
// Make newform 70.4.e.c in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_70_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_70_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_70_4_e_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_70_4_e_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
        poly := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_70_e();"
function MakeCharacter_70_e()
    N := 70;
    order := 3;
    char_gens := [57, 31];
    v := [3, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_70_e_Hecke();"
function MakeCharacter_70_e_Hecke( : Kf := false)
    N := 70;
    order := 3;
    char_gens := [57, 31];
    char_values := [[1, 0], [0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 4;
    raw_aps := [[0, 2], [1, -1], [0, 5], [-1, 19], [2, -2], [-8, 0], [52, -52], [0, -26], [0, -67], [69, 0], [332, -332], [0, -196], [353, 0], [-369, 0], [0, -88], [-582, 582], [350, -350], [0, 467], [-291, 291], [770, 0], [-628, 628], [0, -1170], [525, 0], [0, -89], [-290, 0], [1233, -1233], [0, -1553], [0, -1831], [-811, 811], [2170, 0], [48, 0], [0, 792], [-1084, 1084], [-46, 0], [0, 369], [-2480, 2480], [-1226, 1226], [0, 660], [-949, 0], [0, -3392], [-268, 268], [-4093, 0], [0, 1978], [3998, -3998], [-3030, 0], [-356, 356], [-2602, 0], [-3156, 0], [-4236, 4236], [0, -6334], [0, 4688], [1856, 0], [4806, -4806], [-3200, 0], [0, 1120], [-6405, 6405], [-4935, 4935], [0, 1446], [8020, -8020], [4978, 0], [-5852, 5852], [-3012, 0], [9443, 0], [9994, -9994], [0, -1456], [0, -6872], [0, 392], [3926, 0], [1785, -1785], [-1591, 0], [-1686, 1686], [0, -7372], [-6637, 6637], [0, -3476], [-2378, 0], [0, 2457], [-8562, 8562], [0, 9842], [0, 8097], [-15259, 15259], [16224, 0], [2977, 0], [-14202, 14202], [14310, 0], [0, 2356], [0, -8291], [-3521, 0], [0, 5432], [-5350, 0], [-10123, 0], [0, -6463], [17640, -17640], [10032, -10032], [5916, 0], [0, -13894], [18427, 0], [0, -18557], [9930, -9930], [0, 7924], [0, 23497], [11131, 0], [11978, -11978], [10059, -10059], [0, 15666], [-2530, 2530], [-12818, 12818], [-15228, 0], [0, 2598], [2548, -2548], [1706, 0], [0, 12567], [9628, -9628], [-4316, 0], [-16402, 16402], [8490, 0], [123, -123], [-23924, 0], [-181, 181], [0, 17812], [15334, 0], [-1337, 1337], [-15112, 0], [0, -25392], [-29231, 29231], [0, 7538], [22125, 0], [0, 4439], [0, -13650], [-11397, 0], [0, -16686], [-22470, 22470], [5625, 0], [0, -17620], [39056, 0], [0, -29738], [-1118, 0], [-13432, 13432], [9349, -9349], [-28008, 0], [-19669, 19669], [31860, 0], [0, -2230], [-14803, 14803], [13257, 0], [21574, -21574], [12990, 0], [24838, 0], [17026, -17026], [0, -6728], [0, 30099], [0, 5626], [-15927, 0], [39124, 0], [0, 25331], [-26385, 26385], [-40770, 0], [0, 14618], [0, -41323], [-22620, 0], [-51978, 51978], [0, -9987], [6588, 0], [-4091, 0], [0, -23640], [-36666, 36666], [41071, -41071], [-31984, 31984], [43894, -43894]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_70_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_70_4_e_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_70_4_e_c(:prec:=2)
    chi := MakeCharacter_70_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_70_4_e_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_70_4_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_70_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![1, -1, 1]>],Snew);
    return Vf;
end function;