
// Make newform 2900.2.a.h in Magma, downloaded from the LMFDB on 07 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2900_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_2900_2_a_h();".
// 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_2900_2_a_h();".
// 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, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-2, 0, 1, 0], [0, -3, 0, 1], [0, 5, 0, -1]];
    Rf_basisdens := [1, 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_2900_a();"
function MakeCharacter_2900_a()
    N := 2900;
    order := 1;
    char_gens := [1451, 1277, 901];
    v := [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_2900_a_Hecke( : Kf := Kf)
    return MakeCharacter_2900_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, 0], [0, 0, -1, 0], [0, 0, 0, 0], [0, 0, 1, 0], [-1, -1, 0, 0], [0, 0, -1, -1], [0, 0, 2, 1], [1, 3, 0, 0], [0, 0, 0, 1], [-1, 0, 0, 0], [-5, -3, 0, 0], [0, 0, 2, 1], [-2, 2, 0, 0], [0, 0, -1, 0], [0, 0, -1, -4], [0, 0, -1, -1], [-2, 0, 0, 0], [-6, -2, 0, 0], [0, 0, 4, 3], [-2, 4, 0, 0], [0, 0, 3, 2], [-5, -5, 0, 0], [0, 0, 5, -2], [-4, -2, 0, 0], [0, 0, -3, 4], [-6, 4, 0, 0], [0, 0, 0, -3], [0, 0, -1, 4], [-12, -4, 0, 0], [0, 0, -1, -6], [0, 0, -3, 0], [-1, 5, 0, 0], [0, 0, -12, -1], [0, 4, 0, 0], [-2, -8, 0, 0], [-4, 6, 0, 0], [0, 0, -3, -6], [0, 0, -3, 8], [0, 0, -8, 1], [0, 0, -1, 5], [14, 2, 0, 0], [-16, -4, 0, 0], [-7, 1, 0, 0], [0, 0, -12, -1], [0, 0, -4, -2], [-4, -2, 0, 0], [5, -11, 0, 0], [0, 0, -6, -7], [0, 0, 4, -1], [-10, 0, 0, 0], [0, 0, -8, -6], [2, -8, 0, 0], [-4, 6, 0, 0], [9, 1, 0, 0], [0, 0, -3, 9], [0, 0, 5, 4], [6, 14, 0, 0], [1, -3, 0, 0], [0, 0, 3, 7], [0, -4, 0, 0], [0, 0, -10, -5], [0, 0, -3, 0], [0, 0, 9, -4], [-5, 5, 0, 0], [0, 0, -13, 5], [0, 0, 8, -5], [1, -1, 0, 0], [0, 0, 1, 0], [0, 0, 3, 4], [-22, 4, 0, 0], [0, 0, -7, 7], [21, -5, 0, 0], [0, 0, 9, 2], [0, 0, 18, 2], [11, 7, 0, 0], [0, 0, 15, -2], [6, -2, 0, 0], [0, 0, 10, -8], [-12, -16, 0, 0], [-14, 6, 0, 0], [16, -10, 0, 0], [8, 8, 0, 0], [10, 14, 0, 0], [0, 0, 2, -5], [16, -6, 0, 0], [0, 0, -13, -2], [6, 2, 0, 0], [0, 0, -15, 3], [10, 0, 0, 0], [0, 0, 8, -5], [0, 0, 17, -6], [15, -13, 0, 0], [0, 0, -3, 6], [23, 9, 0, 0], [-4, -8, 0, 0], [0, 0, 11, 2], [-10, 10, 0, 0], [2, -2, 0, 0], [0, 0, -18, -3], [-4, -14, 0, 0], [0, 0, -15, -8], [0, 0, 10, 2], [0, 0, 1, -10], [-24, -12, 0, 0], [10, -2, 0, 0], [0, 0, 20, -1], [0, 0, -22, -1], [0, 0, -5, 7], [9, 9, 0, 0], [2, 18, 0, 0], [0, 0, 1, 14], [0, 0, 10, -12], [0, 0, 13, 0], [9, 5, 0, 0], [-2, 2, 0, 0], [22, -14, 0, 0], [0, 0, -30, -1], [0, 0, -13, 0], [0, 0, 25, 0], [1, -13, 0, 0], [-4, -12, 0, 0], [0, 0, 26, -2], [0, 0, 5, -8], [0, 0, 4, 11], [28, -6, 0, 0], [-4, -12, 