
// Make newform 3528.2.a.g in Magma, downloaded from the LMFDB on 28 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3528_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_3528_2_a_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_3528_2_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3528_a();"
function MakeCharacter_3528_a()
    N := 3528;
    order := 1;
    char_gens := [2647, 1765, 785, 1081];
    v := [1, 1, 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_3528_a_Hecke( : Kf := Kf)
    return MakeCharacter_3528_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 := 2;
    raw_aps := [[0], [0], [-2], [0], [6], [6], [2], [-4], [2], [8], [-4], [-6], [-10], [-4], [4], [-4], [12], [2], [12], [6], [2], [-8], [0], [14], [2], [18], [12], [-18], [-14], [0], [8], [-8], [20], [0], [4], [-8], [18], [4], [-12], [10], [2], [22], [-2], [6], [12], [16], [-28], [-24], [-20], [-2], [-4], [26], [-22], [-4], [-18], [-6], [6], [4], [-2], [12], [-20], [-6], [-28], [-12], [-2], [-12], [-28], [14], [18], [10], [18], [30], [16], [-10], [-12], [24], [16], [2], [12], [18], [28], [-2], [-18], [38], [0], [-22], [-8], [22], [-30], [32], [-12], [-4], [40], [18], [28], [32], [30], [-30], [16], [22], [28], [-20], [-28], [12], [-28], [30], [-28], [18], [-26], [-10], [8], [22], [12], [-24], [8], [-20], [-28], [-36], [8], [-34], [-14], [-2], [18], [-6], [-8], [40], [10], [-48], [-20], [-34], [4], [-10], [8], [10], [-42], [-10], [-22], [-16], [18], [0], [48], [-12], [-16], [-18], [-26], [12], [2], [-42], [-4], [26], [-34], [2], [-20], [36], [-28], [-2], [-32], [-10], [30], [6], [22], [40], [48], [-16], [-36], [-24], [-24], [-46], [-6], [2], [-2], [-30], [-42], [58], [-24], [-14], [-4], [-48], [44], [-22], [-32], [-56], [-26], [50], [-54], [-6], [2], [-56], [-54], [-24], [-14], [6], [-28], [-6], [-26], [42], [-2], [-34], [-34], [-52], [44], [-48], [46], [26], [24], [42], [-32], [-18], [-40], [-4], [-30], [30], [40], [48], [4], [54], [24], [-14], [18], [16], [-6], [20], [-16], [24], [-40], [-22], [54], [2], [-52], [-54], [-62], [-20], [40], [16], [44], [0], [-14], [12], [34], [0], [26], [-32], [16], [-38], [-66], [20], [20], [76], [-60], [2], [2], [-38], [22], [66], [-22], [6], [38], [-20], [-6], [6], [32], [42], [54], [2], [42], [-56], [28], [-46], [-36], [-48], [-46], [-68], [-14], [24], [-78], [-44], [14], [10], [-34], [12], [-24], [56], [-24], [-30], [64], [-66], [30], [4], [-12], [54], [12], [56], [-4], [68], [-82], [2], [0], [-74], [12], [-8], [-26], [-24], [0], [-18], [-60], [-34], [18], [-34], [-2], [2], [-56], [-40], [-44], [4], [50], [-46], [8], [-38], [-70], [84], [28], [38], [-54], [-8], [32], [-38], [4], [-76], [78], [-24], [-78], [72], [44], [-4], [-28], [76], [46], [-10], [-14], [44], [-42], [-60], [-42], [8], [80], [54], [-10], [28], [-40], [58], [-44], [-14], [84], [0], [54], [18], [48], [-32], [72], [6], [-58], [-66], [-18], [-42], [72], [38], [-66], [-24], [-50], [-70], [52], [54], [4], [44], [6], [8], [-30], [-2], [10], [26], [74], [32], [40], [72], [-80], [24], [-40], [22], [92], [92], [2], [-10], [6], [12], [70], [-10], [-40], [74], [-20], [24], [-54], [56], [-48], [54], [-26], [16], [-58], [64], [24], [60], [6], [-48], [50], [-20], [-6], [-42], [58], [-52], [-6], [68], [-72], [58], [-90], [24], [6], [18], [-38], [-60], [-56], [-76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3528_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_a_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_3528_2_a_g(:prec:=1)
    chi := MakeCharacter_3528_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_3528_2_a_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_3528_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3528_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![-6, 1]>,<13,R![-6, 1]>,<23,R![-2, 1]>],Snew);
    return Vf;
end function;