
// Make newform 2352.3.f.f in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2352_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2352_f_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_2352_3_f_f();".
// 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_2352_3_f_f();".
// 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 := [256, 16, 17, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [120, -17, 17, -1], [8, 289, -17, 9], [-65, 0, 0, -3]];
    Rf_basisdens := [1, 136, 136, 17];
    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_2352_f();"
function MakeCharacter_2352_f()
    N := 2352;
    order := 2;
    char_gens := [1471, 1765, 785, 2257];
    v := [2, 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_2352_f_Hecke();"
function MakeCharacter_2352_f_Hecke(Kf)
    N := 2352;
    order := 2;
    char_gens := [1471, 1765, 785, 2257];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 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, 0], [0, 1, 0, 0], [0, 2, 1, 0], [0, 0, 0, 0], [8, 0, 0, -1], [0, 1, 1, 0], [0, -4, -2, 0], [0, -13, 1, 0], [24, 0, 0, 0], [2, 0, 0, -1], [0, -11, -6, 0], [-37, 0, 0, 1], [0, -14, -4, 0], [-19, 0, 0, 3], [0, -14, 2, 0], [-50, 0, 0, 1], [0, -40, 1, 0], [0, 36, 8, 0], [-1, 0, 0, -5], [54, 0, 0, -6], [0, -11, -7, 0], [29, 0, 0, -2], [0, 2, -17, 0], [0, -36, -6, 0], [0, 12, 7, 0], [0, 6, 18, 0], [0, -35, 13, 0], [46, 0, 0, -5], [19, 0, 0, 3], [-106, 0, 0, -4], [25, 0, 0, 0], [0, 90, -9, 0], [-108, 0, 0, 12], [0, 1, 19, 0], [152, 0, 0, 2], [-120, 0, 0, -5], [0, -36, 24, 0], [-18, 0, 0, -10], [0, 22, 14, 0], [0, -120, 12, 0], [-138, 0, 0, 0], [0, 11, 13, 0], [-10, 0, 0, 14], [113, 0, 0, -14], [8, 0, 0, 14], [0, 36, -16, 0], [186, 0, 0, -14], [0, -132, -33, 0], [0, -78, -9, 0], [0, 35, 9, 0], [214, 0, 0, 16], [94, 0, 0, -14], [0, 120, -1, 0], [0, -56, 35, 0], [0, -66, -24, 0], [-228, 0, 0, -18], [0, -98, 5, 0], [0, 12, -13, 0], [-7, 0, 0, -3], [-124, 0, 0, -16], [0, -97, -27, 0], [0, -188, -13, 0], [0, 25, -3, 0], [0, -110, -28, 0], [0, 107, -8, 0], [-254, 0, 0, -23], [329, 0, 0, -15], [85, 0, 0, 18], [64, 0, 0, -14], [0, -108, -4, 0], [0, -46, -2, 0], [-68, 0, 0, -32], [0, 145, 32, 0], [-91, 0, 0, 3], [-83, 0, 0, -49], [0, -268, -26, 0], [-146, 0, 0, -20], [0, -227, 9, 0], [-232, 0, 0, 20], [0, 251, -4, 0], [0, -206, -34, 0], [-31, 0, 0, 33], [-302, 0, 0, -20], [0, 181, -15, 0], [0, -204, -33, 0], [-30, 0, 0, 33], [270, 0, 0, -30], [-11, 0, 0, 0], [0, 424, -16, 0], [-293, 0, 0, 9], [0, 90, -18, 0], [0, -256, 46, 0], [287, 0, 0, 30], [-86, 0, 0, 37], [-715, 0, 0, 15], [0, -22, 100, 0], [0, 106, -49, 0], [0, 88, 2, 0], [0, 167, 35, 0], [-751, 0, 0, 15], [-150, 0, 0, -4], [424, 0, 0, 1], [0, 138, -57, 0], [178, 0, 0, 28], [235, 0, 0, 51], [0, 155, 56, 0], [0, -108, -27, 0], [0, 142, 14, 0], [-340, 0, 0, 56], [0, 263, 48, 0], [0, -131, -136, 0], [546, 0, 0, -26], [-70, 0, 0, 50], [0, -49, -73, 0], [948, 0, 0, 11], [212, 0, 0, 20], [0, 599, 3, 0], [0, 222, 66, 0], [-172, 0, 0, -67], [-776, 0, 0, -38], [0, -25, -95, 0], [-563, 0, 0, 48], [0, -300, -33, 0], [-298, 0, 0, -43], [0, 71, -111, 0], [-554, 0, 0, -35], [-30, 0, 0, -64], [0, -34, 88, 0], [0, 527, 6, 0], [0, 121, 39, 0], [247, 0, 0, -15], [-294, 0, 0, 54], [-97, 0, 0, -36], [354, 0, 0, 2], [0, 100, 56, 0], [0, -239, -22, 0], [0, -274, 58, 0], [0, 48, 152, 0], [0, -288, -93, 0], [-574, 0, 0, -22], [0, -168, -12, 0], [-700, 0, 0, 59], [-462, 0, 0, -16], [-436, 0, 0, 89], [0, -385, -95, 0], [0, 