
// Make newform 2025.2.a.o in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2025_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_2025_2_a_o();".
// 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_2025_2_a_o();".
// 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 := [3, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    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_2025_a();"
function MakeCharacter_2025_a()
    N := 2025;
    order := 1;
    char_gens := [326, 1702];
    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_2025_a_Hecke(Kf)
    return MakeCharacter_2025_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 := 2;
    raw_aps := [[0, 1, 0], [0, 0, 0], [0, 0, 0], [-2, 1, 0], [1, 0, 1], [-1, 0, 1], [-1, 0, -1], [1, 0, -1], [2, -1, 2], [1, 2, -2], [1, 4, -1], [-3, 2, -1], [4, 2, 1], [-3, -2, -1], [5, -3, -1], [0, -2, 0], [0, 2, 0], [0, 2, 1], [-5, 1, -3], [3, -2, -3], [4, -4, 0], [2, -4, 0], [-6, 3, 0], [3, 0, 0], [8, -2, 4], [-2, -2, -2], [-3, 2, -1], [4, 1, -2], [4, -4, 5], [2, 6, -4], [-3, -5, -1], [6, 0, 0], [-4, 2, 2], [8, 0, 0], [4, -4, -5], [9, -4, 1], [6, 2, 2], [-13, -2, 1], [11, -1, -1], [14, -4, 2], [3, 2, -3], [-3, -2, -2], [15, -2, 3], [5, -8, -5], [-10, 2, -4], [-3, -4, -5], [7, -4, -1], [7, -9, 3], [1, 0, 1], [-11, -2, -4], [11, 2, 5], [10, -6, 4], [-8, 0, 5], [15, -6, 3], [-18, 0, 0], [19, -6, 1], [4, -12, 7], [-5, 0, 5], [-1, -4, -5], [14, -4, 5], [-2, -5, 6], [11, -4, -1], [-10, 9, -2], [7, -2, -5], [11, 10, -5], [-13, -2, -1], [6, 6, -2], [4, 2, -6], [1, 2, 7], [-13, 4, 0], [8, 8, -4], [-1, 14, -1], [11, 2, 1], [8, -6, 4], [0, 8, -2], [5, 2, -1], [8, 2, 5], [6, 4, -4], [-2, 10, -2], [-8, -4, 2], [13, -4, 13], [0, -4, -2], [8, 8, 2], [1, -4, 9], [-6, 2, 4], [-5, -5, -5], [-24, 2, 0], [-5, 14, -3], [-11, 10, -8], [11, 8, -5], [-1, -14, 5], [-8, -8, -2], [13, 0, -3], [-4, -6, 2], [26, -2, 0], [3, 5, -9], [-19, 4, -4], [9, -14, -6], [-1, 3, -11], [-9, 12, -2], [-14, -7, 6], [-2, 4, 4], [7, 7, -5], [-22, 8, 8], [20, 0, 0], [0, 2, 8], [26, -1, 2], [9, -10, 3], [16, 8, -2], [4, -4, -10], [15, 7, 5], [5, 0, 13], [-14, 10, -8], [7, -4, 5], [22, -6, -4], [4, -8, 7], [-17, 13, 3], [11, 15, -1], [19, 6, 1], [6, -10, 0], [2, 4, 12], [-31, 2, 1], [-9, 6, -9], [-27, 6, 3], [4, 0, -10], [22, 2, 1], [19, 6, 2], [0, 6, 0], [1, 3, -9], [22, 0, 0], [10, 6, 2], [18, 1, 12], [-9, -10, 1], [-15, 16, -13], [-1, -10, -4], [15, 10, 4], [32, -4, -4], [-29, 2, -3], [-7, -14, 5], [-6, -2, 6], [6, -12, 4], [-10, 14, -7], [3, -7, 5], [31, 5, -5], [-14, 10, 5], [31, -4, 1], [1, -2, -3], [-3, -8, 3], [-9, 14, 1], [9, -7, 9], [25, -8, 15], [0, -10, -3], [1, -3, -3], [27, 6, -3], [-1, -1, -5], [-22, -14, 2], [-12, -4, -14], [-8, 8, -14], [-14, -14, 0], [-5, 2, 10], [1, 19, -5], [-17, 2, -5], [-2, 3, -6], [-31, 8, -1], [17, -6, -7], [-12, 3, -12], [-30, -2, -2], [-12, -18, 8], [33, 10, -2], [-19, -2, -13], [8, 14, 2], [29, -8, 12], [27, 0, -9], [20, -10, -2], [12, -4, 16], [4, -14, 4], [20, -10, 18], [-16, 14, -4], [16, 9, 0], [-4, 22, -9], [19, 6, 9], [-21, -2, 3], [-35, 8, 9], [7, -12, -11], [-23, -5, -5], [-9, -16, -6], [12, -12, 8], [10, -7, 12], [10, -10, -1], [-2, 18, -14], [-17, 14, -9], [-21, -11, 9], [9, 6, -11], [16, -12, -11], [29, -7, -7], [15, -16, -3], [-6, -4, -17], [-4, 14, 10], [33, -6, 15], [-12, -1, 0], [-10, 20, -4], [-15, 6, -5], [-8, 4, 0], [25, -18, 2], [14, -4, 14], [11, -16, 5], [-17, 20, 11], [-30, 1, -6], [-11, 16, -14], [-20, 14, -10], [6, -4, -4], [-26, -16, 4], [15, 15, -1], [19, 2, -11], [9, 0, -3], [-20, 20, 