0, 0], [-14, -18, 0, 0], [-12, -2, 0, 0], [0, 0, -5, -6], [0, 0, -12, 11], [-7, -9, 0, 0], [0, 0, -21, -8], [-13, 9, 0, 0], [0, 0, 17, 6], [-2, 4, 0, 0], [22, -4, 0, 0], [0, 0, -20, -7], [0, 0, 1, -4], [0, 0, 13, 6], [4, 18, 0, 0], [28, -8, 0, 0], [-2, 16, 0, 0], [0, 0, -19, -2], [0, 0, -11, 16], [-34, 10, 0, 0], [33, 13, 0, 0], [0, 0, 24, 3], [0, 0, -1, -17], [-27, 17, 0, 0], [0, 0, -13, 10], [0, 0, -3, 1], [32, 4, 0, 0], [0, 0, 15, 8], [0, 0, 15, 8], [0, 0, 5, -12], [11, -7, 0, 0], [-34, 4, 0, 0], [6, 24, 0, 0], [0, 0, -31, -3], [6, -24, 0, 0], [0, 0, 1, 14], [0, 0, 24, -8], [0, 0, 9, -8], [31, -7, 0, 0], [0, 0, 23, 7], [0, 0, 5, -6], [8, -10, 0, 0], [0, 0, -15, 0], [-18, 12, 0, 0], [0, 0, 0, 13], [36, 6, 0, 0], [-10, 16, 0, 0], [-42, -4, 0, 0], [0, 0, 24, -7], [4, 18, 0, 0], [-6, -14, 0, 0], [12, 14, 0, 0], [-12, -14, 0, 0], [0, 0, -15, 10], [-4, -18, 0, 0], [0, 0, -13, 12], [-11, -13, 0, 0], [0, 0, 14, 6], [0, 0, 7, 9], [0, 0, 2, 25], [-38, -12, 0, 0], [0, 0, 26, 5], [0, 0, -5, 0], [0, -16, 0, 0], [-36, -16, 0, 0], [0, 0, -20, -4], [0, 0, 11, 6], [11, 17, 0, 0], [-16, 0, 0, 0], [0, 0, -3, -18], [0, 0, 8, -4], [-32, 4, 0, 0], [0, 0, -12, -16], [0, 0, 2, -18], [0, 0, -37, -4], [44, 2, 0, 0], [-24, 20, 0, 0], [0, 0, -6, -15], [-34, 0, 0, 0], [-9, -7, 0, 0], [0, 0, 38, -2], [-3, 23, 0, 0], [0, 0, 24, -13], [30, 14, 0, 0], [15, -5, 0, 0], [0, 0, -22, 5], [-26, 8, 0, 0], [0, 0, 27, -2], [0, 0, 9, 8], [-33, -13, 0, 0], [-26, 8, 0, 0], [0, 0, 15, -8], [8, -6, 0, 0], [0, 0, 13, 2], [0, 0, 25, 10], [-12, 20, 0, 0], [-44, 0, 0, 0], [-30, 0, 0, 0], [0, 0, 3, -18], [0, 0, -9, -18], [54, -10, 0, 0], [0, 0, 22, 7], [-33, 7, 0, 0], [0, 0, -15, 6], [-20, 22, 0, 0], [0, 0, 18, -11], [-62, 4, 0, 0], [-23, 5, 0, 0], [0, -24, 0, 0], [0, 0, -1, 2], [0, 0, 23, 8], [14, -14, 0, 0], [0, 0, -15, 2], [-36, 4, 0, 0], [-5, -11, 0, 0], [0, 0, 25, -2], [-4, -6, 0, 0], [0, 0, 19, 6], [-38, -8, 0, 0], [0, 0, -10, 20], [-18, 14, 0, 0], [0, 0, 7, -14], [-6, 0, 0, 0], [40, -14, 0, 0], [0, 0, 31, -2], [0, 0, 10, -11], [-30, -28, 0, 0], [0, 0, 47, -4], [-36, -16, 0, 0], [0, 0, -2, 19], [-14, -18, 0, 0], [-24, -2, 0, 0], [0, 0, 9, 16], [0, 0, 4, -4], [0, 0, -8, -20], [0, 0, -5, -6], [0, 0, -1, 6], [-18, 20, 0, 0], [0, 0, -30, 7], [0, 0, 13, -20], [-43, 1, 0, 0], [48, 14, 0, 0], [-36, -6, 0, 0], [0, 0, 15, 16], [0, 0, -8, 12], [-38, -24, 0, 0], [0, 0, 2, 3], [0, 0, -2, -12], [-29, 19, 0, 0], [0, 0, -14, -3], [0, 0, -17, -22], [0, 0, -7, -6], [0, -8, 0, 0], [14, 26, 0, 0], [42, 20, 0, 0], [0, 0, 19, -6], [-22, -34, 0, 0], [0, 0, -21, 14], [2, -2, 0, 0], [0, 0, -13, -6], [-21, -3, 0, 0], [0, 0, -25, 2], [0, 0, -14, -15], [-44, -16, 0, 0], [80, -2, 0, 0], [16, -10, 0, 0], [0, 0, -5, 12], [0, 0, 