
// Make newform 7623.2.a.ce 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_ce();".
// 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_ce();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, -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_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 := [[1, -1, 0], [0, 0, 0], [0, 0, 1], [-1, 0, 0], [0, 0, 0], [-2, -1, 1], [2, 1, -1], [0, 2, 2], [-2, 0, 1], [0, 2, 2], [-4, -1, 0], [-6, 2, 1], [6, -1, 1], [-2, 0, -2], [-2, 1, -1], [4, 0, 2], [0, -1, 0], [6, -1, 1], [-2, 4, 1], [-2, -4, -5], [-6, 1, -1], [-10, 0, -2], [4, -2, 2], [8, -2, 1], [-4, -2, -3], [-6, -3, -5], [-2, 5, 3], [0, 4, 4], [4, 2, 6], [-8, -2, -1], [8, -4, -4], [-8, 2, 2], [2, 2, -3], [0, 6, -2], [0, 4, 4], [-2, -4, 2], [0, 4, 7], [-8, -2, -2], [12, 0, 4], [6, -5, -3], [-6, 6, 1], [4, -4, -3], [-14, -2, -3], [-4, -4, 0], [12, 4, 0], [6, -7, -1], [0, 4, -4], [4, 3, -8], [16, -2, -2], [4, 0, -3], [12, 6, 2], [-8, 4, 4], [-6, -7, -1], [-12, 1, 2], [10, 4, 0], [-16, 0, 0], [-2, -2, 2], [0, -4, -4], [8, 4, 4], [4, 6, -6], [0, -10, -2], [10, -3, -5], [16, -6, 2], [10, -5, -3], [-4, -2, 3], [8, -2, -1], [6, 6, 5], [8, 2, 2], [6, -8, -2], [-10, 11, 5], [-16, -2, -7], [-10, 4, 2], [0, 5, 2], [-12, 0, -4], [-2, -4, 7], [0, 7, 6], [-16, -6, -1], [-14, 10, 2], [-2, -8, 4], [2, -7, -9], [6, 3, 5], [-20, 0, -6], [2, 4, -10], [12, 2, 5], [-12, 8, 4], [2, 10, -3], [-6, 14, 3], [4, 0, 4], [-18, 3, -3], [2, 0, -3], [8, 3, 8], [12, 8, 4], [10, 8, 11], [14, 4, 2], [-4, -6, 6], [-8, 4, 4], [-12, 0, 3], [-12, 14, -1], [4, 10, 6], [36, -4, 0], [8, 0, 8], [-12, 0, 4], [8, -2, -2], [-8, -6, 2], [8, -16, -8], [28, -2, -3], [18, -3, -5], [10, -15, -1], [-8, -6, 6], [-30, 9, -1], [-4, -8, 4], [-24, 6, 6], [16, 4, 2], [12, 1, -10], [2, -6, 1], [6, 14, 11], [-4, -5, 6], [8, 5, 10], [4, -10, 9], [-10, 8, -10], [4, 8, -5], [12, 2, -2], [26, 1, 7], [-12, 8, 4], [28, 7, 10], [4, 14, 2], [-12, 2, 7], [24, 3, 6], [-28, 7, 0], [14, 11, 13], [0, 8, -8], [-8, -4, -12], [-10, 0, -7], [-10, 4, 0], [6, -5, 5], [-18, 7, 9], [2, 14, -2], [-20, 10, 6], [32, 0, 1], [28, -4, 0], [20, 10, 6], [0, -4, -4], [-22, 14, 5], [-14, 4, -2], [-16, -12, -13], [-8, 15, 2], [6, -1, -7], [30, 3, 5], [-12, 7, 2], [-4, 16, 8], [4, -10, -6], [4, -2, 7], [-4, 8, 12], [-4, 8, -12], [-4, 10, -10], [-28, 6, 6], [40, 0, 0], [30, -4, -8], [14, 7, 9], [18, 9, 7], [14, -6, -3], [32, 4, 12], [-14, 4, -10], [4, -21, -6], [-8, -2, -1], [0, -17, -6], [-8, 2, 6], [18, -7, -1], [12, 8, -4], [24, 4, -1], [-8, 4, 4], [22, 6, 18], [18, 8, -6], [12, 2, 6], [-24, 3, 10], [0, 2, -1], [6, -4, -6], [-26, -2, -9], [-12, -12, 0], [6, -13, 5], [8, 4, 16], [0, -2, -2], [-4, -2, 17], [14, 7, -15], [-26, -10, -19], [6, 2, 6], [4, -8, 12], [18, -7, -9], [-4, 2, -18], [24, 8, 0], [8, 10, -7], [8, -4, -12], [22, -8, 9], [2, -6, 6], [-26, -4, -14], [16, 10, -7], [-16, 26, 10], [34, 10, -7], [18, 17, 15], [0, -4, 12], [24, 0, 0], [-8, 4, -12], [-10, -14, 10], [10, 17, -1], [-8, 21, -8], [20, 0, 17], [-36, 5, 0], [24, -8, -8], [20, -12, -8], [-18, -9, -15], [-16, 2, 10], [32, -8, -9], [22, 6, 5], [-14, -11, -21], [0, -16, -8], [-28, 6, -5], [42, 8, 10], [10, -3, -5], [-8, -18, 2], [-4, -10, 5], [8, -2, 14], [4, -12, -16], [-10, 18, -7], [34, 2, 11], [28, 6, -6], [20, 6, 2], [6, 0, 