
// Make newform 350.4.c.j in Magma, downloaded from the LMFDB on 28 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_350_4_c_j();".
// 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_350_4_c_j();".
// 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, 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_350_c();"
function MakeCharacter_350_c()
    N := 350;
    order := 2;
    char_gens := [127, 101];
    v := [1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_c_Hecke();"
function MakeCharacter_350_c_Hecke(Kf)
    N := 350;
    order := 2;
    char_gens := [127, 101];
    char_values := [[-1, 0], [1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 4;
    raw_aps := [[0, -2], [0, 3], [0, 0], [0, -7], [-17, 0], [0, 81], [0, -91], [-102, 0], [0, 90], [129, 0], [116, 0], [0, 314], [-124, 0], [0, 434], [0, 497], [0, 584], [332, 0], [220, 0], [0, 384], [-664, 0], [0, -230], [-361, 0], [0, -1172], [-40, 0], [0, -175], [270, 0], [0, -565], [0, -1206], [1419, 0], [0, 778], [0, 316], [-1750, 0], [0, 48], [38, 0], [3582, 0], [-1927, 0], [0, -2510], [0, 1870], [0, -2019], [0, -1253], [-492, 0], [-4448, 0], [3999, 0], [0, -4880], [0, 4712], [-2756, 0], [697, 0], [0, 707], [0, -559], [3836, 0], [0, 1368], [3803, 0], [5650, 0], [-1650, 0], [0, 5810], [0, 2758], [-186, 0], [-6828, 0], [0, -8214], [6707, 0], [0, 5497], [0, -313], [0, 5479], [4422, 0], [0, -275], [0, -4106], [8260, 0], [0, 9946], [0, -8798], [1838, 0], [0, -2265], [-4136, 0], [0, -10031], [0, -2792], [-9948, 0], [0, -2700], [631, 0], [0, -13139], [-887, 0], [5190, 0], [5712, 0], [-11155, 0], [8499, 0], [0, -5102], [-15694, 0], [0, -9142], [17145, 0], [0, -10184], [-9152, 0], [0, 1084], [0, 19283], [5118, 0], [0, 2918], [-18627, 0], [-12843, 0], [0, 18837], [7742, 0], [614, 0], [0, -17836], [23463, 0], [0, -1860], [0, 10288], [0, 2356], [15318, 0], [-13164, 0], [0, 8959], [0, 16952], [0, -10071], [2829, 0], [-7662, 0], [0, 907], [0, 29646], [0, 11006], [-21098, 0], [21707, 0], [11090, 0], [0, 995], [0, -20792], [0, 10498], [6749, 0], [12400, 0], [0, -16024], [0, 18735], [0, 12844], [25996, 0], [25435, 0], [-3823, 0], [-21090, 0], [0, -7992], [0, 36965], [3305, 0], [0, 2208], [-11551, 0], [0, 9688], [-7014, 0], [6278, 0], [0, -21351], [0, -25967], [0, -8595], [13671, 0], [-11986, 0], [-12951, 0], [0, 12788], [0, 24662], [-4124, 0], [33154, 0], [0, -12050], [0, 15270], [12564, 0], [0, -12976], [0, -15634], [-8896, 0], [0, 33456], [0, 8288], [0, 15398], [22656, 0], [-9149, 0], [27272, 0], [0, 44177], [-2236, 0], [0, -17464], [0, 18336], [0, -46802], [45024, 0], [0, 12786], [0, -33539], [-18200, 0], [0, -32599]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_4_c_j();".
// 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_350_4_c_j(:prec:=2)
    chi := MakeCharacter_350_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_350_4_c_j();".
// 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_350_4_c_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![9, 0, 1]>,<11,R![17, 1]>],Snew);
    return Vf;
end function;