
// Make newform 2275.2.a.j in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2275_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_2275_2_a_j();".
// 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_2275_2_a_j();".
// 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 := [-2, 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_2275_a();"
function MakeCharacter_2275_a()
    N := 2275;
    order := 1;
    char_gens := [1002, 976, 1926];
    v := [1, 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_2275_a_Hecke(Kf)
    return MakeCharacter_2275_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, -1], [0, 0], [-1, 0], [0, 3], [1, 0], [0, -1], [-3, -3], [3, 2], [3, -2], [-1, 3], [2, -3], [6, 2], [5, 0], [-3, 1], [3, -2], [6, -4], [6, 0], [6, 6], [-6, -5], [5, 3], [7, 6], [-9, -3], [3, -1], [1, -9], [6, 3], [8, 0], [6, 10], [4, 9], [9, -8], [-2, 0], [0, -2], [6, -1], [2, 3], [12, 3], [-14, 3], [8, -3], [0, -6], [3, -13], [-12, -9], [-9, 0], [-6, -9], [18, -2], [6, 6], [-18, 4], [-8, 3], [1, -12], [-5, 3], [-24, -2], [4, 18], [9, 4], [12, -6], [1, -15], [18, -1], [18, 1], [15, -12], [12, -2], [20, 0], [-1, 6], [0, 11], [0, 6], [3, -13], [-9, -3], [-6, -1], [-2, -15], [0, 8], [-18, 0], [33, 0], [0, 4], [-13, -9], [0, 6], [-18, -7], [-6, -3], [0, -6], [28, -3], [12, 6], [24, 4], [-3, 15], [12, -4], [-1, 3], [6, -19], [-6, 9], [-18, 4], [8, 12], [-26, -6], [15, -8], [24, -2], [-14, 15], [6, 2], [0, 3], [-6, 7], [15, -15], [-14, 18], [-6, 16], [-26, 9], [6, 16], [-15, 7], [-12, 4], [-14, -12], [14, -15], [-27, -6], [-12, 0], [12, -4], [-9, 10], [17, 18], [-10, 12], [-9, 11], [-15, 11], [-9, -14], [-22, -12], [14, 9], [16, 0], [-18, 4], [-16, 12], [2, 0], [21, -4], [4, 6], [18, -23], [-6, -8], [-21, 4], [-15, 3], [-7, 24], [6, 25], [24, -2], [-1, 21], [9, -14], [-12, 9], [-6, 3], [-42, 0], [-13, 21], [12, -21], [24, 4], [7, 6], [-13, -6], [39, -3], [31, 15], [30, 2], [29, -3], [30, -4], [9, 6], [12, -24], [-6, -10], [-2, -6], [24, 18], [38, 9], [-36, 2], [-27, -9], [18, 26], [14, 12], [0, -20], [6, 3], [-18, 22], [-2, 0], [12, -22], [15, -12], [-27, -12], [24, 0], [27, 7], [-24, -15], [33, 5], [-18, 2], [3, -10], [12, -21], [18, 18], [0, -17], [-9, 17], [18, -12], [20, -6], [12, 15], [12, 13], [12, 7], [10, -6], [15, -24], [-22, 27], [16, 18], [-24, 20], [-16, -6], [42, 10], [-22, 9], [10, -15], [8, 6], [18, -15], [-4, 18], [3, -25], [0, -17], [-6, 13], [43, -12], [17, 21], [2, 30], [-15, -25], [16, -9], [30, -22], [-47, -12], [-9, 23], [-15, -22], [12, 3], [-16, -21], [-33, -24], [15, -9], [-42, 2], [18, -5], [32, -15], [17, 3], [8, -36], [-21, -7], [-12, 30], [-49, -9], [-27, -8], [30, -7], [-22, 24], [-19, -24], [-36, -19], [40, 0], [-12, -30], [21, 29], [-5, -27], [-11, -30], [-12, -10], [12, -2], [18, 9], [19, -18], [6, -12], [0, -22], [30, -24], [30, 10], [-3, 24], [0, 16], [-27, -20], [66, 3], [-24, 19], [24, 12], [-24, 21], [-40, 0], [12, -28], [18, 6], [-15, 29], [-6, -18], [-12, 32], [-33, 28], [42, 14], [-48, -12], [2, -6], [-16, 24], [26, -18], [-45, -1], [0, 19], [-43, 9], [-9, 19], [-22, -9], [9, -22], [16, -36], [-30, 2], [-6, 2], [2, -18], [6, -20], [12, -18], [11, -12], [-60, 0], [-24, 6], [11, -27], [11, -18], [3, 32], [19, -9], [14, -21], [-18, -28], [10, -36], [-30, -34], [-9, -3], [-8, 33], [24, 26], [50, 3], [-10, -39], [-6, -24], [-13, 18], [68, -3], [20, -24], [30, -3], [-2, -21], [-24, 12], [36, 24], [-6, 18], [30, -36], [-12, 24], [-17, -15], [-50, -24], [-33, 16], [25, 24], [12, 0], [8, 6], [-48, 5], [9, 34], [-36, 22], [-30, -16], [-27, 19], [-7, -18], [48, -4], [2, -6], [-12, -4], [-24, 25], [17, 15], [-6, -21], [-6, -43], [-3, -12], [-33, 6], [3, 0], [-28, 3], [-6, 2], [-24, -15], [-42, 20], [23, -24], [-36, -3], [48, -9], [36, 32], [39, -18], [54, 7], [-20, -36], [39, 13], [24, -31], [49, -21], [-3, 10], [4, 42], [-32, 30], [54, -17], [-32, 18], [0, 32], [-56, -9], [-4, 0], [-17, -27], [-6, -8], [-48, -20], [4, -51], [75, 8], [-34, 33], [-27, 11], [16, 27], [15, -51], [38, -9], [54, -22], [8, 54], [-14, 30], [-26, -21], [-12, -22], [-12, -18], [26, 36], [6, 3], [30, 0], [-2, 33], [42, -27], [-27, -37], [-18, 3], [-11, -15], [48, -15], [54, 1], [-22, 36], [-4, 42], [6, 51], [-27, 17], [-15, 31], [24, 2], [12, -21], [-62, -6], [-42, -2], [-39, -30], [-6, 25], [-34, -45], [-11, -12], [-63, 7], [-4, -9], [-47, 0], [42, 10], [30, -27], [-54, -31], [15, -12], [-32, 9], [-3, 0], [-12, -10], [42, -10], [-16, -18], [-48, -18], [14, 0], [-9, -23], [-18, 46], [-78, -9], [30, 7], [7, -63], [-21, -15], [14, 3], [8, 60], [26, -9], [12, 11], [-38, -18], [15, -7], [-24, 2], [28, 30], [60, 12], [-55, -12], [12, 12], [6, -12], [9, 0], [0, -18], [21, 33], [39, -14], [-52, -21], [39, -13], [45, 23], [12, -33], [48, 6], [18, -53], [-23, 3], [27, 7], [-40, 12], [33, -16], [9, -8], [-6, 18], [7, 66], [12, 23], [24, -42], [-20, 18], [54, 4], [0, 9], [-27, -48], [-8, 18], [-54, 13], [0, 35], [34, -12], [15, 11], [33, -16], [-24, -18], [59, 9], [-42, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2275_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2275_2_a_j();".
// 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_2275_2_a_j(:prec:=2)
    chi := MakeCharacter_2275_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_2275_2_a_j();".
// 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_2275_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2275_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 0, 1]>,<3,R![-2, 0, 1]>],Snew);
    return Vf;
end function;