
// Make newform 3675.2.a.bj in Magma, downloaded from the LMFDB on 26 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3675_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_3675_2_a_bj();".
// 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_3675_2_a_bj();".
// 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, -3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-2, 0, 1], [2, 2, -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_3675_a();"
function MakeCharacter_3675_a()
    N := 3675;
    order := 1;
    char_gens := [1226, 1177, 2551];
    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_3675_a_Hecke(Kf)
    return MakeCharacter_3675_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], [0, 0, 0], [0, 0, 0], [2, 0, 0], [2, -1, 1], [0, 1, -1], [-2, -1, 1], [2, -1, -1], [2, -2, -2], [-2, 3, 1], [0, -4, 0], [0, -3, 1], [0, 0, 4], [-4, 2, 2], [6, -3, -1], [-4, 0, -4], [2, 2, -2], [4, -2, -2], [2, 0, 0], [6, 1, -1], [4, 2, -2], [-4, 4, 0], [-4, -1, -1], [10, -5, -3], [8, 1, -3], [0, -2, 2], [10, 3, -1], [-2, -4, 0], [2, 3, -3], [-4, 2, -2], [-8, 0, 4], [14, -5, -3], [-6, -1, -3], [-6, 8, 0], [8, -2, 2], [6, -1, 1], [12, -2, 2], [-4, -2, -6], [-4, -5, -3], [10, 0, 0], [2, 0, 4], [-2, 4, 4], [-4, 2, -6], [2, 3, 1], [2, 7, 1], [8, -8, 0], [0, -8, 0], [8, -6, -2], [-2, -4, 0], [10, -3, 3], [-6, -4, -4], [6, 2, 2], [-20, 4, 0], [-4, -1, 1], [-10, 1, 1], [4, 7, 3], [2, -7, -5], [0, 8, 4], [10, -10, -2], [4, 6, 2], [-4, 1, -1], [-4, -6, 6], [0, -10, 2], [10, 9, -1], [6, 5, -1], [16, -4, -4], [8, 4, 4], [-6, -1, -5], [10, -6, -2], [0, -9, -7], [-18, -4, 4], [16, 4, -4], [16, 0, 0], [4, 2, -2], [16, 4, 0], [-10, -2, 6], [2, 9, 7], [-18, -2, 6], [18, -10, -2], [-16, -6, 2], [-26, 4, 0], [-6, -4, 4], [2, 5, 3], [18, -1, 1], [2, 7, -5], [-6, -8, 8], [-12, -2, 2], [4, -1, -1], [12, -2, 2], [8, -6, -2], [0, 6, 2], [-8, -8, 4], [2, 0, 0], [12, 2, 6], [-8, -4, 0], [4, -7, 5], [28, 1, -3], [4, 0, 0], [2, 4, 8], [-20, 2, 2], [6, 1, 11], [16, 10, 2], [-10, 2, -6], [4, 8, 8], [-14, 17, 7], [-16, 18, 2], [0, 5, 11], [-10, 12, 4], [2, -4, -4], [0, 0, -8], [-4, 6, 2], [26, -3, 3], [-22, 7, -3], [4, 2, 6], [-6, -6, 10], [-20, 2, -10], [-4, -2, -6], [34, 1, -5], [-2, -12, 4], [-6, 14, -6], [0, 4, -8], [12, 7, -7], [-14, -1, 3], [-2, -3, -1], [10, 6, -10], [2, 0, -8], [4, 2, -2], [-8, -2, 10], [10, -9, -7], [12, 0, -8], [10, 3, -5], [12, 4, -12], [24, 12, -4], [-20, 1, 9], [-6, 8, 0], [-4, -3, 11], [-4, 16, 0], [12, 3, 5], [6, -8, 0], [2, 7, 13], [-26, 0, -8], [4, 6, 10], [-22, -5, -9], [-22, -4, 0], [-4, 4, 4], [-6, -7, -9], [-20, -9, 9], [2, 1, -1], [2, -5, 11], [12, -10, 10], [-16, -3, 5], [-16, 4, -4], [-4, -14, 10], [-16, 20, 12], [-10, -12, 4], [12, 8, 8], [-12, 21, 5], [-2, -1, -7], [-16, 1, -7], [-26, 9, 5], [34, -13, -3], [4, 14, -10], [-4, 6, -10], [-6, -3, 11], [24, 0, 0], [-12, -4, -12], [-14, 17, 7], [2, 8, 8], [12, -3, -13], [34, -4, -4], [-2, -2, -14], [-14, 8, 0], [-28, -10, 6], [30, -7, -9], [-36, -5, -1], [0, 10, -2], [6, 10, -6], [-16, 18, 6], [-18, -14, 14], [-12, 18, 2], [-4, -10, -2], [4, 14, 6], [16, -13, -3], [-26, -3, -3], [0, 3, -9], [4, 6, -6], [-4, -4, 4], [-10, 4, -12], [-4, -2, -6], [34, -5, 5], [-38, 3, 7], [-12, -2, 2], [-4, 1, -15], [-2, -23, 5], [-24, 1, -1], [2, 0, 0], [4, 6, -10], [4, 3, -3], [20, -14, -2], [14, 8, 0], [42, -9, 1], [22, 5, 19], [-26, 18, 10], [-28, 10, 2], [28, -13, -11], [-2, 21, -5], [-6, 7, -13], [-2, 16, 0], [10, -23, -9], [-20, -18, 2], [-20, 21, 9], [0, 4, 12], [-20, 8, -12], [-4, 8, -8], [-14, 12, 12], [24, 4, 0], [8, -5, -5], [14, -19, -11], [-6, 