
// Make newform 3025.2.a.i in Magma, downloaded from the LMFDB on 17 April 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_i();".
// 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_i();".
// 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_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 := [[0, -1], [1, 1], [0, 0], [-2, 3], [0, 0], [-3, -2], [-1, 1], [-3, 6], [4, -5], [3, -1], [-3, 0], [7, 2], [3, 0], [6, 0], [-1, 8], [-2, 7], [7, -4], [8, -5], [8, 0], [-8, 10], [-12, 1], [-1, -3], [15, -3], [-15, 5], [0, 1], [-2, 5], [7, -1], [1, 2], [-5, 5], [11, -14], [-1, 1], [-2, -5], [-9, 9], [13, -6], [-4, 8], [3, 0], [14, -12], [-2, -3], [15, -6], [-1, 4], [-12, -1], [2, -5], [-13, -5], [0, -8], [12, 5], [5, -5], [-17, 0], [-9, 16], [3, 13], [-9, 13], [12, -17], [-8, 1], [3, -15], [-3, -15], [2, -8], [21, 0], [19, 7], [3, -10], [5, 4], [-7, 20], [9, -1], [-2, 16], [7, -25], [-3, -10], [0, -2], [15, 1], [-3, -10], [18, -12], [-6, -9], [17, -24], [-11, 20], [-2, 4], [12, -3], [-19, 10], [-21, 2], [18, -8], [12, -4], [23, -5], [2, 20], [9, -18], [5, 10], [12, -15], [-17, 10], [7, -16], [25, -15], [-14, -14], [-4, -7], [-26, 21], [-7, 10], [-21, 20], [31, 4], [-9, -12], [-11, -12], [23, -10], [12, 11], [-16, 14], [18, -1], [-18, 10], [-10, -8], [18, 15], [26, -3], [13, -2], [-8, 13], [-28, 1], [-12, -15], [1, 9], [17, -13], [-9, 0], [-11, 7], [-2, 15], [-10, 4], [-21, 9], [-22, -10], [3, -16], [12, 15], [-3, 0], [30, -12], [35, -14], [-23, 19], [15, -15], [-33, -10], [17, -12], [-6, -14], [-11, -20], [22, 5], [-32, 20], [4, -28], [7, -14], [-17, -15], [-7, -24], [34, -3], [-30, 7], [-13, 15], [-15, -4], [-12, 0], [5, 0], [16, 11], [-31, 6], [21, -11], [36, -12], [-17, 0], [28, -20], [8, 12], [10, -26], [27, -39], [18, -41], [-45, -3], [37, 10], [0, 10], [35, -22], [-24, 12], [2, 5], [-9, -14], [23, -32], [-24, 24], [-18, 0], [-4, -12], [19, 22], [0, 24], [23, -40], [-7, -20], [-2, -4], [-13, -15], [32, -5], [15, 26], [-8, 4], [-3, 15], [-37, 3], [27, -14], [-6, 30], [-42, -1], [22, 5], [43, 5], [36, -5], [-11, -18], [2, -4], [8, 0], [-13, -20], [-12, 6], [-36, 12], [-10, -14], [-12, -5], [-13, -6], [6, 32], [1, 16], [21, -12], [-16, -14], [24, -15], [-7, 4], [8, 20], [-27, 27], [44, -16], [17, 0], [-3, 5], [22, -25], [-35, -2], [18, -30], [21, 6], [-2, 23], [-33, 33], [-58, 11], [38, -15], [31, 9], [-39, 33], [-26, -8], [-22, 5], [-32, 44], [-27, -24], [-26, 52], [-8, 35], [-3, -30], [37, -10], [30, -37], [30, -54], [35, -10], [32, -50], [8, -12], [13, -25], [24, -27], [28, -43], [33, -35], [29, -8], [-40, -20], [7, 9], [1, -38], [-25, -30], [56, -19], [46, -22], [24, -24], [23, -25], [44, -15], [33, 24], [33, 20], [-17, 10], [-27, 7], [15, 4], [32, 6], [4, -41], [33, -11], [42, -35], [15, 18], [8, 30], [-4, 6], [51, 8], [12, -22], [-8, -19], [6, 29], [12, -15], [-49, 33], [6, 10], [-30, -26], [-57, 20], [28, 10], [3, -36], [-22, 31], [24, -58], [7, -30], [-33, 40], [30, 1], [-22, 28], [32, -32], [24, 6], [7, 36], [-9, -25], [29, -2], [44, -38], [-23, -4], [-48, 25], [3, 26], [25, 22], [62, -30], [-45, 6], [-18, 54], [-35, 30], [-21, 21], [-1, 40], [-24, -16], [33, 9], [-2, 45], [-7, 20], [39, 19], [32, 0], [42, -15], [43, -30], [56, -33], [32, -20], [1, -9], [-12, 33], [-24, 33], [27, 1], [17, 40], [-3, -8], [-24, 40], [3, 15], [-11, -6], [4, -23], [-3, 0], [-23, -26], [50, 20], [23, -12], [-33, 28], [-6, 36], [-48, 11], [69, 0], [7, 45], [-3, -38], [-6, 38], [24, -33], [-23, 1], [-17, 66], [35, -53], [-15, -35], [-32, 55], [-33, 4], [-29, -38], [-20, 30], [6, 8], [-12, 10], [34, -30], [66, -27], [-42, 5], [-6, 53], [63, 0], [-27, 42], [-71, 19], [-38, 60], [5, -15], [-74, -9], [-22, 38], [42, -52], [-52, 30], [-33, 12], [51, -27], [-39, 30], [48, -50], [-42, 40], [-22, -6], [28, -34], [32, -65], [52, -15], [0, -42], [55, 16], [-35, -20], [12, -15], [-61, 20], [-47, 29], [-3, -15], [-53, -18], [33, -55], [-51, 8], [28, -55], [8, -15], [-33, -30], [73, -9], [44, 12], [-15, -18], [10, -20], [-12, -25], [-29, 52], [16, -4], [24, -69], [-57, -20], [-49, 34], [-9, -7], [-46, 3], [-47, -18], [-10, -46], [15, 2], [28, -40], [-83, 25], [-4, 48], [2, -47], [77, 11], [-47, 65], [-39, 24], [-56, 7], [-42, 35], [-46, 4], [-16, -53], [-38, 45], [82, -10], [7, -26], [28, 18], [-61, -19], [87, -24], [-21, 15], [-18, -25], [-48, 72], [21, -60], [34, -32], [4, -13], [68, -28], [-3, -24], [-42, 30], [37, 10], [-37, -24], [19, -3], [-20, 10], [-53, 20], [48, 5], [-40, 60], [26, 11], [34, 21], [81, -36], [51, -12], [-47, 0], [24, -62], [13, 10], [8, -48], [-62, 14], [-45, 57], [-43, 50], [15, -2], [28, 35], [6, -8], [47, -10], [-73, -4], [31, -21], [-33, -18], [-79, -5], [-21, -38], [-75, 24], [-27, 50], [-61, 32], [40, -42], [0, -24], [-18, 4], [-25, 65], [-18, 60], [23, -71]];
    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_i();".
// 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_i(:prec:=2)
    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_i();".
// 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_i( : 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, 1, 1]>,<3,R![1, -3, 1]>,<19,R![-45, 0, 1]>],Snew);
    return Vf;
end function;