4, -4], [-27, 9, 0, 0], [0, 0, -5, 18], [-36, 20, 0, 0], [9, 19, 0, 0], [0, 0, 23, -5], [0, -34, 0, 0], [0, 0, 1, -4], [0, 0, -27, 4], [0, 0, -17, -3], [23, -31, 0, 0], [0, 0, 39, -2], [55, -15, 0, 0], [0, 0, 21, 3], [0, 0, -31, -4], [2, 16, 0, 0], [-78, 4, 0, 0], [0, 0, -28, 2], [0, 0, -20, 1], [40, 2, 0, 0], [32, 20, 0, 0], [0, 0, 43, 0], [0, 0, -27, -8], [-32, 12, 0, 0], [15, -25, 0, 0], [-36, 12, 0, 0], [0, 0, 9, 3], [30, -14, 0, 0], [19, 35, 0, 0], [0, 0, -9, 1], [48, 8, 0, 0], [0, 0, 39, -4], [0, 0, 22, 4], [6, -38, 0, 0], [60, 16, 0, 0], [0, 0, 3, 0], [0, 0, -7, -12], [0, 0, 4, -2], [-10, 22, 0, 0], [0, 0, 15, 17], [-38, -18, 0, 0], [0, 0, 19, -18], [-23, -5, 0, 0], [0, 0, -25, -10], [8, 38, 0, 0], [0, 0, -28, -21], [-20, -8, 0, 0], [0, 0, 15, -12], [0, 0, -11, 4], [0, 0, 22, 12], [-24, -32, 0, 0], [30, 26, 0, 0], [0, 0, -25, 1], [19, -41, 0, 0], [4, -40, 0, 0], [0, 0, -63, -2], [-69, -1, 0, 0], [0, 0, -43, 10], [52, -18, 0, 0], [0, 0, 27, 3], [26, 34, 0, 0], [0, 0, 18, 27], [60, -18, 0, 0], [0, 0, 3, 6], [-17, 25, 0, 0], [36, -16, 0, 0], [0, 0, 2, -13], [0, 0, -5, -6], [0, 0, 30, 16], [18, 40, 0, 0], [0, 0, 25, -6], [-62, -18, 0, 0], [0, 0, 15, 20], [0, 0, 38, -7], [0, 0, -38, 9], [0, 0, -22, -3], [28, -16, 0, 0], [-3, -1, 0, 0], [-48, -24, 0, 0], [0, 0, 22, 25], [-24, 8, 0, 0], [30, 24, 0, 0], [0, 0, 40, -18], [33, 3, 0, 0], [57, -11, 0, 0], [0, 0, -7, -10], [-4, -16, 0, 0], [0, 0, 25, -13], [12, -30, 0, 0], [0, 0, 41, 7], [0, 0, -39, 6], [0, 0, 20, -5], [-32, -2, 0, 0], [0, 0, 36, -3], [-3, -7, 0, 0], [0, 0, -1, 19], [0, 0, -25, -10], [0, 0, 25, 14], [26, 38, 0, 0], [0, 0, -9, 9], [3, -23, 0, 0], [0, 0, 39, 6], [21, -43, 0, 0], [0, 0, -1, 11], [-40, 28, 0, 0], [12, -50, 0, 0], [28, -14, 0, 0], [84, -4, 0, 0], [-14, 24, 0, 0], [0, 0, -32, -23], [0, 0, 45, -16], [0, 0, 6, 12], [48, -24, 0, 0], [-20, -28, 0, 0], [0, 0, -24, -12], [-54, 4, 0, 0], [0, 0, -1, -40], [-38, 22, 0, 0], [0, 0, 19, 5], [0, 0, -41, 3], [0, 0, 41, -12], [-16, -34, 0, 0], [0, 0, 52, 5], [24, 18, 0, 0], [29, 1, 0, 0], [0, 0, 40, 5], [0, 0, 23, -16], [0, 0, -29, -6], [-60, 16, 0, 0], [0, 0, -17, -6], [0, 0, -15, 4], [35, 23, 0, 0], [0, 0, -18, 26], [0, 0, -6, 12], [0, 0, -13, 18], [66, -2, 0, 0], [28, 2, 0, 0], [-49, 21, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2900_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2900_2_a_h();".
// 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_2900_2_a_h(:prec:=4)
    chi := MakeCharacter_2900_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_2900_2_a_h();".
// 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_2900_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 0, 1]>,<7,R![-2, 0, 1]>],Snew);
    return Vf;
end function;