
// Make newform 2070.2.d.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2070_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2070_d_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_2070_2_d_a();".
// 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_2070_2_d_a();".
// 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, 0, 0, 0, 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_2070_d();"
function MakeCharacter_2070_d()
    N := 2070;
    order := 2;
    char_gens := [461, 1657, 1891];
    v := [2, 1, 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_2070_d_Hecke();"
function MakeCharacter_2070_d_Hecke(Kf)
    N := 2070;
    order := 2;
    char_gens := [461, 1657, 1891];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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, 0], [0, 0, 0, 0], [0, 2, 0, 1], [0, 0, -2, 0], [0, 3, 0, -3], [0, 2, -2, 2], [0, -2, -4, -2], [-2, -3, 0, 3], [0, 0, -1, 0], [-2, 4, 0, -4], [6, 0, 0, 0], [0, -1, 2, -1], [4, 2, 0, -2], [0, -3, -6, -3], [0, 2, -2, 2], [0, -1, 12, -1], [0, 6, 0, -6], [-2, 1, 0, -1], [0, -1, -2, -1], [0, 4, 0, -4], [0, -4, -2, -4], [-2, 4, 0, -4], [0, 1, 0, 1], [12, -2, 0, 2], [0, 0, -6, 0], [-2, -8, 0, 8], [0, -10, -2, -10], [0, 5, -8, 5], [-14, 1, 0, -1], [0, -12, 0, -12], [0, 0, -2, 0], [16, 4, 0, -4], [0, 6, 0, 6], [0, 12, 0, -12], [0, 9, 0, -9], [-10, 4, 0, -4], [0, 5, 14, 5], [0, -2, -4, -2], [0, -2, 6, -2], [0, 2, 6, 2], [-16, -4, 0, 4], [-2, -13, 0, 13], [8, 2, 0, -2], [0, 12, 4, 12], [0, -16, 2, -16], [14, 0, 0, 0], [-12, 6, 0, -6], [0, 0, -16, 0], [0, 13, 4, 13], [-6, -3, 0, 3], [0, 10, -12, 10], [-14, -2, 0, 2], [-2, 6, 0, -6], [12, -9, 0, 9], [0, 0, -18, 0], [0, 4, -12, 4], [-22, 2, 0, -2], [4, 4, 0, -4], [0, -6, 22, -6], [16, -10, 0, 10], [0, -15, 6, -15], [0, 17, -4, 17], [0, 18, 0, 18], [2, 14, 0, -14], [0, 0, -18, 0], [0, 6, 6, 6], [-12, -6, 0, 6], [0, -6, -18, -6], [0, 14, 12, 14], [-14, 8, 0, -8], [0, 10, -12, 10], [0, 14, 0, -14], [0, -12, 18, -12], [0, 15, 2, 15], [-2, -15, 0, 15], [0, 16, 4, 16], [-12, -13, 0, 13], [0, -4, -14, -4], [12, -12, 0, 12], [-18, 0, 0, 0], [-32, -5, 0, 5], [6, -3, 0, 3], [0, 6, 0, -6], [0, 12, -10, 12], [0, 0, 0, 0], [0, 8, -12, 8], [-20, 2, 0, -2], [0, 12, 18, 12], [10, -2, 0, 2], [0, 12, 14, 12], [0, 25, -4, 25], [16, 8, 0, -8], [0, 24, 6, 24], [-4, 16, 0, -16], [12, 18, 0, -18], [0, -10, 8, -10], [6, -12, 0, 12], [-12, -20, 0, 20], [0, 5, 10, 5], [2, -24, 0, 24], [0, 6, 4, 6], [0, -19, -12, -19], [0, 7, -12, 7], [20, -4, 0, 4], [-30, 9, 0, -9], [0, 16, 2, 16], [0, 2, -24, 2], [0, -4, -18, -4], [-26, 10, 0, -10], [-4, 12, 0, -12], [0, 12, 0, 12], [0, -3, 10, -3], [0, -8, 20, -8], [-2, -15, 0, 15], [38, 2, 0, -2], [-20, -20, 0, 20], [0, -13, 14, -13], [0, -4, 24, -4], [0, -18, 18, -18], [-36, -7, 0, 7], [22, 9, 0, -9], [0, -24, 6, -24], [0, 23, 8, 23], [0, -6, 0, -6], [-16, 2, 0, -2], [0, 27, 0, -27], [-34, 1, 0, -1], [14, -10, 0, 10], [0, 4, -26, 4], [0, -1, -10, -1], [-20, 16, 0, -16], [0, 4, 8, 4], [-46, 2, 0, -2], [0, -3, 38, -3], [30, 4, 0, -4], [-26, 14, 0, -14], [0, -5, -24, -5], [0, -9, -10, -9], [0, -1, 20, -1], [2, -8, 0, 8], [4, -2, 0, 2], [18, 8, 0, -8], [0, 20, -4, 20], [0, -7, -28, -7], [-22, 20, 0, -20], [0, 24, 0, -24], [0, 8, 10, 8], [0, 2, 