
// Make newform 7800.2.a.by in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7800_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_7800_2_a_by();".
// 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_7800_2_a_by();".
// 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 := [26, 0, -11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-6, -6, 1, 1], [-6, 1, 1, 0], [6, 1, -1, 0]];
    Rf_basisdens := [1, 2, 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_7800_a();"
function MakeCharacter_7800_a()
    N := 7800;
    order := 1;
    char_gens := [1951, 3901, 5201, 7177, 4201];
    v := [1, 1, 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_7800_a_Hecke(Kf)
    return MakeCharacter_7800_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, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [2, 1, 0, 0], [0, 1, -1, 0], [-1, 0, 0, 0], [0, 1, -1, 1], [0, 0, 0, 0], [2, -1, 1, -1], [2, 0, 1, 1], [0, 0, 0, 0], [0, -1, -1, 1], [2, -1, 0, -1], [2, 0, -1, -1], [2, 0, -1, 0], [0, 1, 1, -1], [2, 0, 2, -1], [2, 1, -2, 0], [4, 0, 2, 0], [4, 3, 0, 1], [-2, -2, 1, -1], [0, 1, -2, -2], [6, 4, -1, 0], [2, -3, 2, 1], [0, -5, 2, 0], [2, -2, 1, -3], [-2, 0, -4, 2], [6, -1, 3, 1], [6, -2, 2, 0], [-2, -4, 2, -2], [8, 0, 1, -1], [4, -6, 3, -1], [4, -2, 3, -2], [-2, -1, 3, -1], [6, -1, -3, 2], [-4, 2, 2, 2], [4, 2, 1, 3], [2, -5, -1, -1], [6, 0, -1, 2], [6, 0, 4, 0], [0, 0, 3, 1], [0, 1, 1, -3], [0, -2, -2, -2], [-4, -3, 2, -2], [4, -2, -5, -2], [-6, -4, 1, -3], [-14, -4, -1, 3], [0, -2, 1, 3], [10, 2, 1, -2], [2, -4, -2, 0], [4, 3, 1, 5], [-4, -1, 0, 5], [-6, -2, 2, 4], [0, 8, -3, 1], [14, 0, -2, -2], [0, 0, -6, 0], [6, 2, -1, 3], [-12, -2, 2, -4], [4, -8, 3, -3], [-4, -6, 1, -2], [8, 6, 0, 0], [-4, 4, -5, 6], [2, -5, 1, -5], [-4, 4, -4, 2], [-2, 4, -7, -1], [24, 4, -1, -2], [-4, 0, 6, 0], [2, 8, -1, 1], [-2, -1, -5, 3], [2, 0, 0, 4], [4, 2, -3, 0], [-6, -6, 3, -2], [18, -4, 6, -4], [-2, -4, -2, 2], [-16, -4, -2, 6], [-2, -12, 5, 0], [6, 8, -3, 3], [8, 3, 3, 1], [-8, 6, -5, 0], [6, 2, 8, -6], [-8, -2, 1, -7], [14, 0, 4, -2], [14, 2, 1, -6], [14, 10, -3, -1], [-14, 3, -5, 3], [10, -1, 1, 3], [-2, -5, -4, -7], [4, 3, 0, 2], [6, -3, 1, -2], [2, 1, -6, 6], [-10, -5, 3, 1], [-16, 5, -2, 3], [10, -1, -6, -4], [8, 0, 1, -7], [-8, 6, -4, 2], [-16, -4, 0, -6], [26, -1, 5, -2], [2, 4, -6, 8], [14, 2, 