
// Make newform 7400.2.a.k in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7400_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_7400_2_a_k();".
// 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_7400_2_a_k();".
// 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 := [-1, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 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_7400_a();"
function MakeCharacter_7400_a()
    N := 7400;
    order := 1;
    char_gens := [5551, 3701, 1777, 1001];
    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_7400_a_Hecke(Kf)
    return MakeCharacter_7400_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]];
    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, 0, -1], [0, 0, 0], [-2, -1, 1], [0, -3, 0], [-1, 3, 0], [2, -2, 2], [2, 2, -2], [-3, 1, 0], [-3, 1, 0], [6, 0, 1], [1, 0, 0], [-5, -2, 1], [0, -2, -4], [-4, -3, -1], [2, -5, 3], [-2, -2, -4], [4, 2, -3], [0, 6, -5], [-2, 3, -1], [2, 3, 0], [-1, 1, -2], [4, -5, 3], [4, 0, -4], [-2, -2, -6], [-2, -5, 3], [-6, -2, -3], [5, -5, 8], [14, -2, -2], [2, -4, -4], [-2, 9, 1], [-2, -6, 0], [-2, 6, -5], [-3, 3, 4], [-2, -1, 1], [4, 8, -2], [-2, 5, -3], [-10, 0, -4], [-2, 4, 7], [6, -1, -1], [8, 2, 8], [4, 7, -1], [2, -8, 5], [4, -6, 2], [10, 5, -3], [0, -2, -2], [16, 1, -2], [-4, 5, -5], [4, 0, -2], [-16, -3, -5], [3, -9, -2], [-2, 2, -1], [4, -6, 6], [-8, -2, -2], [-2, -6, -6], [0, -11, -1], [12, -6, 10], [-8, 5, 1], [8, 6, -3], [4, -8, 0], [-12, 8, -2], [24, 2, 4], [13, 0, 5], [23, 1, 2], [14, -4, 6], [-8, -10, 6], [-4, -6, 2], [18, 3, 0], [28, -2, 0], [16, -2, 0], [-8, 0, 2], [14, -5, -5], [2, 6, -8], [4, -11, 7], [-25, -4, -1], [0, 12, -4], [21, -1, 2], [0, -9, 1], [6, 4, -6], [16, -12, -2], [5, 6, -1], [-1, 17, -2], [0, 6, 10], [3, -10, -7], [11, -3, 6], [-4, -3, 8], [-2, 2, -10], [-18, 4, -2], [-2, 4, 12], [13, -11, 0], [-4, 14, 2], [5, 5, -8], [12, 0, -4], [-3, 3, 8], [2, -6, 0], [-10, 4, 1], [6, -7, 11], [22, 3, -5], [18, -12, 2], [4, 2, -3], [20, 0, 6], [8, 4, 9], [2, -20, 12], [20, 0, 8], [29, -2, -1], [-6, -2, 4], [-14, 10, -6], [18, 7, -8], [-2, -13, 11], [-22, 2, 1], [-17, 1, 2], [8, 11, 9], [25, 8, 1], [12, 5, 8], [-25, -3, 2], [7, 2, 5], [2, 4, -2], [-9, -3, -2], [11, -3, -2], [-1, -2, -11], [-5, -15, -6], [25, 2, 11], [10, -13, 21], [-8, 2, -10], [8, -12, 4], [16, -2, 7], [-16, -2, 1], [12, -1, 15], [-14, -12, 1], [26, 1, -7], [-4, 13, 4], [-20, 7, 1], [2, -1, -5], [-8, 20, -13], [-7, -14, 3], [-10, 8, 0], [-24, -1, -13], [4, -7, 17], [37, 3, -6], [-6, 12, -10], [24, -7, 12], [-6, 17, -15], [16, 8, 0], [14, -4, 10], [-1, 1, -2], [24, -8, -10], [16, -12, -5], [-2, 18, -14], [-10, -26, 18], [4, 12, -8], [-6, 4, 12], [6, 2, -13], [-8, 12, -8], [-18, -15, 5], [-10, -10, 6], [-4, 10, 2], [-4, 10, -14], [10, -2, -5], [-23, 4, 1], [8, -18, 8], [-12, 8, -8], [2, -23, 18], [14, 10, 1], [12, 10, 11], [18, 6, 2], [-20, 15, -5], [3, -21, -8], [-2, 0, 0], [-18, -13, 10], [4, 30, -7], [-22, -4, 2], [15, -1, 8], [-14, 16, -5], [7, -12, -17], [22, -13, -3], [8, 4, -4], [-4, 8, -10], [-16, 14, 10], [-16, 17, -21], [14, 0, 6], [-12, -16, 4], [-2, 4, 4], [24, -2, 23], [10, 6, -2], [-4, 8, -10], [14, 5, -5], [2, 3, -5], [-10, -28, 2], [-28, 6, -22], [-24, 3, -5], [-28, 10, -18], [36, 8, 4], [-4, 14, -4], [6, -7, 15], [24, -5, 2], [-2, 7, 15], [-18, 10, -24], [-26, 18, -14], [7, 0, 15], [-14, 6, 17], [-36, 6, -13], [26, -5, 19], [6, -8, 2], [5, -24, -11], [-17, -3, 4], [-4, -34, 9], [-30, 14, 8], [28, -2, -13], [0, -4, -4], [44, -1, 15], [-2, 12, 0], [22, 36, -20], [29, 