
// Make newform 28.6.a.b in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_28_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_28_6_a_b();".
// 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_28_6_a_b();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_28_a();"
function MakeCharacter_28_a()
    N := 28;
    order := 1;
    char_gens := [15, 17];
    v := [1, 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;

function MakeCharacter_28_a_Hecke( : Kf := Kf)
    return MakeCharacter_28_a();
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]];
    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], [26], [16], [-49], [8], [684], [-2218], [-2698], [3344], [-3254], [4788], [-11470], [13350], [-928], [1212], [13110], [34702], [-1032], [10108], [62720], [-18926], [11400], [88958], [19722], [17062], [45904], [-136012], [-69156], [-146414], [-80186], [274800], [180742], [-209678], [17242], [59358], [-336344], [464588], [314792], [285724], [-709148], [-617148], [237828], [-133512], [270446], [875102], [-347620], [-425380], [481592], [-6042], [1804], [-1611526], [-987096], [893510], [365946], [1404594], [1099680], [814948], [-1699056], [-1365082], [-715846], [217726], [1507080], [12502], [-647432], [-935978], [705942], [-1143040], [-2364018], [726240], [136180], [-1169070], [-4280], [2447960], [-904514], [-4230344], [4553996], [-3985406], [552420], [38190], [-3924754], [598386], [4615974], [-61560], [-3797266], [-2288520], [-4759756], [-4367150], [5449942], [1669660], [70768], [5660830], [-1449484], [4075040], [986100], [5983420], [-3493728], [2157108], [-6658170], [5952230], [-6396810], [-5518512], [2021590], [8146778], [-11982426], [1395816], [1967842], [-3188974], [1675002], [-10063464], [1727978], [-16952256], [10194162], [15745190], [-332690], [3597200], [-14638930], [-13838614], [-14035812], [16106274], [4800752], [-17656472], [6592250], [9771780], [-18866316], [8670182], [7934818], [26260034], [22076292], [8492452], [-19071288], [-14683224], [16426488], [-24435672], [-295566], [-473842], [23324062], [15558280], [6668430], [-12246108], [-29149466], [7588486], [-5988490], [817960], [2519628], [-16100600], [-25816732], [-15427012], [3605166], [4069946], [7251112], [2374138], [30355818], [1533380], [-29237868], [44822748], [-36290640], [32535016], [-44667594], [15668018], [-20117480], [-8845184], [-31436374], [-9521584], [10652046], [27293082], [10476372], [1882304], [27151832]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_28_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_28_6_a_b();".
// 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_28_6_a_b(:prec:=1)
    chi := MakeCharacter_28_a();
    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_28_6_a_b();".
// 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_28_6_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_28_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![-26, 1]>],Snew);
    return Vf;
end function;