
// Make newform 7280.2.a.bc in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7280_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_7280_2_a_bc();".
// 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_7280_2_a_bc();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_7280_a();"
function MakeCharacter_7280_a()
    N := 7280;
    order := 1;
    char_gens := [911, 5461, 1457, 4161, 561];
    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_7280_a_Hecke(Kf)
    return MakeCharacter_7280_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, 1], [1, 0], [-1, 0], [-4, 0], [1, 0], [-2, -1], [4, -1], [4, -1], [6, 0], [4, 2], [2, 2], [-2, 3], [0, 4], [0, -4], [6, -1], [-4, 1], [-6, 2], [4, -2], [-8, 1], [6, 2], [0, -4], [8, 2], [14, -1], [6, 4], [2, -2], [0, 0], [-8, 0], [-2, 3], [10, 0], [4, -3], [-16, 0], [-10, 0], [8, 4], [6, 3], [8, 5], [-2, 0], [4, 2], [8, -6], [-10, 0], [-12, 0], [-6, -6], [-16, -2], [6, 0], [-6, 2], [0, 2], [12, -2], [-8, -6], [16, -2], [-2, -2], [-6, 6], [-8, -3], [-2, 1], [-8, 6], [6, -1], [-4, -9], [10, 2], [12, -6], [-2, 5], [10, -6], [24, 1], [-10, 0], [-16, -4], [-8, -6], [14, -5], [10, -4], [4, 0], [-6, 6], [-24, 2], [-2, -6], [14, 0], [8, 5], [8, -6], [-2, -1], [-12, -6], [24, 2], [-2, -4], [6, -4], [-6, 0], [-10, -9], [16, 0], [14, -9], [16, 7], [6, 3], [0, -10], [0, 8], [-14, -2], [6, -12], [30, 2], [-24, 0], [-16, -3], [12, 8], [16, 0], [20, 6], [20, 0], [8, -4], [-26, 2], [-22, 2], [-24, -3], [14, -9], [-8, 6], [18, -2], [-16, 5], [10, 0], [36, -2], [14, -10], [24, 2], [-2, -12], [-32, 0], [26, 4], [40, 0], [-6, -4], [-10, 0], [-12, 3], [0, 9], [26, 0], [8, -6], [8, -6], [-10, 1], [-28, 0], [-2, 16], [-22, -4], [6, -4], [20, -8], [-28, 1], [6, 6], [-2, 3], [0, -12], [-16, -6], [-2, -12], [20, 12], [24, 6], [-32, 4], [6, 1], [-18, -11], [22, 3], [-2, -8], [-16, 4], [38, 0], [-6, -8], [-20, -1], [14, 7], [12, 11], [4, -2], [2, -6], [4, 10], [6, 4], [-34, 3], [8, 14], [-8, 16], [-6, -14], [-30, 4], [8, 2], [16, 6], [8, 10], [0, 12], [-8, -2], [-10, -11], [14, 1], [-10, -8], [-28, 2], [2, -10], [-24, 0], [8, -10], [22, -4], [8, 0], [-32, 4], [22, 6], [-6, 2], [-18, -4], [12, -4], [38, 2], [-16, 2], [30, 4], [-16, -16], [2, -6], [4, -18], [-10, 1], [-32, -10], [26, -10], [48, 0], [24, -10], [6, -3], [38, 8], [32, -12], [-22, 10], [-2, -7], [0, 2], [2, 16], [28, -10], [22, 3], [-4, -14], [-20, -14], [-26, -10], [-8, 18], [6, 5], [50, 2], [-2, -5], [46, 4], [48, 6], [-10, 11], [8, 6], [-18, 4], [18, 4], [12, -15], [-34, 6], [4, -12], [56, 2], [50, 0], [8, -4], [-30, 8], [-22, 6], [28, -13], [0, 12], [-20, 10], [-18, -1], [12, -11], [26, -12], [-8, 10], [-6, 14], [14, -6], [20, 