3, -6], [12, -6, 1], [2, 12, -15], [10, -23, 22], [-22, 6, 3], [6, 12, 8], [2, 30, -17], [0, 14, -18], [14, -17, 11], [-16, 11, -17], [18, 7, -1], [5, -7, -8], [-12, -11, -16], [-1, 5, 6], [37, -15, 2], [30, 1, 7], [-14, 11, 13], [-18, 20, 0], [-4, -17, 5], [8, 25, -18], [-10, -7, 13], [-42, 11, -4], [28, -15, 2], [-34, 1, 11], [-7, 17, 2], [-7, -3, -12], [-48, 14, 2], [-34, 14, 7], [-16, -20, 12], [20, -24, -10], [34, -5, 17], [-16, 0, 9], [-35, 13, -22], [29, 9, 4], [-27, 25, 12], [0, 8, -24], [7, 12, 7], [-20, 28, -4], [8, -6, -5], [14, 10, 7], [18, 2, -6], [-14, -10, -6], [47, 3, -4], [5, 22, -1], [-24, 3, -15], [20, 0, 4], [0, -7, 11], [-60, -6, 0], [-37, 9, -18], [2, -6, 10], [8, 10, 2], [0, -18, -4], [20, 16, -14], [23, 14, 1], [0, -5, 11], [-28, -12, 8], [-40, 19, -16], [-3, 11, -26], [-36, 2, -9], [-4, -12, -8], [22, 0, 0], [15, 27, -24], [-7, 5, 2], [20, -24, 12], [13, -5, 16], [-4, 22, 19], [-35, 16, -3], [-10, 0, 2], [-2, -13, -9], [-44, 14, -10], [4, -20, 18], [14, 37, -27], [18, 0, 28], [42, -15, 19], [4, 14, 6], [20, -14, 2], [54, -2, -3], [-2, -24, 22], [22, 22, -10], [-6, -12, 0], [22, -25, 11], [24, -23, 13], [-28, -11, -7], [16, -11, 13], [-6, 19, -21], [-14, -24, 18], [18, 16, 2], [-3, 24, 9], [12, -18, -10], [-4, 21, 23], [26, 1, 27], [-42, 6, -18], [24, 0, 8], [-16, -42, 22], [-12, -10, 16], [1, 17, -8], [-32, -17, -9], [1, -15, 8], [-32, -25, 5], [0, -29, 5], [38, 3, -18], [-64, 3, -10], [-55, 3, -6], [6, -10, -4], [48, 1, -3], [31, 29, -14], [29, 16, 5], [36, 2, -2], [-40, 12, -6], [6, 27, -2], [-8, 4, 5], [62, 2, 2], [-14, 17, 18], [-22, -10, -22], [-3, 11, -32], [48, 4, 6], [-18, -6, -1], [-22, -4, -16], [14, -19, -25], [-8, -26, 7], [14, -14, -5], [-12, -8, -4], [-8, 20, 12], [-20, 6, 3], [80, -1, 3], [50, 5, 17], [38, -14, 14], [40, 12, 14], [-6, -1, -7], [-7, 2, -9], [33, -7, -8], [-22, 4, 11], [-7, 5, 4], [34, -8, -12], [12, -1, 35], [12, 15, -17], [-33, -15, -10], [20, -17, 1], [-51, 23, -4], [-10, 7, 6], [-18, -6, -6], [40, -22, 2], [-26, 4, -16], [1, 1, 10], [-25, -31, 2], [-22, -10, 22], [46, 19, -14], [14, -26, -5], [-17, -5, 8], [14, 7, -4], [30, -14, -13], [58, -4, 18], [41, -8, 29], [42, -4, 10], [15, 15, -24], [30, -2, 35], [12, 30, -2], [-64, 6, 2], [-2, -44, 2], [-24, -24, 4], [-18, -40, 14], [-15, 1, -8], [4, -7, -3], [36, 8, 24], [2, 5, -31], [17, -23, 10], [-30, 20, 2], [-45, -18, 9], [-13, -15, -2], [10, -10, -24], [-6, 24, -25], [10, -14, -19], [-10, 38, -26], [-28, -13, -21], [-2, -43, 35], [48, 8, 11], [36, 18, -12], [62, -2, -5], [18, -29, 27], [-16, -36, 17], [4, 6, 16], [36, -14, 16], [-22, 7, -11], [-38, -12, 4], [22, 0, 3], [-10, 3, -30], [28, 21, -8], [2, -32, -18], [60, 19, -5], [-40, -6, 8], [-59, -9, -6], [30, 32, -20], [7, 23, 20], [28, 3, 7], [12, 44, -17], [-10, -42, 15], [-1, 41, -20], [31, 32, -5], [-30, -31, 9], [-36, 23, -5], [-20, -22, 24], [-38, -6, -22], [16, -4, 20], [-18, 24, -28], [-28, 8, 18], [-64, -1, 3], [34, -12, -16], [11, 37, -10], [-16, -22, -21], [28, 8, -17], [20, -36, 36], [40, 21, -15], [-10, 32, 16], [50, -19, -3], [-12, 30, -21], [-30, 3, 24], [0, 2, 7], [76, -6, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7400_2_a_k();".
// 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_7400_2_a_k(:prec:=3)
    chi := MakeCharacter_7400_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_7400_2_a_k();".
// 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_7400_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-7, -4, 2, 1]>,<7,R![-4, 10, 7, 1]>],Snew);
    return Vf;
end function;