568, 50, 0], [0, 311, -65, 0], [0, 80, 4, 0], [0, -480, 86, 0], [42, 0, 0, 36], [690, 0, 0, -62], [0, 768, 12, 0], [-727, 0, 0, -69], [0, 298, -118, 0], [433, 0, 0, -67], [308, 0, 0, -88], [941, 0, 0, 39], [0, 350, -44, 0], [0, -61, 26, 0], [0, -116, -175, 0], [122, 0, 0, -52], [-784, 0, 0, -70], [-1303, 0, 0, -2], [0, 212, -95, 0], [-714, 0, 0, -24], [0, -64, -50, 0], [919, 0, 0, 26], [0, -611, 57, 0], [115, 0, 0, -22], [0, 416, -71, 0], [1318, 0, 0, 31], [0, 576, -6, 0], [164, 0, 0, 86], [-1117, 0, 0, -66], [0, 119, -112, 0], [0, -112, 64, 0], [151, 0, 0, 81], [76, 0, 0, -59], [0, 12, 80, 0], [0, 167, 87, 0], [145, 0, 0, -60], [0, -52, 169, 0], [125, 0, 0, 21], [0, -92, -256, 0], [934, 0, 0, 52], [0, -476, -43, 0], [-582, 0, 0, -28], [0, -409, 79, 0], [775, 0, 0, 14], [0, -614, -4, 0], [0, -673, 106, 0], [144, 0, 0, -39], [1026, 0, 0, 28], [0, -904, -104, 0], [262, 0, 0, -71], [0, -616, -110, 0], [-185, 0, 0, 33], [714, 0, 0, 46], [0, -642, -108, 0], [0, -364, -152, 0], [1148, 0, 0, 71], [0, -227, 145, 0], [0, 756, 104, 0], [0, -348, -200, 0], [0, 226, 74, 0], [0, 382, 35, 0], [0, 684, -92, 0], [312, 0, 0, -87], [108, 0, 0, 114], [0, -372, -170, 0], [169, 0, 0, -49], [0, 456, 33, 0], [492, 0, 0, -5], [0, -112, -251, 0], [0, -1036, -26, 0], [0, 647, -133, 0], [-448, 0, 0, 99], [0, 1044, -96, 0], [-2264, 0, 0, -32], [-398, 0, 0, -8], [947, 0, 0, -125], [0, 241, -27, 0], [470, 0, 0, -166], [1940, 0, 0, -63], [0, 950, -83, 0], [1470, 0, 0, 50], [0, -496, -200, 0], [-886, 0, 0, -34], [0, 1103, -20, 0], [1002, 0, 0, -48], [211, 0, 0, 51], [0, -204, -126, 0], [239, 0, 0, -138], [362, 0, 0, 80], [0, -289, -355, 0], [0, -282, -78, 0], [0, -527, -172, 0], [468, 0, 0, 51], [852, 0, 0, 162], [0, -754, -38, 0], [26, 0, 0, 200], [0, -155, -193, 0], [0, 371, 64, 0], [-1062, 0, 0, 86], [0, -1440, 48, 0], [0, 664, -154, 0], [0, -61, 146, 0], [0, -158, -406, 0], [178, 0, 0, -126], [-796, 0, 0, -22], [439, 0, 0, -21], [0, 504, 168, 0], [18, 0, 0, 6], [0, 95, -13, 0], [0, 416, -131, 0], [-634, 0, 0, -7], [-1235, 0, 0, 131], [0, -443, -45, 0], [0, 474, -75, 0], [0, -1308, -57, 0], [-281, 0, 0, -147], [498, 0, 0, -123], [0, -563, -119, 0], [0, 361, 379, 0], [0, 318, 174, 0], [0, -215, -3, 0], [636, 0, 0, 183], [0, 978, 258, 0], [-806, 0, 0, -120], [-1220, 0, 0, -107], [0, 541, -127, 0], [-90, 0, 0, 40], [0, 1417, -18, 0], [-185, 0, 0, -40], [0, -876, 255, 0], [0, 792, -164, 0], [1152, 0, 0, -141], [-449, 0, 0, -129], [-1493, 0, 0, 57], [0, 508, 152, 0], [0, 1708, -16, 0], [-407, 0, 0, 96], [0, -442, -254, 0], [0, 577, 235, 0], [0, -1188, -44, 0], [1374, 0, 0, 108], [1073, 0, 0, 76], [2704, 0, 0, -29], [0, 253, -122, 0], [0, -370, -494, 0], [2740, 0, 0, 10], [0, 104, 259, 0], [534, 0, 0, 114], [0, -2024, 20, 0], [-847, 0, 0, -159], [0, 986, 1, 0], [0, 1271, 203, 0], [0, 700, 80, 0], [0, -866, -286, 0], [0, 1355, 161, 0], [0, 180, -128, 0], [224, 0, 0, -13], [2668, 0, 0, 33]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2352_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2352_3_f_f();".
// 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_2352_3_f_f(:prec:=4)
    chi := MakeCharacter_2352_f();
    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_2352_3_f_f();".
// 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_2352_3_f_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2352_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![1764, 0, 111, 0, 1]>,<11,R![-90, -15, 1]>],Snew);
    return Vf;
end function;