
// Make newform 1216.3.e.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1216_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1216_e_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_1216_3_e_l();".
// 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_1216_3_e_l();".
// 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 := [29, 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_1216_e();"
function MakeCharacter_1216_e()
    N := 1216;
    order := 2;
    char_gens := [191, 837, 705];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1216_e_Hecke();"
function MakeCharacter_1216_e_Hecke(Kf)
    N := 1216;
    order := 2;
    char_gens := [191, 837, 705];
    char_values := [[1, 0], [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+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 := 3;
    raw_aps := [[0, 0], [0, 1], [4, 0], [1, 0], [14, 0], [0, -3], [23, 0], [10, 3], [1, 0], [0, 9], [0, 6], [0, -6], [0, 6], [68, 0], [-26, 0], [0, 15], [0, 3], [40, 0], [0, -3], [0, 6], [-7, 0], [0, -18], [32, 0], [0, -24], [0, -18], [-14, 0], [0, -24], [0, 3], [0, -3], [0, 18], [0, 12], [56, 0], [-19, 0], [122, 0], [82, 0], [0, 42], [-242, 0], [-214, 0], [0, -6], [0, 30], [0, 54], [0, -18], [67, 0], [0, -18], [142, 0], [-263, 0], [0, -15], [0, 48], [0, -15], [304, 0], [2, 0], [-89, 0], [0, -30], [-370, 0], [0, -36], [394, 0], [0, -18], [403, 0], [-206, 0], [0, -48], [56, 0], [0, 15], [0, 78], [265, 0], [77, 0], [0, -75], [0, -81], [0, -72], [392, 0], [-410, 0], [-481, 0], [109, 0], [-50, 0], [0, -93], [0, 39], [0, -48], [-578, 0], [658, 0], [0, -144], [0, 66], [8, 0], [0, -93], [0, 120], [0, 36], [0, 60], [182, 0], [0, -78], [317, 0], [172, 0], [-146, 0], [662, 0], [130, 0], [0, -72], [-280, 0], [-460, 0], [-263, 0], [0, 54], [0, -42], [0, 15], [640, 0], [0, 6], [844, 0], [0, 78], [0, 54], [458, 0], [-85, 0], [488, 0], [-610, 0], [0, 60], [0, -18], [0, -186], [-200, 0], [530, 0], [-256, 0], [826, 0], [0, -108], [170, 0], [25, 0], [1054, 0], [0, -231], [0, 99], [0, -54], [0, -177], [0, 162], [668, 0], [700, 0], [-1292, 0], [-1061, 0], [-83, 0], [-296, 0], [548, 0], [0, -114], [0, -78], [-188, 0], [-1183, 0], [-1177, 0], [0, -189], [0, 21], [0, 33], [737, 0], [0, 147], [-122, 0], [-1499, 0], [0, -177], [0, -15], [0, -72], [-536, 0], [0, 228], [-496, 0], [0, 108], [0, 201], [302, 0], [-1480, 0], [0, -180], [0, 3], [0, 84], [-515, 0], [671, 0], [1601, 0], [0, -291], [-1192, 0], [0, -156], [514, 0], [0, -318], [0, -30], [0, -48], [0, -168], [394, 0], [0, 342], [1162, 0], [0, -261], [0, -315], [-1934, 0], [-571, 0], [0, -54], [-910, 0], [-958, 0], [-1238, 0], [0, 90], [370, 0], [-65, 0], [0, -159], [0, -342], [0, -360], [871, 0], [1552, 0], [0, -15], [0, 414], [0, -24], [-1649, 0], [0, -114], [-142, 0], [0, 18], [0, 345], [1550, 0], [0, 270], [875, 0], [1354, 0], [383, 0], [1045, 0], [0, 294], [0, -384], [0, -105], [0, 36], [-484, 0], [1726, 0], [1447, 0], [0, -327], [1217, 0], [0, 15], [-1429, 0], [-1196, 0], [-626, 0], [0, 135], [0, -240], [0, 102], [1423, 0], [0, -186], [0, 54], [-2156, 0], [0, -174], [0, 204], [0, -360], [1231, 0], [0, 345], [-584, 0], [0, -132], [0, -174], [0, -90], [-2692, 0], [244, 0], [0, 129], [0, 234], [0, -270], [1070, 0], [2335, 0], [-910, 0], [1888, 0], [1736, 0], [0, 300], [0, 342], [-2788, 0], [-1367, 0], [0, 441], [0, 288], [-962, 0], [-2798, 0], [0, -222], [0, -150], [1375, 0], [2620, 0], [323, 0], [-1259, 0], [0, 360], [1000, 0], [-1420, 0], [-1310, 0], [0, -462], [0, -6], [2162, 0], [0, 60], [0, 450], [70, 0], [0, 357], [-1186, 0], [0, 45], [0, -219], [1385, 0], [0, -345], [1996, 0], [0, -63], [0, -255], [-535, 0], [-1445, 0], [0, -198], [-2243, 0], [-310, 0], [0, -195], [0, 0], [-1306, 0], [0, -606], [-431, 0], [769, 0], [0, 315], [2930, 0], [-1130, 0], [-793, 0], [0, -174], [2185, 0], [0, -96], [-2930, 0], [-544, 0], [0, 318], [0, 402], [0, 258], [802, 0], [0, -138], [-3146, 0], [0, -309], [-3262, 0], [1442, 0], [0, 309], [-869, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1216_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1216_3_e_l();".
// 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_1216_3_e_l(:prec:=2)
    chi := MakeCharacter_1216_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1216_3_e_l();".
// 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_1216_3_e_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1216_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![29, 0, 1]>,<5,R![-4, 1]>,<7,R![-1, 1]>],Snew);
    return Vf;
end function;