
// Make newform 625.2.a.d in Magma, downloaded from the LMFDB on 09 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_625_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_a_d();".
// 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_625_2_a_d();".
// 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];
        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_625_a();"
function MakeCharacter_625_a()
    N := 625;
    order := 1;
    char_gens := [2];
    v := [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;

function MakeCharacter_625_a_Hecke(Kf)
    return MakeCharacter_625_a();
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]];
    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], [1, 1], [0, 0], [-1, 3], [-1, 1], [1, -4], [5, -4], [4, -3], [3, -3], [-6, 2], [2, 0], [3, 0], [0, 4], [6, -4], [10, 1], [2, 4], [-1, -8], [-4, -3], [-5, -4], [8, -7], [-5, 3], [3, -6], [0, 8], [2, 1], [-9, -1], [13, -2], [-10, 3], [6, -11], [11, 3], [-9, 6], [12, -3], [-4, 2], [17, 2], [-11, 12], [-12, -1], [3, 8], [13, 0], [3, -3], [11, 4], [6, 6], [9, -18], [-2, -7], [-3, 0], [10, 8], [-10, -4], [-6, -3], [-3, 10], [-5, 3], [4, -2], [-13, 6], [14, 5], [0, 0], [-9, -3], [7, -10], [6, -6], [8, 7], [20, -10], [-13, 0], [-25, 6], [-13, 20], [9, -20], [3, 7], [-20, 21], [0, 9], [0, 13], [11, -11], [25, -6], [-2, 0], [9, -2], [9, -18], [16, -14], [-3, -14], [18, 0], [-10, 3], [6, -2], [27, -1], [-1, 7], [-8, 12], [0, 9], [-22, -6], [-29, -2], [-3, 10], [-24, 12], [-5, 3], [-15, 15], [-18, -6], [-13, 26], [-9, -16], [-23, 5], [15, -7], [-8, 7], [26, -2], [-27, 20], [18, 8], [-9, -12], [9, 0], [33, -1], [-33, 0], [8, -18], [14, 6], [-15, 6], [16, -1], [-3, 24], [-9, -2], [25, -6], [19, -22], [-6, 3], [-21, 0], [4, 2], [-8, -10], [24, -12], [-18, -6], [-31, -7], [-19, 28], [-41, -4], [12, -30], [-15, -2], [19, -22], [-4, -19], [20, -25], [30, -6], [5, -12], [-13, 17], [-22, 7], [-33, 30], [1, -8], [-29, 18], [6, 23], [-5, 21], [11, -24], [23, 4], [34, -5], [1, -13], [-15, -24], [-3, -15], [15, -30], [-21, 40], [5, 16], [-3, 27], [5, -25], [-27, 18], [-16, -4], [-9, 6], [-36, 18], [-44, 18], [1, -17], [8, 27], [-20, 26], [-11, 2], [27, -36], [16, -21], [-8, 25], [-11, 0], [26, -26], [-48, -3], [39, -4], [25, 20], [-7, 14], [-5, 6], [-41, 26], [3, 0], [-3, 29], [11, -26], [-39, 27], [-15, 26], [48, -8], [12, -30], [20, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_625_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_a_d();".
// 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_625_2_a_d(:prec:=2)
    chi := MakeCharacter_625_a();
    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_625_2_a_d();".
// 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_625_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_625_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 1]>,<3,R![1, -3, 1]>],Snew);
    return Vf;
end function;