
// Make newform 1035.2.a.c in Magma, downloaded from the LMFDB on 02 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_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_1035_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_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 := [[-1], [0], [1], [4], [-4], [6], [2], [-4], [1], [10], [-8], [2], [-2], [-8], [0], [6], [0], [6], [8], [4], [10], [16], [12], [10], [-10], [-14], [-12], [12], [-10], [2], [0], [-8], [10], [4], [6], [-16], [-14], [20], [-8], [-2], [-24], [-2], [-24], [-6], [6], [24], [-4], [24], [12], [6], [2], [-20], [-14], [12], [-30], [0], [2], [16], [-10], [-6], [-24], [-26], [-28], [-12], [14], [6], [12], [-26], [12], [14], [-6], [-8], [28], [10], [4], [-32], [14], [30], [10], [10], [28], [-10], [0], [14], [-16], [12], [14], [-18], [34], [-32], [-20], [-40], [24], [-16], [4], [-16], [-22], [-6], [8], [-34], [-28], [-2], [-12], [-30], [44], [-38], [36], [-46], [-28], [10], [16], [-6], [-6], [-4], [8], [26], [16], [-8], [6], [-4], [14], [-46], [6], [4], [-20], [-34], [-42], [-36], [-12], [-6], [12], [8], [-24], [-38], [30], [42], [38], [-16], [-2], [-10], [44], [-22], [16], [52], [-2], [0], [-26], [-14], [-52], [8], [54], [-30], [4], [-48], [-40], [8], [16], [-18], [-2], [14], [-4], [42], [-32], [-4], [50], [-8], [-40], [-2], [-30], [-58], [4], [-34], [40], [6], [-16], [2], [-4], [-62], [36], [-2], [16], [4], [6], [-54], [24], [-2], [22], [-24], [-22], [-60], [-6], [-44], [-4], [66], [-20], [-22], [-22], [-30], [-30], [56], [38], [-16], [6], [-30], [36], [-2], [0], [28], [62], [-4], [18], [26], [-28], [12], [-60], [50], [32], [38], [-8], [6], [38], [56], [-66], [44], [-12], [-10], [10], [-4], [-4], [0], [-34], [12], [16], [6], [-24], [8], [58], [54], [48], [12], [36], [20], [64], [14], [42], [60], [-48], [4], [36], [-24], [18], [10], [-72], [18], [38], [-24], [-2], [-24], [46], [2], [12], [12], [38], [-46], [2], [-28], [54], [-30], [80], [30], [14], [8], [-10], [16], [-62], [-8], [-28], [62], [2], [-12], [16], [-64], [-72], [-74], [68], [12], [-2], [-58], [32], [38], [-66], [36], [-30], [-12], [-74], [46], [8], [54], [-16], [20], [-2], [62], [-16], [-12], [-12], [-22], [-44], [-34], [-72], [-26], [56], [46], [2], [4], [56], [-38], [48], [-12], [-2], [46], [60], [78], [-46], [8], [10], [-30], [52], [-68], [40], [6], [78], [54], [80], [-28], [28], [68], [62], [-78], [-38], [4], [54], [42], [-78], [80], [30], [-24], [70], [4], [0], [-42], [4], [-54], [-6], [-76], [-34], [-30], [24], [-84], [-78], [-72], [18], [78], [-24], [36], [12], [82], [-10], [-44], [66], [-32], [-84], [-24], [-90], [-56], [-26], [-48], [56], [-34], [18], [34], [6], [-6], [-64], [-6], [44], [64], [-80], [22], [-56], [56], [42], [46], [24], [-68], [-8], [-26], [64], [90], [44], [74], [-34], [50], [52], [42], [-30], [8], [-62], [-74], [0], [-48], [-62], [54], [-12], [12], [78], [-62], [52], [-88], [-30], [40], [22], [-38], [-56], [-72], [-62], [78], [84], [22], [60], [4]];
    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_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_1035_2_a_c(: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_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_1035_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1035_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<7,R![-4, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;