3, -1], [-2, -6, 6, -4], [-2, 0, 9, -3], [12, -4, -1, 2], [10, 7, -3, 1], [-10, -10, 0, -2], [-24, 1, -2, 2], [16, -1, 2, 2], [-6, 0, -3, -4], [-4, -8, -1, -6], [4, 2, 2, 0], [-18, 3, 0, 6], [18, -10, 4, -2], [0, 1, 7, -7], [8, -8, 7, 0], [0, 10, -2, 4], [4, 4, -2, 2], [-2, -2, 2, 6], [-10, 7, -7, 3], [-10, -7, 5, 5], [6, 4, -8, -4], [-8, 6, 1, 3], [-6, -4, 0, -4], [12, 6, 4, 0], [4, 5, 1, 7], [-14, 0, -7, 0], [-12, 10, -10, 0], [18, 4, -3, -1], [-2, 6, -4, -2], [-12, 10, -2, 4], [4, 2, 1, -1], [28, 7, 1, -3], [-24, -4, 4, 6], [-10, -2, 5, 2], [-14, 7, 3, -1], [6, -4, 0, 0], [-4, -10, 9, 6], [2, -8, 0, 0], [-8, -14, 3, 0], [0, 4, 4, 10], [12, -5, 9, -9], [6, 2, 2, -4], [-12, -8, 0, -12], [34, 1, -3, 0], [2, 8, 0, -2], [-26, 8, -13, 8], [-12, -8, 5, 7], [-12, -5, 2, -9], [-12, 5, 5, 3], [16, -9, 11, -1], [-8, -9, 4, -8], [6, 2, 3, -4], [-10, 4, -2, 4], [-18, -4, 4, -2], [-8, -6, 4, -4], [-2, 1, 7, 7], [-8, -2, -10, 6], [-8, -6, 8, 0], [0, -7, -2, 6], [-18, -3, 8, -3], [-22, -2, -7, -1], [30, -1, -3, -2], [2, 4, 1, -8], [-8, -7, -3, 5], [16, -2, 1, -5], [16, -12, -7, -1], [0, 14, -1, 2], [-18, 6, -11, -6], [-30, 7, -1, -1], [-20, 4, -3, -1], [10, 8, -2, 8], [0, -15, -1, -9], [-6, -12, 2, -1], [6, -2, 4, 4], [-4, -12, 2, 4], [-8, 7, -12, -6], [-30, -9, -1, -1], [22, -4, 4, -4], [-24, 6, -6, -4], [6, -9, -3, 2], [14, 6, -2, 0], [0, -12, 1, -9], [2, 9, 6, -6], [-32, -8, -3, 7], [-18, -4, 4, -4], [20, 0, -5, 0], [-6, 10, -1, 2], [18, 2, -9, -7], [-20, 6, -11, -1], [-2, 9, -7, -3], [-10, 8, -10, 0], [8, 5, -6, 11], [-16, 4, -6, 10], [10, 0, -3, 12], [-6, 1, 9, 5], [2, 9, -1, 0], [-40, 4, 2, -2], [20, 13, -1, 7], [6, 6, 0, 4], [14, -2, 2, 10], [0, -6, 5, 10], [-18, 7, -11, 9], [16, -4, -4, -3], [6, -15, 9, -7], [-20, -15, 3, 3], [12, 9, -7, 1], [-4, 7, -7, 2], [4, -13, 3, 1], [4, -2, 2, 6], [-12, -16, 4, -2], [18, 1, -6, 3], [-16, -5, -10, 10], [-6, -4, -5, -11], [22, 6, 5, -1], [0, -12, 5, 3], [2, -4, 9, -8], [30, 2, -5, -8], [-14, 6, -6, -2], [-24, -4, 5, -9], [22, 4, -8, 0], [-22, -12, 3, -12], [-8, 14, -7, 0], [-12, 3, 1, 9], [12, 8, -6, -8], [0, 14, -11, 12], [0, 2, 11, 9], [-34, 7, -1, 1], [-32, -15, 3, 3], [32, -1, 9, -5], [4, -10, 14, 0], [-36, 6, -7, 7], [4, 5, 5, -8], [-4, -10, 1, 3], [-18, 3, 3, 3], [24, -14, 16, -2], [-6, 4, -2, 2], [-34, 2, -9, 7], [-18, 14, -9, 4], [22, -4, 8, -10], [-12, -12, 7, 