
// Make newform 1035.2.a.d in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1035_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_1035_2_a_d();".
// 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_1035_2_a_d();".
// 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_1035_a();"
function MakeCharacter_1035_a()
    N := 1035;
    order := 1;
    char_gens := [461, 622, 856];
    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_1035_a_Hecke(Kf)
    return MakeCharacter_1035_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 := [[0], [0], [1], [-3], [4], [0], [3], [-8], [-1], [-9], [-5], [-9], [-7], [4], [2], [-13], [3], [-14], [13], [13], [-4], [0], [1], [8], [10], [-9], [8], [-11], [-16], [-1], [-18], [4], [-2], [-3], [14], [-8], [-3], [-2], [22], [24], [0], [0], [2], [4], [-2], [-18], [7], [10], [4], [4], [8], [-13], [20], [-18], [-18], [23], [17], [-23], [18], [-24], [-5], [-19], [0], [8], [25], [-30], [7], [10], [16], [-1], [6], [2], [31], [-6], [-8], [23], [30], [8], [-30], [25], [0], [4], [-20], [-37], [-24], [-6], [11], [-35], [-26], [-20], [13], [-6], [2], [-3], [21], [-21], [-30], [12], [-20], [-10], [20], [-21], [11], [6], [-22], [-20], [-14], [-18], [4], [-41], [-10], [14], [33], [-4], [38], [20], [13], [-12], [-16], [-12], [-14], [-34], [45], [-26], [8], [6], [22], [7], [-37], [51], [27], [48], [-16], [45], [41], [6], [46], [-27], [-27], [39], [-25], [10], [-16], [-37], [-27], [6], [-46], [18], [43], [-54], [-22], [-30], [-34], [46], [53], [40], [24], [-57], [10], [-10], [8], [22], [10], [12], [25], [-15], [-33], [8], [-28], [10], [24], [-3], [-22], [7], [-32], [54], [-36], [-49], [15], [-16], [16], [-22], [-50], [48], [-41], [24], [-4], [3], [-15], [33], [4], [-54], [-52], [49], [48], [-39], [-40], [14], [10], [46], [2], [0], [-4], [-44], [46], [-58], [70], [-64], [-47], [-8], [54], [-42], [32], [36], [11], [-10], [-50], [9], [-73], [-56], [5], [50], [-41], [19], [-12], [-5], [6], [59], [-28], [-55], [-12], [48], [-6], [11], [1], [12], [10], [54], [-21], [65], [-3], [63], [-74], [-50], [-24], [8], [-22], [-52], [50], [-51], [-50], [-2], [7], [-62], [16], [-33], [14], [-64], [-60], [-74], [23], [-69], [-21], [-51], [24], [62], [-42], [50], [-61], [16], [53], [-31], [35], [-34], [-2], [12], [-70], [-47], [22], [18], [26], [-50], [60], [40], [8], [-19], [23], [-42], [24], [-54], [10], [39], [44], [2], [6], [-12], [-20], [-68], [-51], [-54], [-15], [51], [-16], [26], [16], [46], [-60], [50], [-42], [2], [12], [44], [-10], [-40], [-33], [90], [-49], [-45], [35], [54], [10], [-15], [-2], [-22], [23], [-2], [-38], [-92], [24], [69], [15], [56], [77], [-40], [0], [-28], [-58], [-46], [-22], [-35], [-10], [-90], [-30], [-66], [-47], [73], [13], [72], [66], [-3], [37], [-38], [7], [-17], [-74], [-46], [12], [18], [76], [66], [-13], [13], [62], [-36], [22], [26], [14], [57], [-12], [65], [-53], [-58], [12], [28], [68], [67], [18], [-41], [24], [98], [-18], [86], [-32], [-16], [40], [66], [67], [30], [61], [-11], [-28], [58], [-3], [86], [-40], [-17], [10], [96], [8], [-87], [-49], [14], [-17], [-27], [-57], [-71], [-35], [-30], [-32], [60], [34], [-64], [36], [36], [13], [-50], [-60], [54], [87], [-72], [-96], [6], [24], [-84], [-94], [-24], [64], [31], [-52], [-87]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1035_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1035_2_a_d();".
// 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_1035_2_a_d(:prec:=1)
    chi := MakeCharacter_1035_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_1035_2_a_d();".
// 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_1035_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1035_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![3, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;