
// Make newform 882.3.s.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_s_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_s_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_s_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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_s();"
function MakeCharacter_882_s()
    N := 882;
    order := 6;
    char_gens := [785, 199];
    v := [3, 4];
    // 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_s_Hecke();"
function MakeCharacter_882_s_Hecke(Kf)
    N := 882;
    order := 6;
    char_gens := [785, 199];
    char_values := [[-1, 0, 0, 0], [-1, 0, 1, 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, -1, 0, 0], [0, 0, 0, 0], [0, 3, 0, 0], [0, 0, 0, 0], [0, -9, 0, 9], [1, 0, 0, 0], [0, 12, 0, -12], [23, 0, -23, 0], [0, 12, 0, 0], [0, 0, 0, 24], [0, 0, 47, 0], [55, 0, -55, 0], [0, 0, 0, 33], [23, 0, 0, 0], [0, -3, 0, 0], [0, -36, 0, 36], [0, -60, 0, 60], [104, 0, -104, 0], [0, 0, 97, 0], [0, 0, 0, 69], [0, 0, 65, 0], [-113, 0, 113, 0], [0, 0, 0, 21], [0, 96, 0, 0], [-104, 0, 0, 0], [0, 105, 0, -105], [119, 0, -119, 0], [0, 84, 0, 0], [0, 0, 49, 0], [0, 0, 0, -69], [113, 0, 0, 0], [0, 15, 0, 0], [0, 48, 0, -48], [103, 0, 0, 0], [0, 120, 0, 0], [0, 0, -104, 0], [0, 0, 152, 0], [-56, 0, 56, 0], [0, 0, 0, 3], [0, -108, 0, 0], [0, 3, 0, -3], [55, 0, 0, 0], [0, 45, 0, 0], [0, 0, 151, 0], [0, 0, 0, 12], [0, 0, -160, 0], [-208, 0, 0, 0], [40, 0, 0, 0], [0, 159, 0, -159], [377, 0, -377, 0], [0, -243, 0, 0], [0, 0, 0, -9], [0, 0, -130, 0], [0, 0, 0, 36], [0, -333, 0, 0], [0, -72, 0, 72], [0, -201, 0, 201], [-520, 0, 520, 0], [0, 0, -113, 0], [0, 0, 0, -324], [0, 0, 89, 0], [0, 0, 0, 48], [-233, 0, 0, 0], [0, 177, 0, -177], [-151, 0, 151, 0], [0, 288, 0, 0], [73, 0, -73, 0], [527, 0, 0, 0], [0, -300, 0, 300], [112, 0, 0, 0], [0, 219, 0, -219], [0, -360, 0, 0], [0, 0, -319, 0], [-209, 0, 209, 0], [-433, 0, 0, 0], [0, 276, 0, 0], [0, 261, 0, -261], [113, 0, -113, 0], [0, -252, 0, 0], [0, 0, -319, 0], [0, 0, 0, -543], [65, 0, 0, 0], [0, -339, 0, 339], [367, 0, 0, 0], [536, 0, -536, 0], [0, -60, 0, 0], [0, 0, 0, -435], [463, 0, -463, 0], [0, 0, 0, -192], [47, 0, 0, 0], [0, 525, 0, 0], [0, -300, 0, 300], [0, 0, -239, 0], [0, 0, 0, 144], [49, 0, -49, 0], [0, 0, 0, -417], [0, -171, 0, 0], [0, -600, 0, 600], [-391, 0, 391, 0], [73, 0, -73, 0], [32, 0, 0, 0], [0, -33, 0, 33], [0, -627, 0, 627], [0, 225, 0, 0], [0, 0, 535, 0], [0, 0, 905, 0], [0, 0, 0, -408], [0, 393, 0, 0], [0, 144, 0, -144], [-143, 0, 0, 0], [599, 0, -599, 0], [0, 0, 304, 0], [0, 0, 0, -207], [0, 0, -343, 0], [272, 0, 0, 0], [0, -300, 0, 300], [679, 0, 0, 0], [0, -681, 0, 681], [0, -633, 0, 0], [0, 0, 0, 12], [0, 0, -433, 0], [737, 0, 0, 0], [0, 96, 0, 0], [0, -900, 0, 900], [-247, 0, 247, 0], [0, 0, 0, -228], [1048, 0, -1048, 0], [0, -177, 0, 0], [-641, 0, 0, 0], [-25, 0, 25, 0], [0, 0, 535, 0], [0, 0, 0, 663], [-41, 0, 41, 0], [-1114, 0, 0, 0], [0, -204, 0, 0], [-1127, 0, 0, 0], [0, 153, 0, -153], [0, 0, 170, 0], [0, 0, 0, 828], [0, -237, 0, 237], [1114, 0, 0, 0], [0, 69, 0, 0], [0, 0, -1592, 0], [0, 0, 0, 723], [0, 0, 383, 0], [0, 0, 0, -408], [-527, 0, 0, 0], [0, 936, 0, -936], [-1414, 0, 1414, 0], [0, -969, 0, 0], [-224, 0, 224, 0], [0, 0, 0, -1140], [329, 0, 0, 0], [0, 39, 0, 0], [0, 0, -311, 0], [0, 0, 0, 252], [145, 0, -145, 0], [0, -219, 0, 0], [487, 0, 0, 0], [0, 972, 0, -972], [0, 135, 0, 0], [0, 0, 0, -732], [-1681, 0, 0, 0], [0, 852, 0, 0], [0, 831, 0, -831], [0, 1356, 0, -1356], [0, 0, 625, 0], [0, 0, -943, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_3_s_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_s_a(:prec:=4)
    chi := MakeCharacter_882_s();
    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_s_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_s_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![324, 0, -18, 0, 1]>,<11,R![26244, 0, -162, 0, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;