8], [28, -2, -7, -1], [-32, -16, 2, 6], [-14, -2, -3, 10], [8, 7, -10, -10], [12, -6, -11, 3], [-10, -8, 2, 6], [-4, 4, 13, -2], [4, -4, 8, 2], [24, -6, 9, 1], [-4, -11, 13, 0], [-8, -12, 0, 4], [-40, -8, 6, 0], [-10, -4, 12, -12], [28, 2, 7, -4], [-26, -2, 7, 8], [-22, 9, -16, 10], [28, 11, -5, -9], [12, 3, -5, -4], [28, 8, -3, -1], [-18, -13, 9, -3], [-14, 0, 8, -2], [-12, 3, 2, 10], [8, 4, -3, 3], [-14, 9, -11, 1], [38, 14, -4, -2], [-20, 0, -9, 13], [0, 14, -7, -4], [28, 5, 8, -8], [-14, 15, -5, 10], [14, 13, -10, -5], [2, 1, -1, -15], [-14, 4, -14, 14], [-16, -8, 13, 3], [-22, 1, -7, 11], [-12, -1, 6, -8], [-24, -16, 4, -4], [-30, -8, -1, -15], [2, 15, -4, 4], [2, -4, -3, -12], [2, -4, -8, -4], [-10, -6, 4, -8], [-20, 4, 3, -1], [-14, -9, 1, -11], [-20, 18, -6, 8], [-20, 0, 2, -10], [22, -12, -4, 0], [-16, 8, -2, 4], [-4, 0, 4, -12], [-42, 4, 11, 5], [8, -10, -5, 2], [8, 10, 0, -4], [2, -6, 0, 2], [-6, -2, -13, 5], [-16, -8, 6, 14], [20, -8, 13, 4], [34, 4, 2, -5], [24, 4, 9, 7], [10, 8, 11, -3], [-32, 16, -4, 8], [-12, -13, 1, -9], [-16, 12, 7, 9], [2, -1, 1, 11], [32, -6, 10, -10], [8, 0, 11, 2], [6, -7, 5, 1], [-22, -23, 1, 5], [36, 4, -2, 10], [-28, 5, -12, -2], [-22, 17, -3, 9], [-4, 15, 9, 1], [-6, -10, 5, 0], [-34, 16, -10, 6], [14, 19, 3, -7], [24, 12, 2, 1], [-50, 8, -12, -2], [-20, -8, 0, -16], [-22, -6, 7, 2], [-2, -11, -6, 4], [-20, -9, -3, -2], [12, -25, -4, -7], [14, 6, 1, 11], [-30, 4, -8, 8], [-20, -4, -10, 10], [-2, 2, 5, -3], [34, -12, 9, -1], [22, 13, 0, 6], [-8, -2, -5, -18], [-50, -5, 6, 0], [-20, -8, 0, -4], [-26, -13, -1, -11], [-26, -9, 1, -7], [6, 20, 0, -4], [-38, -4, 2, 12], [8, -5, 9, 5], [-34, 0, -9, 15], [26, 6, 1, 6], [12, -24, 0, -8], [-26, 2, -1, 6], [14, -4, -12, 0], [-16, 0, -1, -10], [-30, 12, -16, 10], [6, 10, -4, 2], [-8, -3, -15, -1], [4, -1, -3, 9], [-2, -9, -1, -2], [8, 0, 0, 12], [36, -6, 11, -4], [-8, -4, 11, -3], [10, -7, -12, -10], [14, -9, -17, 3], [-50, 6, -5, -4], [24, 11, -5, -11], [16, 8, -14, 0], [-32, -3, 2, 12], [50, -11, -3, -2], [4, -14, 7, -11], [-14, -5, 2, -16], [22, -12, 0, -4], [4, -9, -2, 11], [-14, -16, 2, -9], [-14, 8, -12, 8], [-30, -8, 9, 10], [-18, 12, 14, 0], [-70, 4, -4, 0], [-24, -12, -8, 10], [-10, 4, -12, 11], [22, 6, 3, 7], [16, 2, -4, 12], [8, 0, -3, -8], [30, 13, -6, 0], [-52, -11, 3, -5], [48, 0, -5, -1], [4, 8, 4, 8], [14, 