
// Make newform 225.4.a.m in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_225_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_225_4_a_m();".
// 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_225_4_a_m();".
// 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 := [-10, 0, 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_225_a();"
function MakeCharacter_225_a()
    N := 225;
    order := 1;
    char_gens := [101, 127];
    v := [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;

function MakeCharacter_225_a_Hecke(Kf)
    return MakeCharacter_225_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+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 := 4;
    raw_aps := [[0, 1], [0, 0], [0, 0], [15, 0], [0, 20], [35, 0], [0, 28], [91, 0], [0, -36], [0, -20], [-147, 0], [370, 0], [0, -140], [335, 0], [0, -56], [0, 28], [0, -280], [427, 0], [15, 0], [0, -20], [-70, 0], [-876, 0], [0, -168], [0, 0], [-1085, 0], [0, 420], [1540, 0], [0, 252], [-811, 0], [0, -512], [260, 0], [0, -420], [0, -840], [-784, 0], [0, -680], [-797, 0], [1225, 0], [-275, 0], [0, 84], [0, 1260], [0, 560], [-1687, 0], [0, 700], [3625, 0], [0, -552], [3199, 0], [887, 0], [3535, 0], [0, -28], [-5019, 0], [0, -1140], [0, 140], [-623, 0], [0, 840], [0, 1176], [0, -916], [0, 140], [-308, 0], [-7485, 0], [0, -300], [525, 0], [0, 2016], [-2695, 0], [0, -1680], [-2555, 0], [0, 2360], [-5088, 0], [-11485, 0], [0, -2024], [-9674, 0], [0, -1764], [0, -2820], [8505, 0], [-6775, 0], [9241, 0], [0, -2856], [0, -1380], [-4795, 0], [0, 2220], [6881, 0], [0, 4060], [6418, 0], [0, 420], [-8155, 0], [14749, 0], [0, -3528], [0, -2920], [-1030, 0], [0, 2940], [11940, 0], [0, 3836], [0, -840], [14125, 0], [0, -1280], [1279, 0], [0, 5180], [0, 1400], [0, -3500], [10745, 0], [-9423, 0], [9080, 0], [0, -2640], [0, 2436], [0, -1980], [587, 0], [-15995, 0], [0, -6468], [0, 3668], [0, 6040], [3143, 0], [2660, 0], [-6670, 0], [0, 3492], [5579, 0], [23617, 0], [0, 9960], [-14280, 0], [0, -3920], [0, -3808], [0, 1820], [25438, 0], [15150, 0], [0, 6636], [0, -624], [-20552, 0], [0, -2660], [17281, 0], [0, -2940], [30415, 0], [18550, 0], [24016, 0], [0, -3920], [-7772, 0], [-3865, 0], [0, -5040], [-7441, 0], [0, 8148], [6125, 0], [0, -6328], [0, -860], [28623, 0], [0, -8400], [1135, 0], [0, 316], [-1974, 0], [0, 13300], [36645, 0], [0, -3416], [20104, 0], [0, 680], [-21975, 0], [0, -280], [34915, 0], [0, 15624], [17880, 0], [0, 5480], [-25829, 0], [0, -980], [40005, 0], [0, 9520], [0, 13692], [0, -16976], [-20540, 0], [0, 3360], [0, 8968], [0, 11088], [-18283, 0], [-13230, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_225_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_225_4_a_m();".
// 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_225_4_a_m(:prec:=2)
    chi := MakeCharacter_225_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_225_4_a_m();".
// 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_225_4_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_225_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-10, 0, 1]>,<7,R![-15, 1]>],Snew);
    return Vf;
end function;