
// Make newform 3025.2.a.v in Magma, downloaded from the LMFDB on 07 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3025_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_3025_2_a_v();".
// 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_3025_2_a_v();".
// 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 := [5, 5, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [0, -3, -1, 1]];
    Rf_basisdens := [1, 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_3025_a();"
function MakeCharacter_3025_a()
    N := 3025;
    order := 1;
    char_gens := [727, 2301];
    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_3025_a_Hecke(Kf)
    return MakeCharacter_3025_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 := [[-1, 0, 0, -1], [0, 1, 0, 0], [0, 0, 0, 0], [-3, 0, -1, 1], [0, 0, 0, 0], [-1, 0, 2, -1], [-1, 1, 0, 1], [4, -1, 1, 0], [2, -3, -3, -1], [0, -1, 3, 0], [1, -2, -2, 1], [1, -1, -2, 3], [-3, 1, 0, -3], [-3, -2, 2, 1], [-1, 3, -1, 1], [2, 1, 0, -1], [-5, 2, -3, -3], [2, -2, -1, 2], [2, 0, 2, 3], [-6, -2, -8, 3], [4, 2, 6, -1], [-2, 3, -5, 2], [-5, -1, -3, -1], [3, -2, 6, -4], [1, -2, 2, 2], [-8, -1, -6, -1], [5, -4, 2, -2], [1, -7, 6, -2], [2, 2, 0, 3], [1, -1, 5, -7], [-12, -1, -12, 2], [-5, -4, -6, -1], [9, 2, 9, -1], [-2, 9, 2, -3], [-10, 5, 0, 0], [1, -1, -4, 0], [-1, 2, 4, -1], [7, 1, 1, -2], [-8, 0, 3, -3], [-8, 0, -13, 5], [5, 6, 2, 0], [9, -2, -2, -3], [4, -4, 12, -1], [-11, 3, 1, 1], [4, -7, 2, -7], [-1, -3, -7, 4], [14, -4, 0, 1], [-6, -2, 0, 2], [-11, 7, 8, 0], [-1, 5, 13, 5], [5, 0, 7, 1], [-8, 1, -17, 3], [4, 4, 13, -6], [0, 1, -4, -8], [11, -1, 13, -7], [-2, 8, 4, 1], [-3, 0, 0, -1], [2, -10, -1, 1], [4, -2, -7, 12], [-3, 3, 13, -2], [-16, 0, -9, 5], [-13, 0, -11, -7], [-14, 5, -8, 1], [-5, 5, 3, 3], [-13, 5, -5, -8], [0, -1, -1, -5], [-10, 0, -8, 4], [3, 4, -7, 2], [-4, 2, -1, 3], [5, -6, 10, 3], [17, 0, 5, -3], [-5, 8, 4, 2], [-24, -1, -5, -4], [2, -7, 10, -2], [-4, 4, 7, -2], [-4, -5, 8, -3], [-1, -10, -4, 2], [-11, 6, -7, -2], [-1, 13, -6, 1], [19, -10, 5, 8], [-7, -7, 2, -5], [2, -6, 7, 1], [-6, -1, 6, 1], [1, 5, -2, 12], [1, 5, 3, 14], [-24, 8, 3, 3], [-11, 5, -6, -6], [19, -4, 11, 0], [-16, 5, 1, 3], [0, -9, 8, 2], [0, 0, -5, 6], [-2, -9, 23, -5], [11, -3, -2, -6], [-6, 6, -6, -3], [9, -1, 10, 6], [-4, 1, 5, -3], [-20, 4, -3, -9], [20, -6, 17, -3], [-17, 13, 1, -6], [6, 12, 5, 2], [-19, 13, -9, -2], [20, -11, -5, -6], [-14, 5, -21, -4], [-7, -11, -1, -5], [7, -6, 2, 1], [-2, -7, -3, -9], [22, -11, 1, 12], [3, 9, -5, 2], [-23, 3, 1, -1], [-11, 14, -6, 13], [10, -7, 11, -9], [9, -13, -9, -1], [6, 7, -14, 11], [12, 1, 4, 12], [-3, 0, 24, -4], [5, 2, 2, -12], [20, 1, -7, 2], [-7, 0, -19, 16], [11, -3, 27, -3], [-23, 0, -23, -3], [-1, 11, 13, -12], [-13, -7, -18, 6], [9, -10, -8, 2], [-1, -3, 1, 9], [-11, 5, -16, 20], [-9, 13, -12, 11], [-8, -3, -11, 4], [1, -1, 7, -7], [15, 14, 24, -10], [8, -8, 25, -2], [21, 1, -1, 10], [-8, 10, 4, 