
// Make newform 1900.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1900_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_1900_2_a_e();".
// 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_1900_2_a_e();".
// 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 := [-2, 0, 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_1900_a();"
function MakeCharacter_1900_a()
    N := 1900;
    order := 1;
    char_gens := [951, 77, 401];
    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_1900_a_Hecke(Kf)
    return MakeCharacter_1900_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], [2, 1], [0, 0], [2, -2], [-2, 0], [2, 3], [2, -2], [-1, 0], [6, 0], [2, -6], [-4, -2], [6, -3], [-2, 4], [-2, -2], [2, 2], [-2, -5], [8, 4], [-8, -4], [2, 1], [8, -2], [-6, 6], [-4, 4], [2, 8], [2, -6], [6, 3], [4, 0], [6, -7], [-6, -1], [-6, 2], [2, 5], [6, -11], [0, 4], [6, -10], [-6, 8], [-8, 0], [4, -10], [-6, -2], [-6, -6], [-2, -9], [-6, -7], [16, -4], [-18, 0], [4, 0], [-2, -1], [-6, -8], [-16, -4], [-20, -6], [2, 15], [-22, -1], [-16, 4], [-10, 0], [16, 4], [-2, 20], [-8, 4], [-2, -3], [2, 4], [-14, 6], [18, -4], [-14, 2], [2, -12], [14, 8], [22, -1], [2, 7], [-14, 0], [-18, 0], [10, 3], [4, 2], [-22, -3], [-6, 12], [-6, -4], [18, 10], [-10, 8], [14, -10], [6, 11], [28, 4], [-10, -5], [-6, 16], [18, -10], [-18, -8], [18, 6], [4, -4], [-14, -4], [8, -18], [-6, -15], [-24, 4], [-6, 20], [-22, -6], [-2, 12], [6, 8], [-2, -8], [2, 8], [-10, 0], [-6, 13], [24, -8], [-10, 12], [14, 12], [-14, 14], [34, 4], [-6, -9], [0, -8], [-18, 13], [-22, 14], [10, -1], [-2, -2], [-26, -8], [6, 0], [-2, -12], [14, 16], [-16, -8], [-18, 0], [2, 5], [-26, -6], [26, 14], [18, 0], [2, 4], [-10, 8], [-2, 16], [-18, -8], [6, 14], [-12, 16], [10, 16], [-6, 3], [-10, 11], [26, -9], [-10, 4], [44, -4], [2, -12], [10, -20], [38, 6], [-34, 4], [12, -16], [10, 17], [0, -10], [-14, 4], [8, -16], [16, -16], [10, 7], [-2, -31], [-2, 7], [6, 32], [0, -14], [10, 0], [18, 2], [22, -5], [2, 30], [4, -28], [-42, 0], [6, -15], [4, 20], [-6, -7], [30, -5], [16, -20], [42, 2], [42, -5], [2, -13], [24, 14], [24, -8], [18, 8], [26, -10], [-2, 20], [14, 4], [-10, -3], [2, 16], [-4, 22], [-22, -9], [22, -1], [-20, 22], [22, 6], [-14, 6], [-14, 30], [24, -16], [6, 8], [-10, 36], [-38, 4], [-16, 20], [-24, 8], [-16, 12], [-12, 32], [18, 1], [-12, 4], [-2, 32], [-4, -34], [26, -1], [-6, -17], [18, -12], [-18, -28], [-14, -15], [42, -5], [-26, -2], [4, -12], [-18, 21], [10, 12], [-12, -22], [30, -24], [-18, -18], [34, -3], [34, 16], [2, -28], [-14, 12], [18, -18], [-38, -10], [36, 14], [2, 23], [22, 10], [22, 4], [-6, -26], [-10, -20], [-34, 7], [-42, -12], [-4, -10], [-10, -16], [8, 12], [-26, -12], [6, -5], [-36, 12], [10, 8], [-2, -10], [6, 20], [6, 25], [22, -32], [14, -28], [-24, 24], [2, -14], [10, 20], [18, -1], [30, -16], [18, 15], [48, -12], [-6, 7], [-40, 12], [58, 0], [-8, 0], [16, -18], [-14, -40], [38, -12], [-6, 14], [-4, -8], [2, 8], [-10, 4], [-8, 46], [-34, -3], [16, 24], [-22, 30], [-26, 2], [-10, 39], [-16, 0], [30, -2], [44, -6], [-44, 0], [6, 6], [-2, 36], [18, 40], [26, -28], [50, -6], [-38, -24], [-28, -24], [-30, -16], [10, 5], [-2, 3], [2, -34], [6, 37], [6, 31], [-36, -32], [70, -1], [18, 36], [-48, 0], [22, -18], [28, 16], [6, -17], [-26, -28], [-26, 32], [-34, -5], [22, -10], [2, -16], [-42, 3], [-2, -46], [34, 14], [-18, 18], [-10, -28], [-62, 4], [30, -13], [6, -20], [2, 48], [-6, 12], [-2, -36], [-26, -12], [30, 0], [18, -17], [32, -8], [54, 6], [26, -16], [-18, -30], [46, -21], [-28, 14], [74, -7], [-44, 4], [12, 10], [2, 16], [-16, 36], [18, -38], [-54, -6], [26, 29], [18, 24], [18, -35], [0, -12], [42, 33], [-10, -29], [-22, -22], [8, 24], [-22, 40], [-14, 22], [12, -16], [-2, 32], [26, 7], [2, -4], [22, 46], [6, 56], [56, 6], [2, 36], [-22, -12], [-12, 18], [-86, 2], [-50, -16], [38, -5], [-10, 6], [42, 16], [56, 0], [-62, 7], [-2, -43], [-14, 42], [-32, -36], [10, -51], [-34, -12], [-2, -34], [6, -8], [-6, 6], [34, -18], [-18, 1], [34, 16], [-26, 0], [18, -19], [-66, -2], [-2, -10], [-8, 2], [-34, 23], [16, 36], [-26, -32], [-10, 27], [48, 22], [-74, -2], [48, -6], [18, 23], [-16, 36], [6, -5], [-6, 2], [-6, 19], [40, -4], [-24, -4], [34, -18], [42, -31], [-42, -10], [-40, 24], [90, 5], [32, -12], [-18, 20], [-62, 1], [26, -38], [14, 39], [-30, -20], [6, -12], [20, -12], [-74, 14], [-34, 2], [76, 8], [-10, 6], [-32, -20], [28, 16], [-18, -32], [-18, -16], [6, -1], [-26, -8], [34, -28], [-6, -22], [-50, 36], [16, -12], [-74, -3], [6, 16], [26, 2], [10, -30], [2, -43], [10, 34], [-14, 9], [-12, 16], [34, -20], [-20, -38], [42, 19], [12, -16], [74, -2], [-48, -10], [-66, 24], [30, -22], [-14, 16], [-18, -33], [42, -5], [14, 46], [28, 48], [10, -30], [18, -24], [-6, -5], [-44, -12], [-2, 53], [-22, -20], [6, 31], [22, 24], [-66, 22], [-54, -24], [44, -24], [30, 33], [26, -10], [54, -9], [42, -18], [-2, -25], [22, -14], [12, 56], [42, -35], [-54, -11], [58, -9], [0, 12], [64, 20], [-58, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1900_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1900_2_a_e();".
// 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_1900_2_a_e(:prec:=2)
    chi := MakeCharacter_1900_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_1900_2_a_e();".
// 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_1900_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -4, 1]>,<7,R![-4, -4, 1]>],Snew);
    return Vf;
end function;