10], [10, -14], [-24, -2], [-16, 3], [-18, 2], [-18, 5], [-16, 10], [16, -4], [-52, -8], [-18, -13], [16, 2], [16, 13], [34, 0], [-40, 7], [-32, 6], [-34, -1], [-62, -4], [28, 2], [-56, -2], [-12, 0], [-8, 14], [0, -8], [-42, -11], [-2, 18], [40, -6], [24, 8], [-52, -9], [4, -6], [-20, 3], [-30, -10], [-10, 5], [-64, -2], [-6, -4], [6, 10], [-12, -16], [-50, -6], [48, 2], [-18, 2], [62, 6], [12, -21], [-8, 6], [-50, 4], [-42, -2], [6, 24], [8, -12], [-26, -1], [30, 1], [28, 4], [-2, -3], [10, -6], [12, 4], [-10, 12], [16, 10], [6, -13], [-8, 18], [-4, 8], [-10, -7], [-30, 16], [-32, -12], [-32, 2], [8, 1], [-16, -10], [-18, 0], [0, -6], [-8, 10], [2, 26], [-38, -2], [20, -16], [42, -6], [-2, -6], [8, -15], [-2, -28], [12, 21], [6, -23], [2, 10], [-24, -14], [-26, -12], [0, -28], [-32, -12], [6, -15], [-46, 8], [-16, -4], [40, 12], [-12, 8], [-42, -12], [24, -20], [58, 10], [0, -21], [-42, -11], [32, 12], [14, -9], [18, -16], [-8, -24], [16, 20], [-6, -28], [12, 11], [-24, -9], [78, 2], [10, 8], [8, 8], [38, -8], [-6, -6], [-32, 14], [-18, -8], [-14, 14], [36, 6], [-16, 6], [-52, 5], [30, -9], [-26, -21], [22, -7], [32, 0], [32, 10], [4, -28], [24, -4], [-50, -1], [-2, 12], [6, 13], [0, 0], [50, -4], [10, -4], [-26, -18], [-32, -8], [2, 24], [4, -16], [18, 2], [68, 0], [-52, -2], [-10, -2], [-4, -15], [-34, -16], [-42, 7], [0, 28], [22, 4], [-82, 3], [-60, 4], [20, 7], [-38, 6], [-32, 8], [82, 0], [-6, 12], [12, 27], [-44, 2], [40, -7], [42, 6], [-18, -12], [0, 8], [82, -4], [12, -20], [40, -14], [48, -6], [-42, 8], [56, -8], [-34, 11], [-68, 7], [24, 14], [-34, 10], [10, -20], [30, -3], [70, -10], [-26, -20], [-48, 8], [22, 8], [-4, 9], [-48, 6], [-56, -5], [-18, -10], [-20, -2], [-80, 6], [-22, 2], [38, -4], [-20, -20], [64, 9], [-64, 1], [-14, -20], [44, 12], [-38, 18], [-68, 6], [-34, 17], [14, 28], [-14, 22], [16, 2], [-66, 4], [-18, 2], [48, 4], [-38, -4], [-30, 2], [48, 16], [-28, -5], [-26, 3], [54, 1], [-8, 18], [20, 14], [58, 6], [-78, 2], [0, 3], [-40, -2], [-74, -8], [-48, -2], [22, -18], [54, -12], [56, 6], [-32, 24], [-2, -2], [38, -16], [32, -2], [-62, 2], [-12, 5], [-40, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7280_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7280_2_a_bc();".
// 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_7280_2_a_bc(:prec:=2)
    chi := MakeCharacter_7280_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_7280_2_a_bc();".
// 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_7280_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 0, 1]>,<11,R![4, 1]>,<17,R![-4, 4, 1]>,<19,R![8, -8, 1]>,<23,R![8, -8, 1]>],Snew);
    return Vf;
end function;