
// Make newform 825.2.p.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_p_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_p_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_825_2_p_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, -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_p();"
function MakeCharacter_825_p()
    N := 825;
    order := 5;
    char_gens := [551, 727, 376];
    v := [5, 2, 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_825_p_Hecke();"
function MakeCharacter_825_p_Hecke(Kf)
    N := 825;
    order := 5;
    char_gens := [551, 727, 376];
    char_values := [[1, 0, 0, 0], [0, 0, 0, -1], [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 := [[-2, 2, 0, 1], [1, 0, 0, 0], [-1, -1, 1, 1], [1, -1, 0, -2], [1, -1, 3, 1], [-6, 0, 0, 0], [2, -2, 0, -5], [-6, 5, -5, 6], [-3, -1, 1, 3], [-5, 1, -5, 0], [-4, -1, -4, 0], [-3, 6, -6, 3], [-2, 2, 0, -3], [3, 0, 8, -8], [5, 0, 5, -5], [0, 0, -1, 0], [-8, 2, -2, 8], [-1, 0, -7, 7], [0, -3, -1, -3], [-5, 1, -1, 5], [0, -2, 0, -2], [0, -8, 0, -8], [-9, 9, 0, 3], [2, -11, 11, -2], [0, -4, 9, -4], [7, -13, 13, -7], [6, -1, 6, 0], [0, 12, -7, 12], [6, -14, 6, 0], [-4, 4, 0, 5], [9, 0, -3, 3], [0, -7, -6, -7], [0, 6, 12, 6], [0, -3, 10, -3], [-7, 12, -7, 0], [0, 1, 0, 1], [0, -6, 1, -6], [-13, 0, 0, 0], [-7, -14, -7, 0], [-15, 4, -15, 0], [0, 0, 14, -14], [1, -1, 0, 12], [-15, 16, -16, 15], [-1, 5, -1, 0], [4, -4, 0, 9], [-7, 0, -11, 11], [-6, 6, 0, 8], [-1, -2, -1, 0], [0, -6, 19, -6], [0, 0, 17, 0], [-20, 8, -20, 0], [5, 5, 5, 0], [-3, -13, -3, 0], [-8, 21, -21, 8], [2, -2, 0, -15], [18, -6, 18, 0], [-14, 14, 0, 9], [-11, 14, -11, 0], [1, -1, 0, -20], [2, -2, 0, -13], [9, 7, -7, -9], [-2, 2, 0, 28], [9, 0, -12, 12], [14, 5, 14, 0], [0, -12, -7, -12], [0, 10, 0, 0], [0, -8, 20, -8], [-1, -17, 17, 1], [-13, 5, -5, 13], [0, -7, 11, -7], [9, -9, 0, -9], [18, -30, 30, -18], [1, 0, -19, 19], [12, -25, 25, -12], [-25, 22, -22, 25], [26, -22, 26, 0], [0, 11, -13, 11], [-1, 1, 0, -22], [12, -3, 12, 0], [3, 0, -8, 8], [0, -24, 16, -24], [13, 0, 21, -21], [0, -5, 0, 0], [20, 0, 8, -8], [-2, 13, -13, 2], [0, -12, -19, -12], [-9, 4, -4, 9], [10, -26, 10, 0], [-19, -2, 2, 19], [18, 5, 18, 0], [-9, 9, 0, 10], [-3, -15, 15, 3], [5, -5, 0, -12], [7, 23, -23, -7], [-10, 10, 0, 3], [6, 0, 0, 0], [12, -12, 0, -2], [4, 0, 4, 0], [25, 0, 11, -11], [-2, 0, 0, 0], [41, 0, -2, 2], [11, -11, 0, -36], [-28, 0, 9, -9], [3, -22, 22, -3], [3, -3, 0, -18], [-10, 0, 2, -2], [0, -1, 1, 0], [-17, 0, -6, 6], [-18, 32, -18, 0], [1, -1, 0, 5], [-5, 15, -15, 5], [-11, 11, 0, -21], [0, 22, -29, 22], [-7, 17, -17, 7], [18, 0, -6, 6], [0, 0, 0, 3], [-10, 8, -10, 0], [0, -28, 20, -28], [-13, 13, -13, 0], [-23, 13, -23, 0], [-16, 16, -16, 16], [-5, 5, 0, 22], [5, 0, 0, 0], [0, -1, -17, -1], [20, 0, -12, 12], [0, -17, 10, -17], [0, 4, -25, 4], [-15, 0, -3, 3], [20, -5, 5, -20], [-19, 0, 9, -9], [14, -14, 0, -19], [-19, -2, -19, 0], [0, -8, -9, -8], [-21, -18, 18, 21], [-1, 1, 0, -32], [0, -22, 20, -22], [-5, -32, -5, 0], [0, -3, 4, -3], [-22, 28, -28, 22], [0, -6, 35, -6], [-18, 1, -18, 0], [-28, 19, -28, 0], [8, -8, 0, -43], [8, -8, 0, -11], [6, -37, 6, 0], [23, -19, 23, 0], [18, -10, 10, -18], [-10, 0, 15, -15], [-10, 19, -10, 0], [0, 11, -25, 11], [23, -27, 23, 0], [0, 3, -4, 3], [0, -24, 24, 0], [-47, 41, -41, 47], [39, -22, 39, 0], [-26, 26, 0, 23], [-14, 14, 0, 5], [34, -44, 44, -34], [-22, 0, -25, 25], [0, 2, -5, 2], [-27, 27, 0, 22], [15, -7, 15, 0], [0, -18, -3, -18], [-7, 0, 18, -18], [14, -14, 0, 10], [-4, 0, -8, 8], [14, 23, 14, 0], [-1, -27, -1, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_p_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_p_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_825_2_p_a(:prec:=4)
    chi := MakeCharacter_825_p();
    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_p_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_825_2_p_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![25, 25, 15, 5, 1]>],Snew);
    return Vf;
end function;