8], [-17, -8, 19, -8], [2, -10, 16, 4], [-11, 5, -17, -9], [13, -2, 10, -18], [-10, 14, -27, 3], [-10, -2, 19, -12], [-26, 5, -6, 10], [-7, 2, -30, 10], [18, 5, -10, -3], [-11, 6, -1, -13], [12, 2, 17, -6], [-14, 19, -4, 5], [9, 5, 22, 11], [3, 7, 5, 10], [-6, -13, -13, 8], [-6, 15, 12, 12], [-7, -7, 5, -3], [-27, -2, -21, 6], [7, 0, 10, -20], [7, -9, 8, 4], [9, 6, 11, 7], [4, -9, -10, -4], [-21, 10, -2, -9], [26, 6, 6, 4], [-14, 6, 4, -20], [9, -17, -4, 8], [22, 11, 11, -9], [5, -5, 9, -18], [-18, 8, -11, -3], [-18, -1, 0, 10], [-19, 2, -37, -1], [36, 0, 5, 3], [-12, 5, 16, 1], [11, -8, -1, 0], [-29, 9, -29, -5], [1, 12, 5, -3], [-3, 7, -3, 6], [-15, -1, -15, -14], [25, 2, -1, 20], [-4, 1, 15, -10], [-15, -9, 5, 8], [-13, -3, 4, -18], [25, 6, -14, 11], [14, 12, 13, -1], [37, -15, -3, -2], [-21, -5, -4, 8], [17, 4, -8, -2], [-9, 3, -30, -7], [-18, 5, -33, 3], [-16, -1, 2, 0], [-3, 11, -22, 16], [9, 8, 20, 0], [-17, 6, -21, 12], [-2, 6, 0, 18], [-24, -2, -25, -1], [-10, -12, -28, 2], [18, -11, 0, -1], [-37, -1, -28, 13], [-16, 0, -43, -1], [2, 5, 15, -8], [20, -3, 33, -18], [-32, 10, 1, -4], [-1, 6, 14, 7], [-15, 3, 14, -10], [17, -7, -7, -7], [8, -6, 2, 15], [-12, 4, -33, -2], [-8, -1, 3, -3], [21, 9, 21, -16], [18, -12, 26, -8], [26, -14, 11, 9], [-22, 9, 21, 8], [17, -4, 25, 11], [23, -8, -4, -15], [-3, 11, -6, 17], [20, 8, -6, 15], [31, 6, 7, 12], [3, -6, 4, -29], [0, 1, 14, -11], [-12, 19, -18, 13], [9, -13, 21, -16], [0, 22, 6, -6], [-31, 16, -1, -4], [-3, 19, -4, 12], [0, 9, 13, 11], [20, 0, -13, 19], [43, 0, 38, -1], [-24, 7, -12, -9], [4, -23, -2, -5], [-24, 6, 6, 3], [32, -10, 9, -15], [-6, 3, 37, 0], [-23, 16, -10, -13], [30, -21, -4, 2], [25, 2, 8, -11], [-20, -7, -18, 14], [15, -15, 11, 10], [12, 10, -4, -1], [-34, 18, 10, -4], [-20, 9, 15, 3], [-21, 12, 3, -10], [37, -8, 7, 2], [-20, 6, -32, 22], [-33, 12, 10, -6], [-18, 5, -18, 7], [12, 9, 11, -6], [20, -24, -5, 7], [19, 14, 3, -1], [-5, 6, -27, 12], [-9, 10, -35, 28], [2, -3, -34, 0], [-26, 18, 9, 5], [38, -16, 26, 11], [-24, 18, 13, -5], [5, 7, 24, 6], [-20, -11, 13, -4], [45, 6, 4, 8], [6, 6, -32, 14], [1, 16, -4, 24], [3, 16, -1, 20], [-28, -4, -35, 12], [-8, 9, 8, 14], [-30, 15, -7, 0], [-1, 14, 1, 20], [13, 16, 19, -5], [-42, -1, 13, -8], [5, -1, 38, 1], [20, -6, 24, 0], [36, -8, 21, -2], [-17, 5, 10, -13], [27, -24, -3, -21], [-5, 18, -23, -3], [-49, 6, 8, -8], [23, -25, 11, 5], [42, -10, 16, 3], [-22, 11, 7, -3], [-18, -14, -12, -4], [21, 9, 31, -8], [-1, -10, -20, -6], [24, -20, -5, 4], [-17, 9, -24, -9], [-20, -7, 17, 4], [25, 2, -17, 12], [40, -2, 4, 20], [14, -31, -17, 9], [-8, 8, -45, 15], [33, -18, 5, -24], [-17, -6, 16, -1], [11, 0, 6, 24], [19, 11, 6, -15], [-15, -1, -34, 23], [11, 3, 18, 6], [-10, -29, -26, 9], [14, 15, -15, 9], [7, 10, 12, 0], [-22, 5, -43, 5], [14, -2, -17, 6], [-52, -3, -8, 3], [-20, -17, -4, -9], [-33, 11, -11, 8], [1, -15, -36, 6], [35, -24, 8, -10], [19, 7, 26, 4], [46, -3, 7, 4], [-27, 