
// Make newform 2535.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2535_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_2535_2_a_c();".
// 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_2535_2_a_c();".
// 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_2535_a();"
function MakeCharacter_2535_a()
    N := 2535;
    order := 1;
    char_gens := [1691, 1522, 1861];
    v := [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_2535_a_Hecke(Kf)
    return MakeCharacter_2535_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+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 := 2;
    raw_aps := [[-2], [1], [1], [-5], [-2], [0], [2], [0], [6], [-4], [7], [2], [-6], [1], [8], [-4], [-12], [-13], [7], [-12], [-15], [3], [-8], [-14], [5], [-18], [-7], [4], [11], [-2], [-11], [-4], [2], [-3], [12], [8], [-15], [15], [-12], [0], [-6], [-22], [12], [11], [12], [17], [15], [8], [-10], [-14], [-14], [-12], [-10], [0], [-22], [-10], [-6], [-29], [10], [12], [5], [-16], [31], [-22], [-31], [-12], [-9], [-1], [-16], [-3], [-6], [-2], [7], [13], [5], [18], [8], [15], [-16], [15], [38], [23], [28], [1], [15], [26], [-18], [-35], [2], [-3], [-4], [-42], [28], [-24], [-4], [-6], [30], [-30], [0], [-29], [-9], [20], [26], [-20], [12], [-2], [-28], [10], [-16], [22], [16], [15], [-6], [-37], [-7], [2], [19], [38], [42], [-24], [-35], [33], [-12], [20], [-37], [40], [23], [40], [9], [-7], [36], [-24], [-28], [2], [36], [34], [-46], [-17], [30], [4], [-45], [22], [20], [-46], [-11], [34], [-9], [-12], [43], [-6], [-6], [20], [-25], [-44], [20], [0], [8], [-52], [-30], [0], [-18], [6], [56], [20], [-60], [-38], [0], [29], [43], [-8], [-22], [22], [-10], [-19], [-32], [16], [-7], [10], [15], [-1], [32], [-32], [59], [-16], [-12], [-30], [-9], [4], [-37], [22], [19], [-54], [-37], [0], [-12], [-32], [13], [11], [-12], [12], [-38], [1], [17], [53], [18], [38], [13], [-44], [36], [-4], [57], [28], [37], [-36], [60], [-70], [11], [26], [-44], [62], [13], [65], [-44], [-29], [-16], [47], [-26], [-24], [21], [26], [-61], [31], [40], [0], [13], [-66], [-61], [-52], [30], [-6], [-4], [-52], [-12], [65], [-18], [64], [-7], [62], [9], [0], [17], [-32], [60], [49], [0], [-24], [-54], [0], [34], [-54], [24], [-66], [27], [41], [-6], [-25], [4], [24], [36], [0], [-61], [-7], [-10], [8], [7], [-24], [-10], [14], [-3], [-36], [-76], [72], [78], [33], [-52], [-34], [51], [76], [-45], [44], [18], [28], [56], [-30], [-55], [-2], [24], [62], [42], [19], [-11], [-52], [40], [-12], [47], [57], [78], [22], [-64], [-38], [-78], [66], [-90], [-17], [-24], [1], [26], [-30], [-75], [-12], [-59], [51], [6], [1], [34], [-27], [15], [-32], [24], [-40], [-46], [-12], [45], [54], [63], [-6], [-67], [-22], [-10], [-53], [26], [-24], [0], [83], [84], [-2], [-15], [52], [-24], [40], [13], [-62], [-56], [-43], [58], [12], [-26], [90], [-12], [48], [-30], [-72], [-26], [-12], [-45], [17], [-58], [-51], [-2], [-82], [-59], [-78], [-56], [-56], [66], [-78], [76], [-25], [12], [23], [-24], [32], [52], [36], [7], [-38], [8], [-69], [-13], [18], [82], [-76], [84], [80], [90], [59], [8], [-72], [-52], [-78], [41], [-28], [33], [48], [90], [-83], [29], [-90], [87], [0], [21], [-42], [84], [-88], [-37], [-6], [-48], [7], [74], [-34], [60], [7], [-22], [-16], [-1], [-24], [4], [-20], [7], [62]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2535_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2535_2_a_c();".
// 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_2535_2_a_c(:prec:=1)
    chi := MakeCharacter_2535_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_2535_2_a_c();".
// 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_2535_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2535_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<7,R![5, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;