
// Make newform 7623.2.a.ch in Magma, downloaded from the LMFDB on 23 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_ch();".
// 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_ch();".
// 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_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], [0, 0, 0, 0], [1, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 0], [0, -1, -2, 2], [-1, 1, -3, 1], [1, -2, -1, -1], [3, 0, 1, 2], [0, 0, 0, 3], [0, 1, 3, -1], [-2, 1, 1, -1], [1, 0, 3, 2], [5, 0, 6, 0], [3, 1, 2, -4], [4, -4, 1, -3], [6, -2, -2, -1], [-7, 5, 0, 1], [-4, -1, -2, 1], [-1, 2, -5, 3], [-4, 2, -3, 5], [2, 1, 2, -2], [4, -3, 7, -5], [4, 1, 3, -5], [-6, 0, -2, -5], [1, -4, -7, 3], [9, -7, 2, 4], [6, -5, 2, -1], [6, -2, -1, -3], [9, 4, -5, 2], [-4, 2, -7, -3], [6, 0, 3, -2], [12, 2, 8, 0], [-2, -5, 1, -4], [-5, 3, -9, 0], [4, 3, -4, 0], [-4, 2, -12, 5], [6, -3, -1, -2], [4, 6, -1, 1], [7, 4, -4, 0], [-2, 2, -12, 6], [0, 4, 5, -2], [7, -6, -1, -1], [9, -4, 0, 3], [-2, 1, 2, -5], [17, -1, 0, -2], [6, -1, 1, 3], [-9, 6, -5, 5], [13, -4, 9, 3], [7, -4, 8, -3], [5, 2, -3, 1], [-6, 3, 6, -7], [4, 4, -8, 4], [-9, -2, 5, -11], [15, 1, 3, 6], [-8, 0, -10, 0], [15, -1, 9, 4], [2, -2, 2, 6], [6, -5, -6, -6], [0, -1, 12, -2], [1, 2, 12, -4], [-5, 2, -2, 3], [7, 5, 10, 4], [10, -5, 11, 3], [-15, 8, 1, 6], [17, -3, -3, 7], [6, -9, -4, 8], [-6, 4, -6, 4], [-11, -5, 1, 0], [22, 2, 2, 1], [13, -1, -6, 0], [-2, 2, 7, 0], [-3, -7, 11, -15], [-17, 2, -11, 3], [10, -4, 8, -5], [20, -1, 14, -2], [-15, 3, -12, -10], [-10, -3, -11, 1], [7, 2, 2, 2], [-23, -2, -14, 1], [-2, -12, -2, 3], [3, -3, 7, -9], [-27, 3, -4, -4], [-2, 10, 6, -8], [3, 6, 19, -1], [-9, -5, 4, 1], [26, 4, 7, -7], [7, 4, 18, -12], [12, -2, -5, 10], [-5, 10, -6, 3], [-1, -8, -16, -2], [3, -2, -5, -6], [3, 4, -8, 13], [10, 1, 15, -12], [11, -9, 9, 2], [13, 5, 1, -2], [0, 1, -2, 15], [12, 8, -7, 4], [6, -9, -2, -9], [-1, 8, -11, 7], [-23, -1, -18, 6], [-11, 14, 1, 4], [-15, 8, 0, -8], [-16, -1, -13, -6], [6, 2, 2, 9], [0, -1, -5, 9], [6, 1, 4, 1], [3, 10, -4, -1], [-17, -4, 10, -12], [20, 7, -1, -4], [10, 13, 15, -9], [3, 0, -1, 10], [11, 10, -3, 7], [10, -13, 1, 10], [7, 11, 0, 7], [18, -4, 1, 1], [10, -9, 3, -4], [-2, 7, 14, 2], [2, 0, 5, -8], [2, 5, 0, 12], [8, -16, 11, -6], [-1, 10, 10, -4], [20, 7, 16, -1], [-3, 11, -17, 10], [-11, -1, -22, 0], [-15, -2, -35, 2], [28, -11, -2, -1], [-16, -4, 1, 7], [12, 0, -3, -4], [-13, -1, -13, 0], [-13, -3, -10, -4], [-13, 8, 11, 3], [17, -10, 5, 9], [21, 11, 16, -7], [13, -6, -15, 5], [-6, 11, -22, 18], [-6, -5, -4, -3], [27, 1, -2, 1], [22, -12, 15, 3], [-16, -2, -32, 1], [-27, 9, 5, -7], [-13, -14, -20, 8], [-8, 2, -21, 1], [12, 3, -6, 10], [26, -8, -10, 0], [4, -11, -7, 11], [-3, 5, -3, 9], [13, -7, 0, 9], [-16, 