12], [-16, 4, -8], [-4, 12, -8], [-22, -8, 10], [30, -18, -1], [-28, -10, -6], [-8, 24, 8], [8, -24, 0], [-4, 7, -14], [4, 4, -8], [-4, -10, 1], [32, -24, 0], [46, 0, -3], [36, -1, -12], [-6, 0, 1], [56, -10, -2], [-2, -11, 3], [-22, -6, -5], [54, -5, -3], [20, 0, -4], [-18, -7, -1], [-2, -1, -7], [32, 0, 0], [-40, -8, 0], [4, -20, -16], [-18, -5, 5], [-10, 0, -13], [-50, 16, 4], [2, -7, 15], [8, -20, -4], [24, -28, -6], [-36, 2, -10], [-20, 16, 5], [-38, -3, -5], [-14, 12, 6], [-8, -24, -16], [34, 5, -13], [-14, -11, -21], [2, 8, 12], [-10, -17, -23], [2, 18, 7], [16, -22, -13], [-10, -4, -6], [-24, -6, 2], [0, 0, -1], [-18, -16, -1], [22, 16, 0], [-16, -12, -12], [10, -7, -9], [44, -13, 0], [-6, -2, -5], [-4, -20, 8], [-12, -8, -12], [44, 6, -6], [-8, -12, -4], [10, -16, -15], [16, -12, -4], [10, 13, 3], [-28, 2, -10], [-6, -20, 15], [12, -4, 2], [-48, 16, 0], [16, 15, -2], [-30, -15, -25], [-42, -10, -3], [-46, 13, 11], [24, -6, -6], [-40, -6, 2], [-24, 36, 12], [-2, -9, 9], [-16, -21, -22], [-36, -8, -1], [-4, 18, 22], [20, -6, 14], [38, -9, 9], [44, 10, 14], [-42, 0, 6], [-24, 18, -6], [22, -12, -13], [-30, 10, -15], [-14, 6, 19], [20, 12, 7], [0, 16, 20], [24, -8, 0], [-12, -20, -8], [10, -4, 0], [44, -2, 2], [18, 2, -13], [-10, 0, -10], [-10, -9, 17], [60, -13, -2], [-16, -16, 16], [-18, -16, -16], [36, 14, 18], [0, -6, 18], [24, -6, -11], [-38, 5, -5], [14, -18, 3], [28, 16, 12], [8, 2, -1], [-2, 0, 19], [60, 9, 6], [2, -20, 14], [36, -14, -2], [68, -16, -4], [0, 4, -2], [16, 8, 24], [-16, 2, -14], [6, 12, 2], [4, -7, 14], [46, -2, -1], [22, 7, -7], [48, 14, 11], [-32, 28, 4], [4, 2, -3], [-26, 14, -5], [38, -1, -7], [-6, 10, -3], [40, -10, -15], [32, -24, 0], [-8, -24, -15], [30, -16, 7], [-24, 24, 16], [8, 20, -1], [-8, 30, 14], [54, 2, 5], [6, -32, -12], [-8, -8, -8], [-16, 28, -12], [-10, 23, 9], [-32, -11, -14], [-44, 14, -6], [8, 0, 0], [-26, 22, 13], [48, 12, 12], [-10, 15, -7], [42, 14, -5], [2, 4, 6], [-8, -13, 16], [14, 14, 23], [30, 3, 5], [0, 4, -4], [-12, 0, 4], [10, 0, 23], [-2, 3, 21], [-24, 0, 0], [-4, -4, 24], [24, 0, 16], [18, -6, -1], [18, -31, -1], [26, -12, 6], [30, -5, -3], [34, -39, -17], [14, 7, 17], [-6, -22, -18], [26, -10, 19], [26, 0, 2], [16, 4, 4], [-24, -2, 14], [-12, -7, -28], [-28, -12, -20], [16, 0, 3], [-32, -20, 4], [-20, 9, 12], [-2, 22, 19], [-4, -28, 3], [50, -4, 3], [12, 9, 2], [18, 6, 27], [28, -2, -6], [-24, -4, 12], [-38, -8, -4], [12, -2, 18], [-20, -10, 18], [2, 5, -13], [-28, -2, -5], [-48, 24, 8], [26, 12, 32], [66, -7, 7], [-4, 16, 12], [-6, -24, 0], [20, -8, -4], [10, 13, 27], [40, -11, 16], [-10, 15, -15], [-8, 2, -22], [-16, 24, 8], [30, -28, -14], [52, 14, -6], [0, 0, -7], [16, -8, 8], [-40, 13, 2], [-30, 20, 8], [32, 12, -20], [0, 6, 13], [-10, -9, -15], [-72, -2, -14], [-84, -6, -2], [-10, 16, 12], [44, -8, 3], [-22, 4, -3], [-36, 34, 14], [-52, -3, -14], [40, 0, -16]];
    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_ce();".
// 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_ce(: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_ce();".
// 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_ce( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![8, -5, -2, 1]>,<5,R![-4, -8, 1, 1]>,<13,R![-64, 2, 8, 1]>],Snew);
    return Vf;
end function;