
// Make newform 525.2.i.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_i_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_525_2_i_i();".
// 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_525_2_i_i();".
// 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 := [16, 20, 53, -27, 50, -3, 8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-57732, 311369, -301543, 65994, -51899, 14556, -7573, 689], [-8164, -1692, -3431, -17462, -1081, -2209, 21, -329], [208424, -19483, 217641, 67606, 37929, 10196, 3659, 1653], [-34052, -94449, 124955, -97726, 14959, -16412, 3357, -2041], [-33126, 187457, -34651, 112020, -4685, 17076, -1195, 1871], [161284, 59659, -136173, 255062, -12261, 42256, -5771, 6159]];
    Rf_basisdens := [1, 1, 75828, 18957, 75828, 75828, 37914, 75828];
    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_525_i();"
function MakeCharacter_525_i()
    N := 525;
    order := 3;
    char_gens := [176, 127, 451];
    v := [3, 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_525_i_Hecke();"
function MakeCharacter_525_i_Hecke(Kf)
    N := 525;
    order := 3;
    char_gens := [176, 127, 451];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 1, 0, 0, 0], [0, -1, 0, 1, -1, 3, -1, 0], [-1, 0, 0, -2, 1, 0, 0, 1], [0, 1, 0, -1, -1, -1, -1, 0], [0, -3, 0, 1, -1, 0, 1, 1], [0, -2, 0, 2, -2, 0, 2, 2], [4, 0, 0, 0, 0, 0, 0, 0], [0, 3, -1, -2, -1, 2, -1, -1], [-3, 4, 0, -2, 2, -3, -2, -2], [1, -1, -1, 2, -2, 0, 1, -1], [0, 2, 2, -2, 1, 0, -2, -1], [-1, 0, -1, -1, 1, -1, -2, -1], [0, 3, 0, -3, 1, -3, 1, 0], [0, -1, -2, 3, -1, 3, -1, -2], [5, 0, 0, 0, 0, 5, 0, 0], [0, 0, -1, 1, 0, 0, 0, -1], [6, 2, 2, -8, 2, 0, -2, 0], [0, -2, 0, 2, -2, 3, -2, 0], [-2, -2, 1, -1, 1, -2, 0, -1], [1, -1, -1, 0, -2, 0, 1, -1], [3, 0, -1, 1, -1, 3, 0, 1], [-3, 2, 2, -6, 2, 0, -2, 0], [0, -1, 0, 1, 1, 3, 1, 0], [4, 2, 1, -3, 3, 4, -2, -3], [5, 2, -1, 1, -1, 5, 0, 1], [0, -1, 1, 0, 1, 5, 1, 1], [0, 2, 2, 0, 0, 0, -2, -2], [-4, 1, 1, 2, -2, 0, -1, -3], [-6, 4, 0, -4, 4, -6, -4, -4], [0, -2, 2, 0, -2, 2, -2, 2], [-10, 1, 1, 4, 0, 0, -1, -1], [-5, -8, 1, 3, -3, -5, 4, 3], [0, -3, -2, 5, -3, 4, -3, -2], [0, 4, 2, -6, -2, 1, -2, 2], [-6, 4, 1, -3, 3, -6, -2, -3], [3, 1, 1, -8, 0, 0, -1, -1], [8, -4, -2, 2, -2, 8, 0, 2], [0, 4, -4, 0, 0, 0, 0, -4], [6, 0, 0, -8, 1, 0, 0, 1], [5, 4, 3, -1, 1, 5, 2, -1], [0, -1, 1, 0, 1, -10, 1, 1], [2, -2, -2, 0, 2, 0, 2, 4], [0, 1, 2, -3, 5, -8, 5, 2], [6, 3, 3, -2, 4, 0, -3, 1], [-12, 1, 1, -4, -2, 0, -1, -3], [0, 6, 2, -8, 2, 0, 2, 2], [-7, 1, -3, -2, 2, -7, -5, -2], [-15, 0, -1, 1, -1, -15, 0, 1], [-2, -2, -2, 0, -2, 0, 2, 0], [0, -2, -6, 8, 0, 5, 0, -6], [-13, -1, -1, 4, -2, 0, 1, -1], [-17, 2, -1, 3, -3, -17, 2, 3], [0, -8, 2, 6, 0, -10, 0, 2], [0, 1, 0, -1, 3, -3, 3, 0], [-14, 4, -2, 0, 0, -14, -2, 0], [0, -5, 1, 4, 1, 5, 1, 1], [7, 3, 3, -2, 0, 0, -3, -3], [0, -4, 5, -1, 4, 0, 4, 5], [10, 2, 2, 0, 2, 0, -2, 0], [0, -2, -2, 6, -1, 0, 2, 1], [0, 4, -2, -2, 0, -8, 0, -2], [4, 13, 1, -2, 2, 4, -1, -2], [0, 0, -2, 6, -6, 0, 4, 6], [10, 6, -3, 1, -1, 10, -2, 1], [0, 2, 2, -8, 1, 0, -2, -1], [0, -9, 2, 7, 3, -1, 3, 2], [-6, 2, 2, -2, -2, 0, -2, -4], [0, 5, -4, -1, -1, 13, -1, -4], [-13, 0, 3, -1, 1, -13, 2, -1], [0, 1, -1, 0, 3, -20, 3, -1], [5, -1, -3, 4, -4, 5, 1, 4], [6, 3, 3, 2, 1, 