
// Make newform 7360.2.a.bc in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7360_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_7360_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_7360_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 := [-3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_7360_a();"
function MakeCharacter_7360_a()
    N := 7360;
    order := 1;
    char_gens := [1151, 5061, 4417, 6721];
    v := [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_7360_a_Hecke(Kf)
    return MakeCharacter_7360_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], [-1, -1], [-1, 0], [2, -1], [3, 1], [-2, 1], [3, -3], [-2, 3], [-1, 0], [0, -2], [-4, 3], [-8, 0], [-3, -3], [4, -4], [0, 2], [6, -4], [6, 2], [-5, 5], [4, 0], [-15, 1], [2, 6], [-4, 8], [-6, 4], [0, 0], [5, -1], [0, 2], [-1, -7], [0, -2], [-8, -5], [-6, 8], [2, -6], [6, -4], [0, -3], [10, -2], [-12, 1], [5, -3], [16, -2], [-8, 1], [12, 4], [-21, -1], [-12, -2], [-8, 7], [-6, 2], [-10, 6], [-3, 1], [-10, 8], [-2, 4], [-4, 0], [12, -2], [-2, 0], [18, -6], [0, -4], [-4, 8], [-6, 5], [6, 6], [-3, -5], [-12, -6], [-4, -1], [22, -4], [-24, 2], [-2, 0], [-18, 4], [13, 1], [12, -8], [8, -9], [-6, 9], [-14, 2], [-16, 5], [-9, 15], [22, -4], [0, 8], [6, -4], [20, 4], [-2, 2], [-11, 3], [0, 0], [9, -13], [1, 9], [-6, 2], [-22, 3], [12, 12], [-32, -1], [-12, 14], [-28, -3], [-16, -1], [-24, -5], [-6, -3], [-22, 8], [-18, -6], [2, -12], [-6, -6], [-30, 0], [2, -6], [12, 6], [16, -12], [18, -9], [12, -18], [6, 0], [10, -8], [-8, 16], [4, -5], [-12, 8], [-12, -12], [18, -8], [7, -7], [2, 0], [3, 15], [6, 6], [-21, 11], [26, -13], [20, -20], [-14, -8], [15, -3], [-2, -11], [26, 2], [-36, 0], [16, -14], [6, -16], [-30, 9], [0, 12], [31, -3], [-22, 12], [-6, 8], [12, 9], [-44, 6], [-36, -3], [25, 7], [12, 1], [-1, -3], [-14, 12], [-20, 12], [-18, 11], [14, -6], [-8, 4], [-30, 15], [14, -24], [-12, 0], [-14, 18], [6, 6], [39, -9], [28, -10], [30, 0], [-10, 4], [6, -18], [-26, 4], [-36, 6], [10, -9], [-6, -12], [28, -6], [-18, 18], [13, -27], [24, -24], [-20, -9], [12, -12], [-20, 22], [-12, -6], [44, 0], [-30, -12], [8, -23], [-9, 13], [-12, -13], [-27, 11], [-10, 2], [21, 3], [0, -5], [15, -15], [-1, 11], [22, 4], [26, 10], [-51, -3], [0, -20], [16, 6], [36, -2], [17, -21], [-7, -19], [-12, -6], [-14, 20], [12, 8], [11, -15], [-56, 1], [14, 14], [9, 3], [22, 9], [0, 6], [-33, -11], [24, 5], [22, -7], [4, -8], [29, -11], [45, -3], [-10, 6], [0, -24], [-20, -8], [-36, 4], [6, -12], [-36, -7], [20, -30], [22, 20], [-30, 16], [24, -30], [-33, -3], [44, 3], [1, 21], [32, -12], [0, -24], [18, -20], [26, 12], [57, 1], [6, -23], [4, -30], [-4, 10], [30, -12], [38, 7], [-36, 2], [24, -12], [-4, 14], [26, 2], [-18, -4], [-30, -3], [-42, -8], [4, -18], [-34, -8], [-33, 3], [14, -11], [-36, 16], [-20, -10], [0, -5], [24, -21], [50, 5], [36, -28], [10, -17], [-17, 13], [20, -4], [21, 5], [-8, 6], [0, 12], [8, 14], [0, -6], [-24, 6], [39, -25], [36, -22], [-2, 0], [26, -28], [-2, -6], [0, -18], [-24, 37], [-40, 26], [6, -33], [37, 7], [-27, 3], [-26, -10], [18, 26], [30, -11], [-4, 20], [6, 12], [54, 2], [1, 23], [-14, -26], [-42, 5], [-46, 8], [5, 15], [-30, -20], [16, 8], [28, -18], [18, 0], [-17, 9], [15, -15], [6, 0], [58, -6], [6, -9], [52, 2], [-44, 8], [-16, 19], [-28, 28], [14, -12], [26, 14], [-3, 1], [16, -2], [-10, -16], [-42, 27], [48, -8], [20, -40], [39, -11], [58, 4], [-8, 23], [-15, 29], [2, -12], [18, 6], [-37, 23], [3, -21], [-18, 24], [30, -30], [-54, 2], [42, 1], [43, -17], [-57, 13], [-4, -6], [-9, -1], [6, -8], [-5, -5], [-4, -9], [-72, -2], [-46, -14], [24, 8], [34, 1], [-10, -30], [45, -27], [28, -23], [12, -6], [19, -33], [24, -6], [72, -1], [-6, -2], [-77, 7], [30, -15], [20, 18], [6, 4], [48, -19], [-22, 40], [45, 17], [13, 15], [-31, 39], [-18, 0], [20, 6], [15, 21], [44, 8], [46, -1], [-44, 13], [-42, 27], [-12, 6], [10, 6], [15, -25], [-64, 0], [-36, -3], [-44, 0], [-12, 5], [19, 21], [6, 16], [11, -33], [-43, 3], [-14, 36], [3, -7], [36, -4], [-16, -34], [-54, -12], [-48, 0], [-38, -8], [-8, 9], [-18, -24], [0, -34], [-68, 18], [-10, 16], [-6, -2], [41, 3], [-50, 0], [-48, -6], [6, -2], [18, -45], [-18, 0], [-30, 16], [-14, 6], [-54, 0], [30, -38], [-15, -5], [37, 5], [50, -4], [9, -3], [56, 0], [20, 6], [-54, 4], [-62, -14], [-18, 0], [21, -13], [-16, -24], [-2, -9], [12, -10], [-6, -6], [17, -1], [36, -42], [-40, -4], [21, -37], [15, 13], [38, 26], [6, -8], [64, -20], [36, 12], [56, 15], [-26, -23], [-26, -28], [42, 3], [-34, 32], [18, -9], [-48, 12], [-44, 12], [-54, 24], [-28, 5], [2, 0], [12, 16], [34, 21], [6, 4], [22, -18], [6, 26], [-4, 0], [0, -21], [-12, 28], [-1, 11], [-44, -6], [6, -25], [-20, 10], [-48, -20], [38, -34], [-15, -7], [24, 24], [31, -47], [20, 5], [30, -28], [-24, 33], [32, -2], [-21, -3], [-6, 35], [-27, 11], [28, 10], [24, 14], [-24, 30], [44, -10], [30, 0], [-24, 36], [6, 35], [-26, 34], [-45, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7360_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7360_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_7360_2_a_bc(:prec:=2)
    chi := MakeCharacter_7360_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_7360_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_7360_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 3, 1]>,<7,R![-1, -3, 1]>,<11,R![9, -7, 1]>,<13,R![-1, 3, 1]>,<17,R![-27, -3, 1]>],Snew);
    return Vf;
end function;