
// Make newform 7280.2.a.bn in Magma, downloaded from the LMFDB on 25 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_bn();".
// 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_bn();".
// 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 := [12, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-7, 1, 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_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], [0, 1, 0], [-1, 0, 0], [-1, 0, 0], [0, 0, 0], [-1, 0, 0], [1, -1, 1], [1, 1, -1], [-2, 0, -2], [-4, 1, -2], [-4, 1, 0], [0, -1, -2], [1, -1, 1], [4, -2, 0], [-2, 0, 2], [2, 0, 0], [-4, 3, 0], [-10, 0, 0], [-1, -1, -3], [0, -2, 0], [4, -4, 2], [-1, -1, 1], [2, 0, 2], [0, -3, -2], [2, 2, 4], [4, 0, 2], [1, 1, 3], [-2, -2, -2], [-2, 0, 4], [6, -2, -4], [-6, -4, 2], [2, -4, 2], [12, 1, -2], [2, -2, 2], [2, 0, 0], [-4, -2, 4], [-8, 3, -2], [-8, -1, 0], [4, -6, 4], [0, -3, 6], [-11, 1, -1], [2, -2, -4], [-12, -1, 0], [19, -3, 3], [8, 9, -2], [-4, 2, 4], [-1, -5, -3], [-6, -2, -2], [-10, 0, -2], [-7, -1, 1], [-8, 0, 2], [-2, 0, 2], [-1, 5, -5], [-8, 6, -4], [10, 8, 0], [-14, 0, 2], [-4, 4, 2], [4, 0, -4], [-8, 4, 2], [8, -2, 10], [15, -5, -3], [-8, 2, 2], [-10, 0, -2], [2, -6, -2], [4, 3, -6], [-10, 4, -4], [4, 0, -4], [2, 4, 0], [10, -4, 10], [8, -7, 6], [-14, 8, -4], [-6, 2, -10], [-16, 0, 4], [2, -10, 0], [2, 0, 6], [14, -6, 2], [3, 9, -5], [-12, 6, -2], [0, -6, -6], [-26, 4, -4], [-6, -2, 2], [-6, 4, 0], [6, -2, -6], [-19, -1, -3], [4, 4, -4], [2, 0, 2], [14, 2, 4], [4, 1, -2], [4, -9, 2], [3, -5, 5], [-13, 7, -7], [-16, 1, -4], [16, 7, -4], [0, 4, 4], [24, -4, 8], [-16, 0, 4], [21, -1, -3], [-8, 0, 10], [-16, 8, -4], [-20, -2, -2], [-10, 4, -2], [-1, 5, -5], [1, 9, -5], [8, -5, 6], [16, 1, 0], [4, 4, 10], [6, -6, 6], [12, 0, -6], [-12, 0, 4], [-14, 10, 0], [16, 1, 0], [6, 4, 4], [33, 3, -3], [24, 3, 4], [2, -6, 6], [-8, 7, -2], [-26, 4, 6], [-8, 1, 0], [4, -10, -2], [-17, 3, -11], [-1, -3, 11], [-12, 8, -2], [6, 4, 4], [-9, 3, 5], [-4, 5, 0], [29, -5, 1], [12, 8, -2], [10, 14, -2], [-20, 3, -4], [10, 4, -8], [-18, 2, 2], [-5, -5, -11], [28, -7, 8], [6, 0, -4], [-4, 13, 2], [2, 12, 0], [-10, 12, 4], [28, -8, 0], [-8, -1, 6], [-15, 7, 1], [4, 0, -12], [-12, -6, 6], [-28, 8, 0], [0, -8, -4], [-8, 8, 6], [20, 0, 4], [-32, -4, 2], [20, -9, -6], [-14, 2, -6], [9, 5, 7], [1, -9, 9], [-18, 6, 12], [-20, 4, -8], [28, 1, -4], [-24, -8, 4], [-11, -7, 11], [16, -1, 4], [-20, 5, -6], [-27, 3, -3], [9, -5, -7], [5, 1, 7], [8, -16, -6], [-32, 5, -4], [-8, -8, 4], [7, 13, -9], [34, 4, -2], [-21, -9, -3], [-39, -1, -7], [-18, -6, -4], [31, -3, 11], [0, -10, 0], [24, -7, -2], [20, 0, -4], [-14, -4, 4], [-26, -4, -6], [24, -10, 2], [12, 2, 4], [-12, 14, -2], [4, -11, 12], [-20, -6, -6], [12, 1, 0], [-18, -12, 6], [-12, 6, 14], [38, -6, 8], [-16, 8, -16], [-4, 8, 10], [44, 0, -2], [-22, 6, -6], [20, -4, 6], [16, -9, 4], [-9, 9, 11], [8, -9, 8], [-4, -7, -4], [-15, -1, -15], [2, 14, -6], [34, -4, 0], [32, -10, 10], [-12, -8, -10], [32, -8, 6], [-16, -7, 8], [4, 10, -2], [-30, 0, -10], [14, 6, 4], [6, -4, -4], [-35, 1, -5], [13, -17, 13], [1, 1, -1], [-22, 4, -14], [-2, -8, -12], [-16, 8, -12], [-14, 6, -8], [-40, 2, -10], [18, -4, -6], [16, -18, 4], [8, 8, 8], [-4, 3, -14], [0, 10, 4], [-4, -4, -2], [0, 17, -4], [17, -13, -7], [-7, 7, 13], [-5, -9, 13], [14, 8, 4], [-3, 21, -13], [19, -17, 