0, -32, -2], [17, 7, -2, 3], [27, 1, 20, -13], [16, 1, 6, -10], [4, 19, 1, -6], [3, -5, 7, -2], [14, 4, 8, -17], [1, 6, 22, -6], [5, -1, 4, 9], [7, 9, -11, 15], [20, 3, 28, -10], [1, -18, 13, -15], [7, -16, -11, -4], [-5, 18, 13, -4], [-11, -4, 1, 9], [-20, 0, -2, -5], [-7, 8, -11, 2], [9, -11, 19, -20], [23, -18, 15, 6], [31, -1, 21, 14], [-18, 4, -5, 3], [15, 0, 18, 0], [9, -11, -9, 1], [-5, 20, -9, 10], [18, 8, 16, -1], [9, -13, -11, -15], [-4, -5, -23, 8], [-9, -11, -34, 3], [6, 9, 30, -4], [32, -4, 18, -5], [-30, 3, -14, -18], [-26, -6, -8, 1], [-4, 1, -3, 6], [9, -13, 9, -10], [-2, 17, 5, -7], [-17, 9, -8, -9], [18, 3, 31, -12], [-23, -2, -17, 18], [-22, -10, 4, 3], [-11, 2, 13, -15], [-3, 3, -19, 14], [-17, 8, -32, 10], [3, -7, 24, -5], [13, -1, -16, 20], [-34, 10, -15, -10], [-21, 10, 4, -7], [-18, -4, -23, -6], [18, -13, -4, -3], [-8, 5, 2, -8], [-5, 4, 12, -7], [5, -19, -6, 2], [-38, 19, -3, -11], [-30, -5, -8, 3], [14, -14, -2, 3], [19, -3, 17, 5], [29, -22, -4, -12], [6, -13, 22, -11], [29, -12, 3, 1], [-12, -3, 18, 8], [-37, 14, -30, -6], [8, 7, 13, -24], [6, -16, -12, -15], [-19, -4, 15, -20], [-7, 10, 8, -8], [-23, 12, 6, 10], [10, -1, 1, 10], [-7, 13, -11, -2], [-8, 8, 7, 3], [12, -2, -9, 15], [14, 10, 36, -1], [5, -7, 7, 11], [9, 16, -26, 18], [8, -10, -3, 6], [-16, 17, 0, 0], [-15, 17, -11, -10], [-1, 3, 16, -4], [19, -3, -12, -5], [-13, 7, 9, -14], [23, -9, 10, -25], [5, -1, -16, -7], [-27, 14, -9, 10], [-26, -5, 23, -1], [18, -12, 18, -3], [-26, -17, -20, 16], [-5, -3, 0, -1], [32, -9, 23, -7], [5, -14, 25, 2], [-12, -4, 8, 6], [28, -21, 4, 3], [-9, 15, -17, -3], [-12, 21, 6, 1], [-17, 5, 9, 15], [-4, -4, -5, 2], [-11, -17, 7, 3], [3, -23, 14, -6], [-1, 0, 0, 16], [-39, 10, -22, 1], [5, 8, 6, 16], [-10, -13, 20, -19], [22, 12, 39, -11], [1, -5, 15, -17], [22, 4, 38, -22], [-3, -16, 5, -14], [38, -4, 34, -4], [21, -22, 7, -3], [-5, 13, 7, -15], [15, 4, -27, 0], [-9, 9, -10, 27], [-21, -8, -24, -8], [17, 2, 8, -14], [-19, 17, 7, 0], [-16, -16, 3, -9], [-4, -14, 15, -13], [15, -13, 10, -23], [-3, -1, -13, -18], [36, -10, -11, -14], [8, -10, -24, 22], [11, 17, 37, 1], [-47, 2, 2, 3], [-1, 34, 3, -4], [28, 3, -21, 1], [40, 3, 7, 11], [15, 2, -1, 11], [-19, -2, 8, 4], [12, -5, 8, -27], [23, 0, 14, -27], [45, -5, 10, 13], [-4, -6, -15, -2], [-4, -8, -16, 1], [-23, 2, 8, 2], [3, 19, 16, -10], [-10, -6, -20, 22], [43, -13, 14, -7], [53, -3, -4, 7], [17, 1, -13, 9], [24, 4, 47, -9], [21, -24, 15, -17], [-18, -1, -26, 24], [37, -11, 33, 0], [43, 0, -3, 20], [25, 9, 35, 6], [17, -8, 23, -6], [-39, -2, -2, 5], [16, -3, 28, -7], [-18, 7, 4, -19], [41, -13, 0, -15], [4, -1, -15, 15], [7, -21, 28, -7], [-28, 0, -20, -5], [-18, 33, -24, 2], [3, 13, 38, 0], [-13, -3, 10, 11], [-18, 3, -5, 13], [-6, -6, -11, -5], [5, -11, 