5], [2, 3, -2], [18, -12, 12], [-19, 1, -1], [2, 2, 8], [20, -2, 9], [-27, 4, -11], [-45, 0, 0], [34, -13, 12], [35, 7, 5], [0, -2, 4], [1, -14, 7], [27, -14, 9], [35, -3, -5], [5, -28, -1], [16, -14, -12], [-1, -8, 3], [1, 14, -1], [12, 20, -9], [15, -19, -13], [-10, -1, 8], [-4, 2, -3], [23, 4, 5], [-3, 12, -9], [9, -12, 9], [-18, 3, 12], [-27, -6, -5], [53, 3, 1], [3, -18, -2], [-21, 24, 3], [-25, 10, 11], [26, -5, -2], [15, -16, 9], [11, -18, -9], [4, -7, 10], [-9, 10, 5], [-13, -26, 8], [27, -12, 15], [10, -2, 11], [-6, -4, -6], [-62, 4, -2], [-8, 6, -16], [-15, 5, -1], [-7, 14, 11], [27, 8, -1], [3, 7, -13], [23, -5, 17], [30, 10, 10], [20, 14, -8], [-25, 20, 5], [-24, 8, -14], [-54, 2, 6], [-3, -2, 18], [-35, -3, -3], [22, -22, 10], [-4, -18, 12], [-28, -3, -8], [-5, -12, 3], [-32, -26, 8], [-24, -12, 8], [5, 7, 1], [13, -17, -5], [-2, -26, 2], [8, -10, 3], [-29, 10, -11], [23, 10, -7], [-7, 22, 3], [31, 0, 7], [15, -24, -14], [-32, 7, 6], [-8, -8, -8], [7, -22, 3], [2, -28, 14], [44, -12, 0], [-14, 16, -17], [-32, -10, -8], [13, 6, 19], [3, 18, 9], [11, -34, -1], [-25, -14, 13], [-8, -8, 19], [-1, -18, -3], [-24, 6, -18], [36, 0, 18], [31, -6, 9], [45, -6, 23], [-40, 18, 14], [43, -6, 23], [-65, 4, -5], [20, 10, 0], [-55, -8, 7], [22, -17, -2], [-11, -10, 8], [-41, 16, -11], [12, -24, -4], [33, 18, 3], [-45, 0, -6], [2, 26, -10], [-6, -7, 8], [21, -22, -9], [-5, 6, -16], [28, 4, -2], [9, 0, 21], [6, 6, 14], [16, -14, 16], [8, -6, -18], [23, -26, -5], [-44, -12, 10], [-9, 4, -1], [-40, 2, 8], [46, -18, 17], [9, -26, 1], [-56, 5, -18], [23, 11, -1], [-50, 14, -2], [-60, 12, -17], [-35, -16, 1], [-16, 6, -6], [-11, -6, 7], [40, 2, 8], [43, 3, 1], [10, -16, -13], [30, 2, 0], [2, -12, 3], [-11, -2, 15], [-28, 0, -20], [36, -4, 12], [12, 12, 15], [-24, 6, 4], [13, -16, -11], [14, 2, -16], [18, 16, -8], [-38, 17, -6], [24, 12, -6], [16, 0, 4], [12, -18, 16], [43, -2, 3], [2, 0, -1], [-30, -5, -4], [27, 0, -8], [31, 4, 7], [30, 22, -12], [30, -24, 6], [-22, -16, -10], [23, -21, 17], [-56, 2, -6], [-3, 20, 0], [-5, -20, 7], [7, 0, 19], [37, -18, -9], [33, -8, -7], [-47, 16, -29], [-3, -33, 17], [-16, -24, 18], [-25, 14, -13], [52, -14, 26], [-8, -13, -8], [18, 0, 9], [10, -8, -10], [-38, 26, -12], [-1, 16, -13], [-18, 18, 12], [-36, 26, -10], [28, -14, 7], [-21, 22, 5], [-7, 12, -10], [9, -12, 15], [-23, 17, 21], [-29, -6, -5], [1, 22, -19], [-1, 18, -19], [61, -10, 5], [-10, -20, 10], [8, 23, 16], [38, 11, -4], [44, 2, -3], [-31, 12, -1], [22, -18, 22], [-3, 26, -25], [46, -24, 4], [-2, 18, -6], [18, 10, 10], [18, -30, 6], [2, 4, 0], [-28, -22, 8], [-46, 16, -9], [-8, 4, -14], [21, -2, -13], [11, 16, 11], [-34, 6, -25], [9, 8, -11], [16, 12, 0], [23, -6, -4], [-12, -21, -4], [8, -22, 8], [10, 10, 12], [60, 6, -6], [-33, 13, 21], [-22, 26, 0], [-11, 36, -9], [25, 18, 4], [31, -10, 1], [-39, -14, -1], [12, -12, 12], [45, 5, -15], [6, 10, -9], [18, 22, -4], [-15, 30, -9], [-8, -18, -20], [-61, -8, 7], [-24, 28, 0], [-21, -1, -9], [28, 16, -2], [-7, 12, -27], [50, -20, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2025_2_a_o();".
// 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_2025_2_a_o(:prec:=3)
    chi := MakeCharacter_2025_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(2999) - 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_2025_2_a_o();".
// 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_2025_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -5, -1, 1]>,<7,R![-3, 3, 5, 1]>,<11,R![12, -8, -2, 1]>],Snew);
    return Vf;
end function;