
// Make newform 1520.2.g.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1520_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1520_g_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_1520_2_g_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_1520_2_g_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 := [25, -5, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-25, 16, 4, 1], [15, 4, -4, -1], [5, 4, 0, -1]];
    Rf_basisdens := [1, 20, 10, 4];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1520_g();"
function MakeCharacter_1520_g()
    N := 1520;
    order := 2;
    char_gens := [191, 1141, 1217, 401];
    v := [1, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1520_g_Hecke();"
function MakeCharacter_1520_g_Hecke(Kf)
    N := 1520;
    order := 2;
    char_gens := [191, 1141, 1217, 401];
    char_values := [[-1, 0, 0, 0], [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, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [-2, -1, 0, -1], [0, -1, 2, 1], [0, 0, 0, 0], [0, -1, -4, 1], [0, 2, 1, -2], [4, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, 3, 0, 3], [-6, 1, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [8, 1, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 3, -4, -3], [0, 0, 0, 0], [16, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [10, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 5, 6, -5], [0, 1, 12, -1], [0, -5, 2, 5], [8, 5, 0, 5], [0, 0, 0, 0], [0, -8, -4, 8], [-24, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 5, -6, -5], [0, 0, 0, 0], [0, -8, -4, 8], [0, -3, -14, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-8, 5, 0, 5], [0, 7, 4, -7], [0, -7, -6, 7], [0, 0, 0, 0], [0, 5, 14, -5], [0, 0, 0, 0], [-2, 7, 0, 7], [0, 0, 0, 0], [0, -12, -6, 12], [0, -1, 16, 1], [0, 0, 0, 0], [14, -3, 0, -3], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1, -18, 1], [0, 16, 8, -16], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [18, -1, 0, -1], [-16, 3, 0, 3], [0, -16, -8, 16], [0, -5, -18, 5], [-28, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [16, 7, 0, 7], [0, 9, 8, -9], [0, 0, 0, 0], [0, 0, 0, 0], [0, 4, 2, -4], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [18, 7, 0, 7], [0, 0, 0, 0], [0, -9, 4, 9], [16, -5, 0, -5], [22, 3, 0, 3], [-22, -1, 0, -1], [0, 20, 10, -20], [0, 0, 0, 0], [0, 20, 10, -20], [0, -5, -22, 5], [44, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-8, 9, 0, 9], [0, 0, 0, 0], [0, -1, -24, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 12, 6, -12], [0, 11, 4, -11], [2, 11, 0, 11], [0, 16, 8, -16], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 7, -16, -7], [0, -11, -12, 11], [0, -20, -10, 20], [0, 11, -2, -11], [0, 0, 0, 0], [-26, -3, 0, -3], [-14, -11, 0, -11], [0, 7, 24, -7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -9, -22, 9], [50, 0, 0, 0], [-10, 0, 0, 0], [0, -5, 22, 5], [-14, 9, 0, 9], [0, -24, -12, 24], [0, -7, -26, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, -11, 8, 11], [28, 1, 0, 1], [28, 5, 0, 5], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-4, -13, 0, -13], [0, 0, 0, 0], [24, -5, 0, -5], [-2, -13, 0, -13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -24, -12, 24], [0, 0, 0, 0], [0, 13, 14, -13], [0, 0, 0, 0], [0, 0, 0, 0], [12, -11, 0, -11], [-22, 7, 0, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -4, -2, 4], [50, 0, 0, 0], [0, 9, 28, -9], [0, 0, 0, 0], [-8, 0, 0, 0], [0, 0, 0, 0], [-12, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 32, -1], [0, 0, 0, 0], [0, 13, -8, -13], [0, 0, 0, 0], [0, 0, 0, 0], [0, -28, -14, 28], [0, -16, -8, 16], [0, 0, 0, 0], [-28, 5, 0, 5], [0, 20, 10, -20], [-32, 1, 0, 1], [0, 0, 0, 0], [-8, -15, 0, -15], [22, -9, 0, -9], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [26, -7, 0, -7], [-24, -13, 0, -13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -20, -10, 20], [0, 0, 0, 0], [-34, -7, 0, -7], [0, 0, 0, 0], [0, 0, 0, 0], [32, 0, 0, 0], [0, 0, 0, 0], [-4, 15, 0, 15], [0, 24, 12, -24], [0, 32, 16, -32], [-14, 13, 0, 13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -20, -10, 20], [0, -11, -32, 11], [0, 15, 22, -15], [0, 0, 0, 0], [36, 7, 0, 7], [0, 0, 0, 0], [0, -32, -16, 32], [-70, 0, 0, 0], [34, -3, 0, -3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-68, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 17, 8, -17], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [38, 7, 0, 7], [0, 0, 0, 0], [-16, -17, 0, -17], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 5, -34, -5], [0, 13, 32, -13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, -17, 0, -17], [38, -1, 0, -1], [-28, -15, 0, -15], [0, -7, 32, 7], [0, -20, -10, 20], [0, 0, 0, 0], [0, 0, 0, 0], [0, -15, 14, 15], [-4, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 13, 34, -13], [26, -11, 0, -11], [0, 0, 0, 0], [0, 0, 0, 0], [-6, 17, 0, 17], [0, 8, 4, -8], [36, -5, 0, -5], [-52, 0, 0, 0], [0, 0, 0, 0], [0, -17, -24, 17], [0, 4, 2, -4], [-16, 15, 0, 15], [0, 0, 0, 0], [0, 0, 0, 0], [0, 32, 16, -32], [0, 0, 0, 0], [0, 0, 0, 0], [-32, -15, 0, -15], [0, 0, 0, 0], [0, -11, 28, 11], [0, 0, 0, 0], [0, 0, 0, 0], [-12, -19, 0, -19], [0, 0, 0, 0], [0, -13, 24, 13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 17, 28, -17], [0, -5, 38, 5], [0, 0, 0, 0], [84, 0, 0, 0], [18, 19, 0, 19], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1, 42, 1], [0, 0, 0, 0], [0, -28, -14, 28], [-2, 19, 0, 19], [0, 0, 0, 0], [-34, 9, 0, 9], [0, 19, 22, -19], [0, 7, 44, -7], [0, 0, 0, 0], [0, 17, -14, -17], [0, 0, 0, 0], [-2, 0, 0, 0], [42, 11, 0, 11], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-14, 0, 0, 0], [0, 0, 0, 0], [0, -17, 16, 17], [0, 0, 0, 0], [-26, -19, 0, -19], [0, 13, -28, -13], [0, 0, 0, 0], [0, 15, 38, -15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1520_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1520_2_g_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_1520_2_g_c(:prec:=4)
    chi := MakeCharacter_1520_g();
    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_1520_2_g_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_1520_2_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1520_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![-12, 3, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;