0, -3, -2], [-6, -2, 6, -4, 4, -6, 2, -4], [0, 3, 6, -9, 1, 13, 1, 6], [-4, 1, 5, -6, 6, -4, -1, -6], [-3, 2, 1, -3, 3, -3, -2, -3], [0, 3, -1, -2, -5, -9, -5, -1], [3, 3, 3, -4, -6, 0, -3, -9], [-11, 2, 2, 2, -2, 0, -2, -4], [0, -2, -2, 4, -2, 2, -2, -2], [8, -2, -2, -4, 3, 0, 2, 5], [0, 4, -3, 3, -3, 0, 0, 3], [12, 4, 6, 2, -2, 12, 8, 2], [-1, -5, -5, 10, -2, 0, 5, 3], [-3, 12, -2, 0, 0, -3, -2, 0], [10, -8, -8, 8, -4, 0, 8, 4], [-4, -3, -3, 8, -8, 0, 3, -5], [-14, 4, 4, 0, 0, -14, 4, 0], [0, -7, -8, 15, -3, -1, -3, -8], [0, -1, 1, 0, -1, 24, -1, 1], [0, -6, -6, 8, -2, 0, 6, 4], [6, 1, 2, -5, 5, 6, -3, -5], [-2, 6, 6, 0, 4, 0, -6, -2], [2, 2, -4, 6, -6, 2, 2, 6], [0, 6, 8, -14, 2, 0, 2, 8], [16, -5, 1, 6, -6, 16, 7, 6], [-17, -7, 3, 2, -2, -17, 5, 2], [2, -4, -4, 6, 2, 0, 4, 6], [0, 2, -2, 0, 2, -8, 2, -2], [0, -3, 0, 3, 1, 13, 1, 0], [-11, 8, -5, 1, -1, -11, -4, 1], [0, -6, 9, -3, 6, 2, 6, 9], [0, -1, -1, 2, 3, 4, 3, -1], [-6, 4, 4, -8, 4, 0, -4, 0], [16, -6, 0, 2, -2, 16, 2, 2], [0, 1, 4, -5, 1, 27, 1, 4], [7, 6, 6, -10, -1, 0, -6, -7], [20, -2, -1, -3, 3, 20, -4, -3], [0, -2, 6, -4, 0, 2, 0, 6], [10, 2, 2, 0, 6, 0, -2, 4], [0, -5, 1, 4, -1, 8, -1, 1], [-2, -9, -9, 12, -6, 0, 9, 3], [0, 14, -2, -12, -2, 12, -2, -2], [8, -9, -9, 8, -8, 0, 9, 1], [0, -3, -8, 11, -7, 1, -7, -8], [-13, -2, -1, 3, -3, -13, 2, 3], [-3, 5, 5, -12, 0, 0, -5, -5], [0, 6, -2, -4, -4, -21, -4, -2], [9, -10, -10, 20, -8, 0, 10, 2], [-4, -18, -4, 6, -6, -4, 2, 6], [0, 7, 4, -11, 7, -13, 7, 4], [-6, 10, 5, 1, -1, -6, 6, 1], [19, -9, -9, 18, -8, 0, 9, 1], [-3, 8, 2, 0, 0, -3, 2, 0], [-3, -12, -5, -1, 1, -3, -6, -1], [-4, -1, -1, -4, -1, 0, 1, 0], [-11, 4, -6, 2, -2, -11, -4, 2], [0, 1, 2, -3, 7, -12, 7, 2], [-34, -4, -4, 8, -2, 0, 4, 2], [16, -14, 1, -3, 3, 16, -2, -3], [11, -4, -4, 6, -2, 0, 4, 2], [-17, 10, -3, -3, 3, -17, -6, -3], [14, 4, 4, -20, 7, 0, -4, 3], [0, -2, 4, -2, 6, -2, 6, 4], [0, 3, 4, -7, 1, -16, 1, 4], [-4, 6, 6, -8, 6, 0, -6, 0], [0, -4, 6, -2, 0, 12, 0, 6], [-24, 1, 1, -14, 8, 0, -1, 7], [-17, 12, -3, -5, 5, -17, -8, -5], [0, 3, 0, -3, 5, -14, 5, 0], [28, 4, 4, 0, 2, 0, -4, -2], [0, -8, 2, 6, -2, -1, -2, 2], [4, 2, 2, -4, 4, 0, -2, 2], [32, -4, -4, 10, -7, 0, 4, -3], [0, 0, 2, -2, 0, 32, 0, 2], [-32, 0, -4, 0, 0, -32, -4, 0], [13, -8, -3, 1, -1, 13, -2, 1], [9, -16, -4, 8, -8, 9, 4, 8], [28, 4, 4, 0, 6, 0, -4, 2], [-20, 11, 11, -8, 7, 0, -11, -4], [-24, 20, 6, -14, 14, -24, -8, -14], [0, 13, 2, -15, 3, 22, 3, 2], [2, 8, 8, -16, 2, 0, -8, -6], [-6, 3, 7, 0, 0, -6, 7, 0], [7, -2, -1, -1, 1, 7, -2, -1], [-10, 12, 12, -4, 7, 0, -12, -5], [0, -6, 8, -2, 6, -8, 6, 8], [22, -24, -8, 12, -12, 22, 4, 12], [11, 5, 5, -10, 0, 0, -5, -5], [24, 9, 9, -32, 8, 0, -9, -1], [-17, 0, -7, 5, -5, -17, -2, 5], [0, -3, 0, 3, 3, -19, 3, 0], [0, -11, 2, 9, -7, 21, -7, 2], [0, -5, -3, 8, 1, -20, 1, -3], [0, 8, -2, -6, -2, 21, -2, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_2_i_i();".
// 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_525_2_i_i(:prec:=8)
    chi := MakeCharacter_525_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_525_2_i_i();".
// 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_525_2_i_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, -20, 53, 27, 50, 3, 8, 1, 1]>],Snew);
    return Vf;
end function;