48, 2], [4, -20, 0, 20], [0, -16, 0, -16], [0, -2, -14, -2], [-12, -10, 0, 10], [0, -2, -16, -2], [0, -30, 6, -30], [0, -15, 6, -15], [20, 14, 0, -14], [-22, -12, 0, 12], [28, -10, 0, 10], [0, -30, 2, -30], [16, -11, 0, 11], [0, 18, 24, 18], [0, -34, -8, -34], [0, -12, -12, -12], [36, 13, 0, -13], [0, 14, 0, 14], [0, 6, -12, 6], [18, -24, 0, 24], [0, 12, -30, 12], [-10, -30, 0, 30], [0, 26, 6, 26], [-12, 13, 0, -13], [6, -6, 0, 6], [-28, -8, 0, 8], [0, -6, -34, -6], [4, -32, 0, 32], [-4, -2, 0, 2], [40, 4, 0, -4], [-18, -30, 0, 30], [0, 10, 38, 10], [6, 15, 0, -15], [0, -20, 4, -20], [4, 11, 0, -11], [0, -6, -46, -6], [0, 22, 20, 22], [0, -2, 20, -2], [4, -23, 0, 23], [0, -8, 34, -8], [0, -15, 34, -15], [4, -24, 0, 24], [-24, 28, 0, -28], [0, 0, -6, 0], [0, 2, -32, 2], [-2, -25, 0, 25], [-30, -6, 0, 6], [0, -17, 24, -17], [0, 4, -8, 4], [-10, -16, 0, 16], [0, -23, -14, -23], [0, 8, 48, 8], [0, 16, -24, 16], [-20, -1, 0, 1], [-4, 24, 0, -24], [0, -18, 34, -18], [-6, 0, 0, 0], [40, -1, 0, 1], [0, -14, 38, -14], [-18, 36, 0, -36], [0, -16, 12, -16], [40, 2, 0, -2], [24, 6, 0, -6], [0, 0, -10, 0], [42, -10, 0, 10], [0, -6, -18, -6], [0, 25, 24, 25], [0, 44, 0, -44], [22, 14, 0, -14], [0, 24, -16, 24], [6, 28, 0, -28], [0, -12, 12, -12], [0, 0, 6, 0], [26, 10, 0, -10], [50, 2, 0, -2], [-30, 28, 0, -28], [0, 18, 2, 18], [0, 4, -32, 4], [-42, -18, 0, 18], [0, 16, 12, 16], [-8, -16, 0, 16], [0, 2, 26, 2], [-28, 10, 0, -10], [0, 36, -14, 36], [6, 21, 0, -21], [10, -26, 0, 26], [-48, 6, 0, -6], [0, 5, -34, 5], [0, 26, 8, 26], [-22, -30, 0, 30], [0, -3, 36, -3], [0, 24, 0, -24], [48, -8, 0, 8], [0, 29, -12, 29], [-28, 22, 0, -22], [0, 24, -4, 24], [10, 2, 0, -2], [0, 14, 44, 14], [-18, 2, 0, -2], [0, 24, -26, 24], [48, -5, 0, 5], [46, 9, 0, -9], [0, -28, -24, -28], [0, 5, 22, 5], [0, 38, 0, -38], [0, 0, -12, 0], [-6, 12, 0, -12], [0, -8, -14, -8], [12, -6, 0, 6], [18, 21, 0, -21], [0, 17, -46, 17], [0, 44, 14, 44], [0, 12, 20, 12], [0, 30, -6, 30], [0, -35, 0, -35], [2, 38, 0, -38], [0, 9, 34, 9], [0, 6, 0, 6], [-18, -3, 0, 3], [-4, 7, 0, -7], [-36, -2, 0, 2], [0, -21, -22, -21], [0, 8, 42, 8], [-58, 2, 0, -2], [0, 15, -34, 15], [0, -22, 10, -22], [14, 18, 0, -18], [0, 0, -28, 0], [0, 28, -14, 28], [0, -16, -24, -16], [26, 4, 0, -4], [14, -32, 0, 32], [40, 19, 0, -19], [0, -8, 24, -8], [38, -26, 0, 26], [0, -26, -44, -26], [26, -1, 0, 1], [0, 18, -40, 18], [-18, 22, 0, -22], [0, 8, -14, 8], [0, 15, 36, 15], [8, 36, 0, -36], [48, -10, 0, 10], [-24, -1, 0, 1], [0, -13, -48, -13], [0, -4, 14, -4], [4, -29, 0, 29], [0, 16, -38, 16], [16, 25, 0, -25], [30, 12, 0, -12], [0, 18, -18, 18], [12, 34, 0, -34], [0, -12, -20, -12], [0, 4, -38, 4], [0, -11, -12, -11], [-26, -36, 0, 36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2070_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2070_2_d_a();".
// 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_2070_2_d_a(:prec:=4)
    chi := MakeCharacter_2070_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2070_2_d_a();".
// 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_2070_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2070_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 0, 1]>,<11,R![-18, 0, 1]>],Snew);
    return Vf;
end function;