15, 1], [9, -3, 10, 14], [49, 9, 25, -12], [-21, 8, -3, 0], [3, 10, -23, 5], [-9, 12, -3, 16], [-33, -5, -19, 2], [1, 11, 33, -12], [-27, 2, 1, -2], [-3, -7, 11, 2], [4, 18, 13, -12], [7, 6, -4, 3], [-20, 4, 14, -2], [-7, -2, -5, 19], [-25, 4, -33, 32], [-3, -19, -15, 16], [28, -18, -7, 22], [-58, -1, -3, 4], [39, 1, 48, -19], [-13, 10, -2, -11], [-14, -5, -30, -13], [8, -28, 1, 3], [12, 0, 37, -3], [-5, 22, -12, 1], [-35, -7, -12, -6], [-9, 25, 12, -9], [-50, 12, 7, 9], [-54, -6, 8, -10], [20, -1, 0, 20], [-21, 1, -26, -8], [-8, 9, -16, -15], [36, -1, 42, -4], [8, 6, -13, 1], [29, 8, 5, -15], [-50, -13, -33, 10], [16, 0, 31, -17], [9, 8, 28, 12], [-6, -20, -5, -6], [13, -10, 46, 0], [48, -10, -13, -10], [70, 5, -5, 10], [-22, 36, -3, 0], [-14, 11, -2, 31], [-20, -24, -7, 14], [46, -15, 34, 7], [-13, -6, 26, -1], [26, -22, -20, -19], [-21, 3, 23, 3], [-28, -7, 10, -11], [-52, -1, -38, 23], [1, 4, -20, -10], [-16, 21, 21, -8], [6, -8, -23, -18], [-40, 21, -41, 1], [10, -3, -16, -18], [21, -1, -10, -11], [-49, 15, -24, 2], [10, -28, -7, 0], [-4, 16, -12, 42], [7, -23, -8, 12], [1, 17, 8, 10], [13, -5, 24, -41], [-34, 2, -39, -14], [-23, 7, -6, -25], [-16, -25, 37, -21], [-1, 8, 23, 2], [-66, 26, -13, -21], [-44, 14, -15, -25], [-17, 26, 3, 21], [61, -3, 5, -15], [-68, -6, -28, 9], [-31, 24, -2, -10], [28, -16, 34, -9], [22, -8, -26, 4], [-29, 7, 23, -9], [-52, -4, -13, 10], [-16, 24, -2, 21], [23, -2, 9, -11], [-3, 18, 42, 1], [37, -8, 36, -20], [36, -27, -9, -9], [24, -23, 14, 13], [-3, -4, 36, -26], [-2, 2, -22, -10], [15, 13, 41, 7], [37, -9, 43, 6], [-5, 2, -20, 2], [27, -12, -32, 9], [14, -24, -17, -9], [60, -25, -8, 5], [20, 21, 36, -10], [-9, 18, 1, -26], [3, -29, 4, -1], [-23, 11, 1, -19], [-23, 18, 2, -13], [30, -24, 7, -6], [13, 0, 23, -30], [-59, 14, 7, 7], [-29, 24, 11, 9], [-17, 6, 24, -29], [-38, -27, 0, 5], [27, 20, 19, -18], [11, -7, -22, 6], [-2, -25, -47, 2], [9, -1, 4, 16], [-26, -5, -2, -4], [3, -12, 25, -9], [-21, -13, 10, -1], [-2, -2, -18, -26], [52, -22, 0, 4], [52, -16, 49, 14], [67, -2, 34, -9], [-16, -24, 33, -9], [6, 30, -1, 0], [40, 8, 1, 9], [37, 17, 15, -4], [-18, 3, -52, 3], [-35, 2, 8, 14], [14, 1, 21, -20], [-20, 3, -35, -5], [-30, 17, -39, 9], [43, -13, 14, 15], [-34, -10, 7, -1], [-13, 10, 7, -4], [24, -26, 15, 10], [-6, 27, 3, -3], [31, -10, -13, -3], [41, -22, -4, -23]];
    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_ch();".
// 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_ch(:prec:=4)
    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_ch();".
// 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_ch( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![5, -5, -4, 2, 1]>,<5,R![-1, 3, 8, -6, 1]>,<13,R![-29, -65, -32, 0, 1]>],Snew);
    return Vf;
end function;