25, 3], [18, -8, 0], [22, 15, -3], [-10, 12, 4], [-8, 4, 4], [12, -4, -4], [-22, 28, 4], [4, 7, 9], [-10, 4, -12], [-48, 14, 2], [10, -4, 20], [24, -12, 4], [14, -1, 17], [0, 2, -18], [38, -8, 0], [-20, 10, 6], [24, 0, 16], [2, 14, -10], [-22, -17, 13], [-14, 20, -4], [36, -6, 6], [-26, 1, 5], [2, 7, 13], [-40, 4, 4], [30, -24, -8], [-20, 15, 1], [52, -4, 4], [-16, 2, -10], [-32, -2, 6], [32, -10, 2], [-22, -9, -1], [20, -18, 2], [24, -17, 7], [22, -7, 1], [18, 6, -10], [20, 9, -1], [-6, 0, -16], [-46, 0, -8], [20, 22, -14], [44, -15, -9], [26, -7, -1], [28, -34, -6], [-46, 11, -1], [30, -16, -12], [62, -13, -3], [-24, 9, -9], [-34, 13, 7], [-2, 4, -4], [-28, 9, -3], [-4, 6, -18], [18, -3, -9], [-42, 18, 6], [0, -12, -16], [-14, -5, -3], [-44, 4, 4], [26, -1, 1], [0, -10, -14], [10, -1, 11], [-50, 4, 8], [10, -8, 12], [60, -20, -8], [40, -20, -4], [-28, -6, -2], [0, -8, -24], [22, 2, -2], [-36, -12, 12], [14, -16, 0], [-12, 14, 10], [-10, -11, -9], [-2, -13, 17], [12, -7, -7], [54, 12, -4], [20, 0, 12], [-22, -3, 11], [8, -26, -2], [16, 16, -8], [-12, -5, 3], [14, -15, -9], [8, 19, -3], [28, 4, 16], [20, 8, 8], [22, 15, -7], [-22, 9, 19], [-24, 4, 4], [-26, -7, -11], [-68, 0, 0], [8, 12, 20], [6, 9, -3], [22, 0, 12], [-30, 4, -12], [-12, -10, -18], [-16, 12, 4], [34, 6, 6], [-2, -18, -10], [24, 4, -4], [-30, 11, -5], [38, -10, -2], [-48, 28, 8], [-42, 16, 8], [-14, -5, -3], [-34, -6, -6], [18, 9, -17], [-48, 4, 8], [-28, -7, -15], [-40, -4, -4], [14, 21, -5], [-26, 2, 10], [-12, 24, 16], [20, 2, -10], [30, 9, 1], [-34, 3, 1], [18, 28, -8], [26, -19, -9], [2, 21, -1], [-48, 14, 6], [0, 26, -10], [-24, -14, 10], [6, -4, -12], [-12, -15, -1], [6, -10, -2], [8, 10, -10], [40, -4, -4], [58, 9, -1], [-12, -29, 15], [-8, -18, -6], [-22, -11, -25], [-18, 8, 8], [-10, -12, 8], [-8, 4, -12], [-4, 22, 2], [28, -3, 19], [-34, 23, 9], [-24, 4, -8], [22, 4, 28], [-16, 4, -12], [-14, 4, 24], [-8, -25, -7], [-36, -6, -18], [-24, 36, 0], [18, 19, -11], [2, -17, 7], [40, -12, 4], [0, -17, 3], [2, -9, 7], [-18, 12, -4], [52, -16, -8], [-24, 8, -4], [32, 9, 7], [-52, 22, 6], [26, 32, -12], [18, -28, -20], [-30, -11, 3], [-30, 11, 11], [-6, 16, 16], [50, -3, -9], [-32, 8, 12], [-40, -12, 16], [14, 12, -4], [-38, 17, 7], [20, 13, 1], [-10, 25, 7], [24, 9, 29], [50, 7, -7], [-64, 4, -8], [22, -7, -9], [-38, -29, 9], [0, -8, -20], [4, 20, 12], [-30, -17, -7], [-4, -18, 14], [-12, 22, 14], [-34, 4, 4], [-16, -21, 13], [54, 8, 0], [4, -6, 14], [42, 16, -8], [-16, 0, 8], [-2, 5, 3], [-8, -7, -7], [20, 4, -4], [34, -18, 6], [-34, 2, -10], [10, -29, -19], [60, -10, -2], [20, -35, -13], [40, 17, 9], [-42, -7, -9], [72, 4, 8], [-38, -8, -16], [20, -28, -20], [12, 12, -24], [-26, 15, 25], [-58, 15, 5], [18, -5, 7], [-16, 16, -16], [20, -10, -2], [28, -13, -1], [10, 4, -12], [-8, 6, 26], [-40, 11, -11], [-4, -2, -2], [38, -22, -6], [34, 1, -17], [-14, -5, 19], [-20, -6, -22], [2, -15, 23], [-4, 21, -5], [-66, 5, -7], [30, -18, 14], [-18, 9, -13], [-28, 20, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3675_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3675_2_a_bj();".
// 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_3675_2_a_bj(:prec:=3)
    chi := MakeCharacter_3675_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_3675_2_a_bj();".
// 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_3675_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3675_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -5, -1, 1]>,<11,R![-2, 1]>,<13,R![8, -4, -6, 1]>],Snew);
    return Vf;
end function;