
// Make newform 1035.2.a.b in Magma, downloaded from the LMFDB on 20 September 2025.
// 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_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_1035_2_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_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 := 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]];
    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], [0], [1], [1], [-2], [-2], [-3], [-2], [-1], [-7], [-5], [11], [-1], [0], [0], [-11], [13], [-8], [5], [-5], [6], [-12], [-9], [-4], [-14], [5], [-8], [15], [-10], [9], [-4], [-12], [-6], [-19], [-16], [12], [17], [-18], [24], [-24], [-24], [16], [0], [16], [2], [26], [15], [14], [-4], [8], [-4], [29], [-18], [26], [8], [-19], [-21], [13], [6], [6], [11], [-9], [-26], [8], [-3], [-12], [11], [10], [12], [11], [8], [12], [-13], [-34], [12], [-11], [-10], [4], [-2], [-11], [18], [-2], [-28], [-1], [8], [32], [21], [25], [2], [-10], [-33], [-12], [-28], [-9], [1], [33], [-18], [42], [-4], [-26], [24], [-11], [-31], [6], [-16], [20], [-2], [4], [24], [-21], [32], [-26], [-13], [36], [40], [-14], [25], [50], [-18], [30], [-14], [28], [27], [28], [-44], [0], [12], [37], [19], [11], [23], [-16], [-22], [37], [-45], [32], [10], [-7], [-53], [-15], [15], [-10], [-16], [29], [45], [-54], [-14], [-22], [-5], [18], [-2], [32], [-56], [26], [5], [40], [-4], [-27], [-22], [24], [52], [-26], [44], [-12], [39], [-9], [-1], [-40], [-20], [0], [-20], [-19], [-18], [55], [-24], [16], [-36], [-3], [47], [-58], [-46], [56], [-14], [-34], [21], [26], [-22], [35], [-23], [-25], [30], [-2], [-50], [-21], [-24], [7], [58], [-50], [2], [-66], [-64], [-44], [-46], [-58], [28], [58], [12], [-34], [-57], [28], [22], [-30], [-8], [-36], [-47], [60], [22], [-49], [21], [10], [-35], [-54], [29], [-21], [-54], [-41], [50], [21], [60], [-21], [-18], [46], [-50], [13], [45], [12], [-52], [-42], [9], [-69], [-49], [-25], [-22], [-2], [-28], [-4], [12], [30], [-22], [31], [-18], [2], [-3], [-30], [26], [-39], [-56], [32], [-18], [14], [-21], [-3], [43], [-3], [64], [-12], [-36], [30], [43], [60], [-71], [53], [15], [22], [20], [-34], [28], [29], [68], [-32], [-50], [8], [-72], [-28], [-62], [3], [-9], [-66], [16], [30], [58], [-39], [-36], [-32], [8], [8], [-30], [78], [-9], [42], [-31], [-3], [72], [36], [-36], [-28], [-6], [-66], [-28], [44], [36], [18], [-28], [-44], [-31], [-80], [-35], [-3], [63], [-54], [-44], [13], [-50], [-22], [-23], [60], [60], [26], [-16], [63], [51], [-42], [53], [42], [-20], [-48], [2], [30], [10], [-71], [6], [6], [-18], [72], [3], [23], [21], [-82], [74], [75], [-67], [-56], [-47], [47], [32], [-66], [42], [18], [-18], [-34], [-69], [3], [-10], [24], [2], [74], [-20], [29], [44], [35], [-17], [-82], [-96], [-32], [-46], [-79], [6], [83], [72], [62], [18], [42], [-100], [-50], [-68], [-94], [-33], [68], [-19], [-33], [58], [-34], [-29], [68], [6], [-93], [-78], [0], [104], [-17], [-89], [46], [-37], [83], [-87], [-15], [-67], [22], [-12], [-36], [-28], [6], [-52], [0], [17], [-18], [-56], [40], [-67], [76], [-78], [-50], [-62], [20], [88], [-68], [84], [69], [4], [79]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1035_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1035_2_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_1035_2_a_b(: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_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_1035_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1035_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<7,R![-1, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;