
// Make newform 7623.2.a.bd in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7623_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_7623_2_a_bd();".
// 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_7623_2_a_bd();".
// 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, -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_7623_a();"
function MakeCharacter_7623_a()
    N := 7623;
    order := 1;
    char_gens := [848, 4357, 4600];
    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_7623_a_Hecke(Kf)
    return MakeCharacter_7623_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], [1, 0], [1, 0], [0, 0], [1, -2], [-4, 0], [-3, 0], [2, 0], [1, 2], [-2, 0], [1, -4], [-2, 4], [-4, 4], [7, -2], [0, 0], [3, 2], [2, 4], [-3, 2], [2, 4], [-1, 6], [-6, 4], [-12, 0], [6, 0], [-8, 4], [6, -8], [0, -4], [-7, 4], [-10, 0], [4, -4], [10, 0], [-6, 0], [-12, 8], [4, -8], [17, 2], [0, -8], [-14, 4], [1, -6], [-14, -4], [-2, 8], [2, -8], [-10, 8], [-14, 4], [-8, -8], [-6, -4], [-8, -4], [8, -8], [4, 0], [-2, 8], [2, 4], [6, -12], [-17, 8], [-15, 2], [21, -2], [9, 4], [-3, -8], [-6, -8], [-19, -4], [12, -4], [17, -2], [-3, 8], [-24, 0], [-8, 0], [-20, 8], [-8, -8], [-6, -8], [-32, 0], [-18, -8], [-28, 0], [1, 14], [15, -12], [8, 0], [10, 0], [-14, 4], [-5, 6], [-8, -8], [-6, 0], [0, 16], [0, -4], [14, 4], [17, 6], [-5, 12], [1, 8], [26, -4], [-7, 12], [12, 4], [-24, -4], [-2, -16], [12, -12], [-11, 6], [27, -6], [2, -16], [8, 0], [-3, -4], [-13, 6], [6, 8], [-14, 8], [5, 4], [43, 0], [-2, 4], [0, 4], [29, -6], [14, -24], [18, -12], [-16, -4], [14, 4], [9, -2], [24, 4], [18, -8], [-17, 6], [13, -12], [-16, -4], [-30, 12], [-4, -12], [-12, 16], [6, -12], [16, -20], [9, -22], [-4, -16], [13, -20], [8, -16], [-22, 20], [8, -20], [0, 20], [-6, -16], [30, 4], [15, -4], [-31, 10], [14, -8], [-26, -4], [18, 4], [9, 8], [1, -18], [-19, -12], [6, 0], [5, 2], [-15, 24], [7, -8], [19, -8], [21, 6], [-5, -8], [-21, 14], [-19, -2], [-31, 20], [-20, 0], [27, -2], [-38, 12], [-40, 8], [34, 8], [-10, -20], [8, -8], [-15, 20], [31, 6], [16, -12], [8, 16], [0, 12], [24, 8], [-21, -12], [30, -20], [10, -16], [22, -16], [17, -18], [-20, 4], [1, 22], [22, -4], [12, -24], [1, 6], [30, -20], [-10, -4], [-11, -16], [-51, 4], [-16, 8], [40, 0], [38, -16], [44, 0], [26, 0], [10, 12], [-14, -8], [15, -20], [-13, 18], [-39, 14], [-34, 20], [-50, 8], [16, 20], [44, -8], [33, -24], [-40, 16], [-18, 12], [-32, 12], [16, 0], [-14, 4], [7, 4], [41, -14], [-1, 16], [-23, -4], [51, -12], [28, -24], [2, 8], [18, 0], [36, -8], [-14, -4], [35, -20], [10, -4], [-17, -2], [24, 8], [-6, -8], [-10, 4], [3, -12], [30, 12], [32, -16], [-8, 20], [47, -8], [11, -14], [16, 0], [36, 12], [-20, -8], [-2, 0], [-30, 0], [-54, 0], [16, 20], [0, -4], [59, 4], [60, 0], [-29, 10], [-30, 12], [-4, 20], [-37, 4], [-36, 0], [-1, -12], [11, 12], [14, -16], [-5, -8], [14, 0], [21, 22], [-36, 16], [24, -28], [-42, 24], [-15, -6], [12, -8], [7, 30], [30, 16], [-28, 8], [-24, 36], [38, -8], [12, -16], [18, 12], [28, -8], [-13, -6], [-18, 24], [45, 8], [18, -4], [-40, -4], [-22, 16], [22, -20], [-64, 4], [36, -16], [-19, 20], [11, -32], [-29, 8], [-17, -2], [26, -12], [-44, 8], [30, 4], [43, 2], [39, 12], [36, 16], [-54, 12], [-19, 28], [2, -4], [-22, 28], [-8, -8], [-47, 2], [0, 16], [16, -36], [-49, 14], [-52, -12], [6, 24], [-32, -8], [32, 12], [-42, 12], [48, -16], [-4, -24], [18, -8], [59, 2], [-19, 0], [-24, 12], [19, -16], [-26, -4], [-54, 4], [-18, 20], [6, -4], [-32, 16], [9, -18], [26, 16], [2, -8], [-16, -28], [18, -8], [26, -16], [8, 28], [21, 16], [-23, 2], [-5, 14], [-14, 40], [54, -20], [-19, -6], [-61, 8], [-22, -20], [46, -16], [-54, -12], [12, -24], [24, 12], [-56, 24], [-19, -2], [36, -16], [0, 24], [42, -20], [8, 0], [0, 8], [-30, -20], [6, 4], [49, 16], [-37, 32], [-20, 0], [4, -32], [-9, -22], [18, -4], [32, -8], [30, -4], [-45, 24], [-27, 2], [-36, 20], [-62, -4], [-35, 36], [2, -12], [20, -20], [15, 18], [-65, -12], [-6, -28], [30, -44], [12, -8], [-10, -24], [-36, 24], [-34, 28], [-77, 10], [-6, -4], [9, -4], [68, -4], [-32, 40], [18, -12], [54, 0], [36, -8], [53, -8], [14, 16], [-11, -4], [24, -4], [-24, 12], [-37, -26], [22, 12], [-11, -18], [36, 8], [-14, -12], [-2, 8], [-22, 28], [45, -4], [-22, 16], [15, -8], [58, 8], [26, -40], [-44, -4], [18, 12], [-28, 40], [67, -8], [11, 6], [-8, 40], [-9, 28], [-63, -10], [-25, -16], [19, -42], [-22, 28], [30, -12], [7, -24], [32, -20], [-20, 36], [29, -10], [-40, 0], [-49, -2], [-43, 22], [-30, 20], [-22, -4], [-83, 10], [-47, 16], [-10, 32], [2, -12], [-52, 20], [6, -12], [-18, -4], [1, -12], [-17, -28], [32, -16], [17, 2], [-87, -2], [42, -8], [54, 12], [-54, 0], [-24, -4], [-85, 4], [-5, 20], [18, 20], [-22, -24], [-52, 24], [-66, -12], [6, 20], [72, -20], [-46, -4], [-58, 20], [69, 0], [29, -40], [-16, 28], [-58, 24], [-22, -8], [-28, 0], [-27, -10], [2, -20], [22, 4], [-70, 20], [-51, 24], [12, -36], [33, 14], [-34, 20], [100, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7623_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7623_2_a_bd();".
// 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_7623_2_a_bd(:prec:=2)
    chi := MakeCharacter_7623_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_7623_2_a_bd();".
// 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_7623_2_a_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 1, 1]>,<5,R![-1, 1]>,<13,R![-13, 0, 1]>],Snew);
    return Vf;
end function;