1], [-44, -1, -2], [-4, 11, 2], [-12, -1, 16], [35, -1, 1], [-12, 2, 4], [-20, 4, -2], [-10, 2, -2], [0, 16, 0], [8, 5, 14], [-5, -1, 17], [-16, 14, -16], [22, 12, 4], [24, -9, 6], [9, 9, 3], [6, -16, 10], [-20, 11, -12], [4, -10, -8], [-20, 19, -4], [4, 0, 14], [12, 4, 2], [-2, 0, 10], [18, 16, -2], [17, -3, -9], [-4, -14, -4], [46, -4, -2], [-21, 9, -1], [-23, 11, -15], [-40, 12, -16], [-14, 10, -16], [-2, -4, -4], [-16, -2, 0], [-27, 3, -15], [-4, -8, -8], [35, 5, 7], [16, -2, -2], [22, -8, 14], [6, 10, 6], [8, -3, -10], [-16, -7, 6], [26, -8, -12], [6, -6, 14], [8, -4, 2], [-19, -9, -11], [-40, 5, 8], [2, 8, -8], [-16, 4, -2], [-7, 1, 19], [12, -8, -14], [11, 15, -3], [23, -11, -5], [-28, 14, 12], [-23, 9, 3], [-14, 8, -8], [12, 8, -2], [-28, 2, 8], [61, -3, -1], [-32, 10, 8], [44, 11, 4], [-2, 12, -12], [-24, -16, -4], [-23, 9, -1], [12, -16, 6], [30, 16, -4], [-11, 13, -13], [24, 8, 2], [46, -8, -8], [-20, -8, 8], [-4, -7, -18], [-15, -3, -1], [-68, 2, -2], [48, -4, 6], [6, -12, -6], [35, -11, 15], [-2, 20, -10], [28, -14, -16], [-1, 17, 3], [-1, 1, -21], [-44, 9, -16], [-4, 4, -16], [-25, 7, -11], [35, -15, -5], [18, -14, -6], [-12, -22, 2], [-44, -4, -4], [16, -14, -6], [5, 1, -1], [-2, 2, -4], [17, 7, -7], [46, -2, 14], [-12, 17, 16], [-48, 2, 10], [4, -16, -12], [-24, 14, 0], [76, 0, 2], [3, 17, 7], [6, -16, 6], [-17, 21, -9], [-44, 10, 2], [-52, 1, 8], [-28, 3, -2], [22, 14, 4], [-14, -14, -10], [22, -14, -18], [24, 14, -4], [32, -4, 10], [-4, -14, 6], [-2, 0, 4], [-10, 6, -6], [20, 14, 8], [-40, 4, 16], [-36, 12, 0], [-24, -4, 2], [16, 18, -2], [-8, 15, -2], [12, 9, 12], [-12, 3, -6], [-8, 2, -14], [28, -9, -14], [-16, 17, -4], [6, -8, -4], [24, 7, 8], [-10, 4, 0], [-24, 20, -20], [-39, 9, 15], [34, -4, -8], [-24, 9, 4], [-6, 20, -12], [6, -8, 12], [8, 23, 0], [-40, 15, -14], [12, -23, 18], [-15, 9, -9], [-60, 3, -8], [26, 2, -16], [-8, -5, 12], [-4, 31, -14], [-22, 20, -16], [0, -10, 4], [-18, 0, 2], [-44, -13, 4], [30, -2, 12], [-30, -8, 0], [-57, 7, -15], [16, -7, -10], [49, 1, -5], [-32, -2, -4], [36, -5, 8], [-6, -8, -4], [6, -22, 2], [-8, -12, 10], [-11, -15, 3], [19, -13, -27], [46, -14, -16], [-12, 22, -18], [-68, 5, 2], [28, -15, 10], [66, -4, 12], [-9, -13, -15], [17, -9, 13], [20, 1, 24], [8, -6, 8], [10, 16, -18], [17, -5, -23], [16, 8, 4], [20, -5, 12], [-66, -6, -4], [12, 16, -10], [-46, -6, -2], [20, 7, 12], [0, -10, -8], [-28, -26, -2], [16, -17, -4], [-22, 10, -24], [16, -11, 8], [38, -8, 4], [-33, -19, 11], [8, -2, -14], [-56, 5, 4], [24, 2, -2], [-5, -11, -1], [-40, -10, 8], [13, 19, 13], [0, -8, 18], [-58, 14, -26], [-32, -8, -8], [2, 8, -8], [-6, -4, 24], [-10, 0, -26], [47, -1, 13], [46, -16, 20], [32, -8, 6], [-12, 8, 12], [20, 16, -8], [-18, 16, 16], [-60, 5, 12], [-70, 0, -4], [10, 4, 0], [-17, 15, 17], [0, 12, 12], [44, 2, -14], [26, 10, 0], [24, 32, -4], [-14, -10, 0], [-40, -4, 16], [-12, 2, 12]];
    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_bn();".
// 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_bn(:prec:=3)
    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_bn();".
// 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_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![12, -9, -1, 1]>,<11,R![0, 1]>,<17,R![18, -15, -1, 1]>,<19,R![8, -7, -5, 1]>,<23,R![-48, -36, 4, 1]>],Snew);
    return Vf;
end function;