14, 5, -6], [-22, 4, 17, 3], [-32, -9, 6, -18], [0, 8, 1, -9], [4, -21, 13, -14], [16, 8, 12, -12], [8, -1, 18, -10], [-22, 0, 6, 2], [4, 0, -18, 6], [4, 4, -10, 7], [4, 10, 11, 8], [4, -22, 7, -13], [48, 2, 3, 2], [16, 0, 14, 2], [38, -16, 11, -10], [16, -2, 8, -18], [-14, -10, 6, 6], [4, 0, 0, 12], [-2, 3, -1, 13], [-30, -6, 8, -6], [40, 10, 5, -8], [-12, -3, 13, 12], [24, -6, -4, -4], [16, -5, 20, -9], [38, 4, 7, 1], [4, -6, 10, -4], [-54, -8, 0, -4], [16, -13, -10, 6], [0, -4, 12, 5], [-30, 10, -14, -6], [-8, 19, -11, 19], [42, -17, 4, -8], [-4, -8, -5, 14], [6, 9, -17, 8], [36, -1, -4, 8], [10, 4, 14, 6], [-32, -6, -9, -4], [-22, -3, 3, 13], [-16, 3, -3, -4], [16, 2, 8, 8], [-36, -9, -9, 5], [18, 11, -7, -11], [6, -9, 17, 5], [48, -12, 14, -10], [-22, 2, -19, 7], [24, 5, -8, 5], [0, 12, 5, -5], [-12, 28, -1, 8], [-42, 17, 3, 5], [-54, 4, 7, -5], [52, -17, 19, -7], [6, 12, 5, -8], [28, -14, 7, 7], [-46, -2, 11, -1], [20, 4, -1, 0], [34, 3, -7, 11], [-2, -9, -6, 5], [48, 20, 0, -8], [12, -16, 10, -2], [-42, -10, -6, 12], [8, -7, -1, -4], [-8, -15, -4, 8], [-26, 4, -1, 10], [-8, 1, 3, -19], [-22, 10, -4, 6], [-6, 12, -2, 12], [34, 18, 0, 8], [58, 8, 1, -7], [-12, 10, -10, -2], [26, -2, -11, 14], [-22, 13, -6, 5], [30, -6, 12, -12], [32, 4, -6, 16], [6, 8, 20, -16], [-18, -32, 4, -6], [-2, -36, 5, -7], [22, 24, -9, 6], [-24, 12, -23, -5], [74, 8, -6, -6], [14, 19, -11, -5], [18, 18, 1, 12], [-22, -8, -11, -12], [12, 6, 17, -8], [-28, 1, 2, -4], [10, 22, -5, 13], [2, 0, -8, -22], [-8, -4, 13, -7], [-12, -20, 9, -25], [28, 0, -9, -16], [66, -7, -5, -1], [24, -8, 16, 14], [32, 0, -9, 7], [-2, 5, 6, 8], [-14, 15, -11, -5], [-20, 21, -14, 0], [36, -17, 4, 4], [26, -2, 13, 2], [-38, -10, 4, -8], [-2, 8, 11, -5], [0, -2, 1, 3], [66, 4, 9, -8], [-20, 19, -10, -5], [42, 8, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7800_2_a_by();".
// 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_7800_2_a_by(:prec:=4)
    chi := MakeCharacter_7800_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(3361) - 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_7800_2_a_by();".
// 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_7800_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, 24, 6, -7, 1]>,<11,R![-4, -26, -20, -1, 1]>,<17,R![8, 36, -26, -3, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;