
// Make newform 7623.2.a.cb in Magma, downloaded from the LMFDB on 18 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_cb();".
// 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_cb();".
// 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, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 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_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, -1, 1], [1, 0, 0], [0, 0, 0], [0, -1, 1], [0, -2, 0], [-4, 1, 1], [2, 2, 0], [4, -1, 1], [-2, 0, 2], [0, 3, 1], [2, 2, 2], [-2, 2, 0], [8, -1, -1], [0, 0, 2], [8, -1, 3], [-6, 0, 0], [4, 1, 1], [-4, 4, 0], [-8, 1, -1], [-4, -4, 0], [-6, 0, 2], [-6, 0, -4], [8, 0, 2], [-4, -6, 4], [0, 2, 2], [-4, 1, 1], [4, -2, 4], [2, 4, 0], [-6, 4, 2], [4, -4, -4], [0, -2, -4], [-8, -8, 4], [12, -1, 1], [6, 4, -2], [-10, -6, 2], [-4, -7, 1], [-2, 2, 4], [-10, -2, -2], [12, 0, 0], [8, -4, 2], [2, -4, 2], [-4, 0, -6], [2, -4, 0], [-2, 2, -4], [-4, -2, -2], [-4, -10, 2], [-8, 2, -2], [0, 0, -2], [10, -4, -4], [8, 7, -1], [0, -1, -3], [0, 5, 1], [4, -5, -3], [-8, 1, 1], [10, 4, -4], [0, 1, 1], [10, -2, -2], [-4, 7, -3], [-4, -3, 5], [10, 2, 6], [-4, -4, -4], [-8, 8, 0], [-4, -6, 0], [-4, -12, 2], [8, 8, -4], [-4, 8, 2], [4, 0, 8], [8, 7, 1], [4, 5, 7], [4, 0, 4], [-4, 2, -2], [-12, 2, 0], [4, 3, -5], [0, 4, -4], [-10, 10, -2], [4, -6, -4], [-16, -4, 2], [6, -12, -4], [4, 9, -7], [-20, -3, -1], [-8, 5, -3], [10, 6, -2], [0, 5, -3], [0, 8, -4], [-6, -14, 6], [0, 0, -2], [2, 10, -2], [0, 13, -3], [20, -5, -5], [14, 0, -6], [24, -4, -4], [20, 5, -3], [-4, 3, -5], [-2, -6, -4], [18, 0, 0], [-8, -1, 5], [-8, -9, 3], [-2, 10, -2], [-20, 0, 0], [-8, 3, 1], [-2, -6, 8], [-14, -4, 4], [0, 4, 8], [0, -6, 4], [8, -3, 1], [-2, -14, -2], [-4, -12, 0], [4, -3, -1], [-4, 5, -7], [12, -2, 4], [-22, 4, 0], [18, 8, 2], [4, 0, 4], [-12, -6, -4], [-10, -6, -8], [4, -3, 9], [-18, 14, 2], [4, 9, -7], [4, 0, 10], [20, -4, 14], [18, -12, 0], [14, -4, 10], [-16, -6, 6], [-10, 0, 8], [-4, -15, 3], [0, 9, 1], [-18, -6, 4], [18, -12, -4], [-30, -2, 0], [-16, -7, 1], [-20, 9, -3], [-8, 7, -3], [14, -2, 2], [-16, -13, 1], [36, 3, 1], [-36, -1, -1], [0, 1, 11], [-4, 7, -3], [-24, -1, -5], [12, -11, -9], [28, 5, 1], [16, 1, -3], [-6, 10, -2], [8, 3, -5], [14, 8, -4], [-22, 12, 4], [4, -4, -4], [10, 2, 0], [34, 2, 2], [-12, 7, -7], [0, 7, -5], [30, -4, -2], [20, 16, -8], [8, -14, 10], [26, -6, 0], [-16, -11, 11], [-10, -4, 4], [-26, 2, 2], [-4, 10, -6], [-16, -1, 1], [-8, -2, -2], [-8, -15, 5], [-16, -8, -2], [-12, -12, 0], [-4, 17, 5], [-6, 4, 12], [6, 16, -8], [0, -9, -7], [16, -11, 1], [-6, -20, 0], [-16, 0, 8], [-36, -2, 0], [-12, -10, -6], [-34, 8, 0], [-38, -6, -4], [-2, 12, -8], [4, -9, 11], [0, 17, -13], [-20, 1, -11], [10, -2, 0], [26, 4, 0], [-6, -6, 2], [-24, -12, -4], [-8, -13, -3], [26, -2, 6], [-2, -16, 6], [20, -10, 0], [2, 0, 6], [-6, 8, 0], [36, 7, -1], [4, -19, -3], [4, 17, 7], [-16, 1, 5], [-28, 1, -5], [0, 22, -4], [18, 4, 0], [20, 4, 2], [30, -10, 8], [-2, -4, -4], [8, -5, 3], [28, 20, -10], [-4, 9, -5], [-12, 4, -4], [-6, -4, 4], [-28, 4, 8], [-20, -1, 7], [-26, -4, 0], [-6, -2, 0], [-10, 4, 12], [4, -7, 7], [-24, 3, 3], [56, 0, 4], [16, 2, 10], [6, -4, 16], [-8, 12, 4], [4, -10, -12], [-34, 4, -10], [-16, 22, 0], [0, -10, -8], [32, 7, -1], [0, 