
// Make newform 25.6.b.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_25_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_25_b_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_25_6_b_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_25_6_b_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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_25_b();"
function MakeCharacter_25_b()
    N := 25;
    order := 2;
    char_gens := [2];
    v := [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_25_b_Hecke();"
function MakeCharacter_25_b_Hecke(Kf)
    N := 25;
    order := 2;
    char_gens := [2];
    char_values := [[-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 := 6;
    raw_aps := [[0, 1], [0, 2], [0, 0], [0, 96], [-148, 0], [0, -143], [0, -839], [-1060, 0], [0, -1488], [3410, 0], [-2448, 0], [0, 91], [-9398, 0], [0, 622], [0, -6044], [0, -11923], [20020, 0], [32302, 0], [0, 30486], [-32648, 0], [0, 19387], [33360, 0], [0, -8358], [-101370, 0], [0, -59519], [-89898, 0], [0, 9752], [0, 79146], [-36830, 0], [0, -5593], [0, 35276], [76452, 0], [0, -72459], [-112220, 0], [-403750, 0], [-446648, 0], [0, -131129], [0, 77282], [0, 198336], [0, 286737], [594460, 0], [-107098, 0], [469552, 0], [0, -26353], [0, 227931], [-865000, 0], [1105652, 0], [0, -560788], [0, -11674], [596010, 0], [0, 242667], [48880, 0], [-110798, 0], [-1643748, 0], [0, 653121], [0, -1064168], [1441090, 0], [-93248, 0], [0, -55149], [-192198, 0], [0, 165942], [0, -1097403], [0, -1188754], [-2373048, 0], [0, 714707], [0, 1062311], [3099852, 0], [0, 1200041], [0, 888706], [2148050, 0], [0, 330927], [259320, 0], [0, -749964], [0, 1119037], [-3159340, 0], [0, -171108], [-88470, 0], [0, -2728369], [4046802, 0], [2712070, 0], [-3717460, 0], [3552502, 0], [-4062048, 0], [0, -3631433], [5410280, 0], [0, 3257622], [509950, 0], [0, 610421], [-4072098, 0], [0, -1011468], [0, 1625486], [3279360, 0], [0, -4265984], [1512652, 0], [6491900, 0], [0, -4308848], [-2673230, 0], [6185002, 0], [0, 3447262], [155502, 0], [0, 6327206], [0, -3538929], [0, -423318], [-4960410, 0], [8965052, 0], [0, -1430399], [0, -3370134], [0, 893047], [-4946200, 0], [-4580998, 0], [0, 3539996], [0, -2548043], [0, -6500139], [-4844060, 0], [6227752, 0], [1532802, 0], [0, 8719122], [0, -2127344], [0, -1505423], [8114620, 0], [2473702, 0], [0, -2885313], [0, 8359851], [0, -3572658], [-8783948, 0], [-16014098, 0], [19135370, 0], [-10293360, 0], [0, -9663224], [0, -2630983], [-28294420, 0], [0, -10493128], [-18966848, 0], [0, -5412829], [19053402, 0], [15782590, 0], [0, 12202437], [0, 16880366], [0, 10994231], [29359670, 0], [31770252, 0], [-2714298, 0], [0, 6293312], [0, -4361774], [10617810, 0], [-16776520, 0], [0, 11009277], [0, 15833821], [-15806380, 0], [0, 7214332], [0, 123951], [41026802, 0], [0, -20900758], [0, -10523784], [0, 3741546], [-6631648, 0], [16897640, 0], [12865310, 0], [0, 5344341], [28245502, 0], [0, -8544594], [0, -11112973], [0, 12077536], [-54831348, 0], [0, -7830599], [0, 8170992], [13757552, 0], [0, -6454869]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_25_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_25_6_b_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_25_6_b_a(:prec:=2)
    chi := MakeCharacter_25_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_25_6_b_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_25_6_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_25_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![4, 0, 1]>],Snew);
    return Vf;
end function;