
// Make newform 7360.2.a.bq in Magma, downloaded from the LMFDB on 25 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_bq();".
// 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_bq();".
// 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 := [-5, -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], [0, 1], [1, 0], [1, -1], [-2, 1], [-3, -1], [-2, 1], [-3, -1], [1, 0], [-2, -2], [5, -3], [4, 0], [-4, -1], [0, -4], [-10, 2], [-6, 0], [8, 2], [-2, -3], [0, -4], [0, 3], [-4, 6], [8, 0], [6, 0], [4, 4], [6, -5], [-10, 2], [2, -3], [-10, 2], [-5, 3], [6, 0], [-4, 6], [-2, 4], [1, 1], [4, -6], [-11, 1], [-8, -1], [-2, 6], [-17, 3], [-4, 8], [16, 1], [6, -6], [-1, 7], [20, 2], [-4, -6], [-8, 7], [6, -8], [10, 0], [4, -4], [6, 6], [2, -8], [4, -2], [-4, -4], [-28, 0], [-29, 1], [12, -6], [14, -1], [-6, 6], [-7, 9], [-6, -4], [-6, 6], [14, -8], [6, -12], [-4, 7], [12, 0], [-13, 3], [-7, -1], [4, 6], [-15, 1], [-8, 1], [-26, 0], [-8, 4], [-2, 4], [-8, -4], [4, -6], [-2, 3], [-24, 0], [-10, 11], [-4, 7], [12, -6], [-21, 1], [-32, 4], [7, -9], [-2, 10], [-11, -7], [13, -7], [37, 1], [9, -9], [-10, 0], [12, 6], [-10, 0], [-8, 10], [-18, 12], [12, -2], [6, -6], [-16, 4], [17, -1], [-18, -6], [18, 0], [-6, -4], [-24, 8], [1, 9], [-16, 8], [4, 4], [-6, -12], [-14, 9], [30, 4], [28, 1], [-4, -10], [-24, 9], [-21, 1], [-28, 0], [-14, 0], [18, -15], [-1, 1], [0, 10], [4, 4], [10, -6], [10, 4], [-23, 7], [20, -4], [34, -3], [38, 0], [2, -16], [3, 3], [-26, -6], [-15, -3], [-22, 7], [-17, 7], [14, 9], [-26, 0], [-8, 0], [1, -5], [8, -6], [-12, 8], [-25, 5], [-18, -8], [0, -12], [12, 2], [32, -10], [20, 5], [30, -10], [-10, -4], [34, -4], [-16, 14], [10, -12], [-26, -2], [-21, -7], [2, -4], [10, 6], [-28, 2], [12, 11], [8, 8], [21, 5], [4, 4], [10, -6], [10, -2], [-44, -4], [-18, 12], [53, -3], [-46, 5], [17, -7], [-14, 7], [-16, -6], [-2, -5], [-9, -9], [-6, -3], [-36, 1], [10, 12], [20, -18], [20, -13], [20, -4], [14, -2], [-22, -10], [30, 7], [-4, 15], [14, 2], [6, -4], [-48, 0], [-18, -11], [-7, 1], [44, -6], [26, 11], [-11, 15], [-6, -6], [14, -7], [21, -3], [-43, 7], [-40, 4], [-28, 15], [-40, -1], [40, 2], [8, -16], [-20, 0], [44, -4], [26, -16], [33, -3], [-22, 6], [10, -24], [-42, 0], [-14, -2], [-14, 13], [-19, 21], [16, 3], [-28, -12], [8, -16], [-2, 4], [-38, 8], [-24, 15], [-23, -5], [-18, 14], [22, 2], [50, -4], [37, -1], [14, 2], [12, -12], [-10, -18], [52, -10], [-6, 12], [-41, -11], [-10, 8], [-30, 2], [6, -8], [16, -23], [-61, -3], [36, 0], [26, 10], [-25, -1], [49, -11], [-25, -3], [44, 8], [-37, 1], [-26, -3], [-40, 0], [12, -9], [10, 6], [-12, -12], [38, 6], [-22, -10], [34, 10], [24, -3], [-34, 2], [10, 12], [-2, -4], [12, 2], [38, 14], [-1, -13], [-38, 14], [-15, 3], [2, 7], [14, -1], [28, -18], [28, -26], [3, 21], [-40, -12], [-14, 4], [-4, 14], [-14, -9], [-32, 18], [17, 11], [14, 0], [34, 11], [22, -8], [-8, 24], [38, -2], [-14, 28], [-38, 9], [14, -7], [-10, 20], [-8, 6], [15, 9], [46, 6], [-8, 0], [-25, -9], [52, -4], [-10, 12], [-24, -2], [68, -1], [26, -26], [38, 0], [13, 7], [40, -20], [8, 0], [-18, 33], [18, -28], [17, 13], [12, -9], [26, -12], [4, 10], [-4, 21], [20, 17], [-22, 8], [-64, 2], [-4, 2], [23, 5], [32, -23], [-22, -19], [10, -10], [28, 1], [46, -8], [-32, 21], [-1, 3], [2, 2], [-48, -2], [16, -32], [31, -27], [-8, -10], [-16, -13], [-19, 1], [-42, 6], [16, 9], [14, -10], [-17, 7], [-52, 2], [4, -15], [-9, -15], [-10, 30], [30, -12], [-57, 3], [-42, 16], [-16, -13], [34, -9], [18, -29], [10, 4], [-22, -6], [18, 9], [20, -24], [-15, 11], [39, -1], [41, 11], [-62, -2], [-56, 18], [-52, -7], [48, -8], [-33, -9], [20, -8], [-33, -9], [38, -11], [-30, 24], [40, 5], [14, -9], [14, 4], [-6, 21], [-28, 8], [22, 2], [-22, -16], [28, 4], [-6, -4], [7, -21], [46, -8], [10, 10], [-18, -22], [14, 24], [8, 2], [-38, 23], [18, 8], [6, -18], [28, -14], [-7, 23], [58, 4], [54, 12], [44, -2], [42, -24], [0, 18], [-46, -13], [-32, 3], [54, -20], [40, 19], [-16, 24], [-6, -14], [-10, 32], [12, -22], [-58, -4], [34, 19], [8, -36], [-9, 17], [-14, -2], [4, 10], [-22, 3], [-22, 26], [12, 4], [-22, 11], [-22, -7], [-40, -6], [-10, -4], [28, -36], [-16, 20], [-3, -23], [37, 15], [34, 24], [53, -13], [-2, 20], [51, 9], [16, 4], [-48, -4], [-66, 12], [37, -7], [2, 0], [40, -20], [11, 1], [18, -12], [4, -6], [-36, -6], [-52, 0], [-73, -1], [8, -16], [20, 9], [-18, 2], [3, -3], [82, 6], [-16, -28], [-8, 2], [-4, -1], [28, 16], [10, -15], [-3, 25], [-18, 12], [-29, 7], [-10, 6], [22, -23], [29, -37], [42, 3], [-50, 6], [-58, 2], [10, -26], [38, 18], [22, -8], [64, -8], [-9, -15], [16, -6], [-4, 5]];
    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_bq();".
// 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_bq(: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_bq();".
// 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_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-5, -1, 1]>,<7,R![-5, -1, 1]>,<11,R![-3, 3, 1]>,<13,R![7, 7, 1]>,<17,R![-3, 3, 1]>],Snew);
    return Vf;
end function;