12, -14], [-4, -21, -1], [-8, 8, -12], [10, 0, 16], [8, 13, -13], [40, 2, -6], [-16, -21, 3], [32, -1, -5], [38, -4, 4], [4, 19, -5], [20, 18, -2], [12, -1, 5], [30, -2, -4], [-8, 18, -6], [16, 14, -8], [-16, 7, -11], [-30, -4, 6], [4, -5, 7], [22, -14, 0], [-20, 0, -8], [-10, -8, 10], [-14, -4, 8], [14, 2, 14], [20, 16, 4], [4, 16, 4], [16, -21, -1], [0, 0, -12], [-52, -7, -3], [18, 8, -8], [52, 4, 2], [4, -10, 10], [30, -4, -8], [10, 8, -12], [-12, -8, -8], [4, -21, 5], [-8, -1, -21], [12, 9, -17], [12, 5, -1], [0, -22, 10], [-44, 8, -8], [-54, -8, 0], [-28, -23, 7], [20, -23, -5], [-46, 8, -6], [30, -12, -8], [4, -13, 13], [0, 4, -8], [22, 8, -8], [18, 2, 14], [-12, -3, -3], [36, -8, 6], [10, 12, -6], [28, 21, -9], [-46, 0, 6], [-28, -20, 12], [14, -12, 24], [-44, -2, -8], [-30, 6, 0], [-30, -2, 4], [16, -8, 0], [22, 4, 14], [-32, -3, 7], [16, 1, 5], [46, -10, 0], [8, 13, -1], [10, 8, 20], [-14, 12, -14], [-12, 22, 4], [-12, -2, 4], [-20, -24, 0], [8, 7, 1], [14, -26, -4], [-38, 2, -6], [36, -16, -6], [24, -2, 14], [-14, 4, 12], [-22, -26, 0], [32, -7, -11], [-20, -13, 13], [-32, 5, -21], [-14, -26, 12], [46, 10, 0], [-36, 5, 5], [-20, 9, -9], [-6, -6, 12], [-52, 4, -2], [-18, 6, -8], [16, -16, 18], [-32, 26, 10], [-30, 14, -10], [8, -5, -3], [4, 16, -16], [8, -8, 8], [-2, 12, 12], [40, 12, -8], [0, -32, 0], [48, 4, -2], [6, -4, -8], [-4, -3, 13], [-36, 1, -9], [42, -4, -8], [12, -4, 0], [-4, 5, -5], [18, 26, -6], [-28, 16, 0], [-22, 6, -8], [28, -1, 19], [4, -9, 1], [-34, 6, -22], [10, 14, 6], [4, 21, 1], [6, 4, -6], [-14, 26, -12], [-48, -9, 3], [-8, -3, -5], [-24, 4, -18], [-8, 18, -4], [32, 20, 4], [18, -4, 16], [12, 6, -8], [-16, -2, 4], [-36, -1, -21], [-48, -14, -4], [-12, -11, -11], [50, -4, 16], [24, -16, 4], [-4, 4, 8], [2, -8, 16], [0, 28, 8], [-16, 31, 1], [-6, 6, -2], [44, -11, -7], [12, 14, -12], [16, 2, -4], [12, 15, 11], [60, 8, -8], [-12, -13, -7], [-62, -6, 8], [30, -2, 10], [2, 22, -10], [-44, -4, 8], [-36, 1, 1], [-4, 4, -12], [-28, -19, 11], [-24, 6, -4], [-16, -10, -18], [62, -8, 0], [2, 16, -14], [-50, 8, 6], [8, 35, -5], [52, -3, -9], [16, 4, 12], [0, 13, 3], [0, -35, 1], [-16, 27, 11], [-36, -11, -9], [10, 20, -12], [62, 4, 4], [64, 13, -9], [0, -12, 18], [-24, -14, -14], [8, 3, -11], [4, -8, 16], [16, -13, -13], [0, 9, 9], [-22, -12, 16], [-6, 16, 2], [-40, -7, -7], [24, -1, 17], [6, -4, 16], [-30, -14, 12], [6, -14, 16], [-48, 6, -2], [48, -10, 4], [20, 5, 17], [-24, 9, -17], [40, 0, 4], [-40, 15, 5], [-56, 15, 9], [-4, 6, 0], [-6, 12, -6], [22, 8, -8], [-20, -24, 2], [16, -13, 27], [12, -27, 15], [10, 24, -16], [-4, -18, -2], [40, 0, 12], [50, -20, 0], [-6, -8, 4], [-8, -8, 20], [-30, -2, 16], [-30, 4, -8], [4, -13, -7], [20, -27, -15], [-22, 4, -14], [2, 8, 4], [6, 12, 22], [4, 6, -20], [20, 15, 5], [-22, 4, 2], [-10, 20, 2], [2, -16, -8], [20, -5, 1], [-34, 4, 10], [16, -21, -3], [-32, 30, 2], [24, 28, -20]];
    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_cb();".
// 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_cb(:prec:=3)
    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_cb();".
// 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_cb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -6, 0, 1]>,<5,R![-2, -15, 0, 1]>,<13,R![-2, -15, 0, 1]>],Snew);
    return Vf;
end function;