55, 0], [0, -154, 0, -58], [0, -220, 0, 41], [0, -138, 0, 290], [0, -164, 0, 36], [0, 15, 0, -176], [-18, 0, -320, 0], [-76, 0, 272, 0], [64, 0, 89, 0], [124, 0, -292, 0], [0, 105, 0, 140], [0, 361, 0, 120], [84, 0, 24, 0], [0, 32, 0, -108], [-232, 0, -208, 0], [-140, 0, -172, 0], [0, -164, 0, 68], [-16, 0, 275, 0], [0, -236, 0, 137], [240, 0, -161, 0], [0, 9, 0, -104], [0, 106, 0, -310], [-494, 0, 16, 0], [412, 0, 32, 0], [0, -160, 0, 201], [0, 388, 0, 72], [-584, 0, 144, 0], [-304, 0, -16, 0], [-112, 0, -360, 0], [0, 233, 0, -4], [192, 0, 212, 0], [0, 18, 0, -170], [0, 308, 0, -40], [612, 0, 112, 0], [-72, 0, 308, 0], [264, 0, -28, 0], [0, 560, 0, 196], [0, -220, 0, 313], [0, 105, 0, 184], [0, 70, 0, -202], [-520, 0, 112, 0], [-212, 0, 216, 0], [-606, 0, -16, 0], [0, -538, 0, 70], [-480, 0, -103, 0], [-124, 0, -380, 0], [0, -120, 0, 231], [0, 98, 0, 546], [0, -304, 0, 47], [488, 0, -236, 0], [0, -9, 0, -192], [0, -48, 0, -176], [0, 0, -363, 0], [-816, 0, 143, 0], [0, -58, 0, -222], [-112, 0, 108, 0], [-288, 0, 95, 0], [0, 210, 0, -574], [0, -20, 0, 392], [448, 0, 283, 0], [60, 0, -352, 0], [0, 503, 0, -28], [560, 0, -137, 0], [0, -63, 0, -484], [504, 0, -204, 0], [0, -150, 0, -434], [144, 0, -485, 0], [-352, 0, -35, 0], [0, 632, 0, -96], [0, 352, 0, 136], [0, 119, 0, -244], [-620, 0, -168, 0], [-76, 0, 664, 0], [-1124, 0, -256, 0], [352, 0, 91, 0], [0, -144, 0, 721], [0, 280, 0, -273], [0, -1, 0, -612], [0, -462, 0, -170], [0, 12, 0, 417], [-1278, 0, -72, 0], [0, -278, 0, 514], [290, 0, -208, 0], [504, 0, 512, 0], [24, 0, -32, 0], [0, 156, 0, -727], [0, 184, 0, -116], [0, -775, 0, -60], [0, -120, 0, 137], [0, 6, 0, 486], [56, 0, 528, 0], [1040, 0, 299, 0], [0, -632, 0, -199], [480, 0, -556, 0], [0, -520, 0, -88], [-168, 0, 120, 0], [556, 0, 52, 0], [48, 0, -432, 0], [0, -240, 0, 199], [0, -607, 0, 360], [0, -724, 0, 687], [-628, 0, -168, 0], [866, 0, 448, 0], [-564, 0, 828, 0], [0, 130, 0, 166], [-384, 0, 71, 0], [0, 1008, 0, 100], [-312, 0, -448, 0], [0, 543, 0, 284]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_3_c_a();".
// 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_882_3_c_a(:prec:=4)
    chi := MakeCharacter_882_c();
    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_882_3_c_a();".
// 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_882_3_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![98, 0, 68, 0, 1]>,<23,R![112, 40, 1]>],Snew);
    return Vf;
end function;