
// Make newform 825.2.n.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_n_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_825_2_n_e();".
// 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_825_2_n_e();".
// 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, -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_825_n();"
function MakeCharacter_825_n()
    N := 825;
    order := 5;
    char_gens := [551, 727, 376];
    v := [5, 5, 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_825_n_Hecke();"
function MakeCharacter_825_n_Hecke(Kf)
    N := 825;
    order := 5;
    char_gens := [551, 727, 376];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1]];
    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 := [[1, 0, 0, -1], [0, 0, -1, 0], [0, 0, 0, 0], [0, 0, 0, 3], [-2, 4, -1, 2], [0, -3, 3, 0], [4, 1, 4, 0], [0, 2, -6, 2], [-7, 0, -1, 1], [1, -1, 0, -3], [7, -4, 4, -7], [-3, 3, 0, 7], [0, -2, 3, -2], [-3, 0, 2, -2], [0, 1, -11, 1], [2, -6, 6, -2], [3, -3, 0, 6], [-2, 9, -2, 0], [5, 0, 6, -6], [-4, 5, -4, 0], [3, -3, 0, -5], [2, 7, -7, -2], [-1, 12, -1, 0], [-3, 0, 9, -9], [-3, -9, 9, 3], [-2, 7, -7, 2], [-10, 10, 0, 9], [0, -12, 3, -12], [15, 0, 0, 0], [0, -4, -2, -4], [1, 5, 1, 0], [3, 0, 7, -7], [11, -5, 11, 0], [3, -3, 0, -4], [-10, 0, -10, 0], [0, -10, 7, -10], [0, -11, 0, -11], [14, -9, 9, -14], [-16, 8, -8, 16], [0, 4, -1, 4], [0, -18, 9, -18], [11, -5, 11, 0], [19, -19, 0, -4], [4, 12, 4, 0], [-6, 0, -6, 6], [-12, 0, 6, -6], [-7, -3, 3, 7], [0, 6, 3, 6], [9, -9, 0, -9], [10, -10, 10, -10], [-2, -5, 5, 2], [0, -14, 2, -14], [6, 0, 3, -3], [15, -6, 6, -15], [-13, 13, 0, 2], [0, 0, 18, -18], [-4, 7, -4, 0], [-15, 15, 0, 13], [11, 10, -10, -11], [7, -8, 7, 0], [0, -5, -19, -5], [12, -12, 0, -12], [-6, 0, -16, 16], [0, 4, -25, 4], [6, 21, 6, 0], [3, 7, -7, -3], [-9, 0, -12, 12], [6, -6, 0, -20], [-7, -1, -7, 0], [0, 8, 16, 8], [-1, 0, 16, -16], [23, -23, 0, -4], [24, -24, 0, -4], [-10, 0, -12, 12], [23, -14, 23, 0], [-24, 28, -28, 24], [-5, 5, 0, 15], [14, 0, 9, -9], [10, -17, 10, 0], [-1, 19, -19, 1], [14, 0, 8, -8], [0, -15, 17, -15], [-18, 21, -21, 18], [10, -10, 0, -21], [13, 0, -9, 9], [0, 17, 0, 17], [-12, 8, -8, 12], [-4, 30, -4, 0], [-3, 0, -10, 10], [13, 0, -6, 6], [13, -26, 13, 0], [-20, 25, -20, 0], [0, 15, -23, 15], [-3, 3, 0, 12], [24, -24, 0, 3], [0, -9, 18, -9], [0, -11, 3, -11], [-36, 36, 0, 21], [11, 21, 11, 0], [10, 14, -14, -10], [0, 6, -31, 6], [30, -30, 0, -12], [-3, -21, 21, 3], [0, -4, 22, -4], [21, 0, -12, 12], [-23, 17, -23, 0], [-28, 28, 0, 17], [-15, 0, -17, 17], [-4, -28, -4, 0], [1, -1, 0, 30], [27, -24, 24, -27], [0, -21, 9, -21], [-14, 0, 8, -8], [0, -20, 35, -20], [-10, 10, 0, 43], [0, 31, -11, 31], [-1, 22, -1, 0], [20, -31, 31, -20], [8, -8, 0, 0], [-28, 0, -26, 26], [-3, 0, 0, 0], [-12, -5, 5, 12], [4, -14, 4, 0], [9, 0, 6, -6], [21, -15, 15, -21], [0, 2, 11, 2], [4, 3, 4, 0], [8, -8, 0, 31], [1, 0, -2, 2], [9, -9, 0, 12], [-35, 20, -20, 35], [-10, -21, -10, 0], [0, -4, -11, -4], [-31, 18, -18, 31], [10, -26, 26, -10], [-26, 0, 3, -3], [0, 14, -1, 14], [-1, -24, -1, 0], [-33, 12, -33, 0], [-2, 1, -2, 0], [0, -6, -10, -6], [33, -33, 0, -36], [1, 3, -3, -1], [24, 13, -13, -24], [-7, 7, 0, 21], [0, 11, 7, 11], [-27, -10, -27, 0], [7, 0, 5, -5], [3, 0, -9, 9], [-1, -33, -1, 0], [0, 12, -49, 12], [-28, 0, -10, 10], [0, -27, 27, -27], [10, -10, 0, -12], [-24, -15, -24, 0], [-46, 33, -33, 46], [-3, -11, -3, 0], [4, 48, 4, 0], [29, -17, 17, -29], [0, 3, 0, 0], [-32, 0, 17, -17], [40, -40, 0, -31], [-9, 0, 23, -23], [0, 38, -2, 38], [-27, 12, -12, 27], [-18, 18, 0, -12], [-33, 0, -25, 25], [-6, 6, 0, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_n_e();".
// 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_825_2_n_e(:prec:=4)
    chi := MakeCharacter_825_n();
    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_825_2_n_e();".
// 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_825_2_n_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, 4, -3, 1]>,<13,R![81, 81, 36, 6, 1]>],Snew);
    return Vf;
end function;