3, -19, 12], [10, 7, 11, 6], [-45, 9, -22, -10], [-46, 14, -6, 3], [-46, -1, -5, -6], [-2, 3, -5, -17], [-14, -7, -20, 18], [-17, -8, 15, -32], [-36, -13, 3, -5], [30, -31, -25, -11], [32, -8, 24, -25], [-15, 7, 5, -1], [-32, 13, -23, 0], [8, 1, -26, 17], [7, -12, 16, -3], [10, -7, -26, 7], [31, 20, 32, -14], [41, -19, 11, 1], [12, 2, -13, -13], [11, 15, 16, -14], [3, -25, -12, 13], [25, -24, 15, -7], [19, 4, 10, 9], [-1, -4, -2, -28], [24, 0, 9, -25], [-18, 21, 13, -6], [-3, -6, -31, 9], [0, -2, -17, 1], [-46, 3, -5, -18], [-8, -15, 40, -15], [8, -5, 21, -15], [-9, -5, 22, -37], [43, -4, 48, 6], [-14, 15, -9, -1], [-11, -23, 4, -6], [24, -12, 4, 26], [-34, -6, -24, 29], [23, 1, 2, -21], [64, -9, 1, 5], [-17, -6, 3, 12], [-52, 8, -32, -21], [15, 1, 2, 13], [-10, -28, -22, 17], [42, -14, 14, 18], [-22, 1, 13, -22], [0, -7, 21, -9], [-3, 8, 9, -9], [35, -12, 20, -3], [24, 4, 29, -15], [-22, -1, 12, -38], [14, -24, -30, -9], [22, 4, 57, -9], [-55, -12, -29, 16], [-43, 5, 1, 19], [-6, -19, -41, -3], [-11, 32, 23, -5], [2, 15, -1, 26], [-15, -13, -47, -6], [-8, -2, 10, -25], [23, -28, -19, 2], [-28, 3, -3, 8], [-22, -8, 21, -35], [10, 11, 26, -18], [-20, 5, -44, -3], [17, -21, -16, 4], [-4, 34, -7, 3], [-18, 2, 0, 29], [-41, -10, -17, -5], [-23, -13, 9, -20], [6, -8, 26, -42], [6, -6, 26, 15], [-3, -3, -3, 29], [-25, 6, 0, -7], [-21, -11, -16, 3], [-36, 27, -15, -13], [12, -14, -9, 13], [17, 16, -35, 16], [-46, -20, -23, 7], [-8, 8, -22, 3], [-8, -7, -3, -22], [-39, 1, -9, 9], [63, -4, 14, 7], [23, -2, 56, -9], [6, 12, 13, -31], [-6, 16, 30, -9], [-9, -30, -8, -2], [26, 12, 16, -9], [-51, 3, -1, 6], [30, -30, -6, 1], [24, 15, 19, 7], [-40, 23, -8, 15], [8, 19, 14, -14], [0, -30, -20, 8], [22, -2, -4, 30], [-23, 2, -26, 9], [-36, 33, -18, 0], [37, -13, 14, -11], [-62, -7, -12, 5], [-5, 17, -26, -4], [-8, 20, 44, 3], [14, -10, 12, -34], [20, -38, 2, 2], [-5, -4, 3, 5], [17, -21, -6, 29], [-16, 31, -15, 10], [-16, 15, -39, 15], [26, 4, 36, 5], [-1, 15, 11, 3], [-41, -1, -5, -9], [6, 10, 3, 0], [16, -27, 18, -14], [42, 17, 30, -26], [-3, 0, 24, -2], [51, 6, 6, 3], [-10, 14, -11, 17], [42, -14, 43, 4], [12, -5, -30, -8], [74, -15, 11, 3], [47, 10, 6, 4], [-11, 17, -29, 7], [32, -41, -7, 9], [-38, -2, -42, -17], [42, -13, 31, -12], [-38, -3, -64, 15], [16, 27, 45, -4], [33, 2, -26, 7], [-38, -3, -30, -11], [-38, -4, 12, -6], [27, -14, 36, -27], [-32, -2, -17, -15], [-29, 5, -47, 17], [56, -13, 24, 27], [-36, -19, 24, -23], [5, 37, 17, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3025_2_a_v();".
// 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_3025_2_a_v(:prec:=4)
    chi := MakeCharacter_3025_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_3025_2_a_v();".
// 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_3025_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -11, -3, 3, 1]>,<3,R![5, 5, -4, -2, 1]>,<19,R![25, -65, 46, -12, 1]>],Snew);
    return Vf;
end function;