
// Make newform 147.3.h.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_147_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_147_h_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_147_3_h_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_147_3_h_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 := [373, -730, 486, -102, 34, -30, 6, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [174191, 13664, 10993, -16648, -2577, -1344, 15, 50], [164120, -174898, -37037, -11602, 13500, 3078, 1809, 62], [-105932, 120543, -6072, 3666, -7514, -257, -704, 140], [36554, -51751, 15883, -469, 3276, -945, 63, -163], [-202912, 346799, -114503, -655, -20310, 7365, 255, 1223], [51101, -160597, 62674, 7667, 12099, -2823, 39, -616], [19023, -45746, 16429, 986, 3373, -808, 29, -152]];
    Rf_basisdens := [1, 46998, 46998, 15666, 6714, 46998, 23499, 5222];
    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_147_h();"
function MakeCharacter_147_h()
    N := 147;
    order := 6;
    char_gens := [50, 52];
    v := [3, 4];
    // 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_147_h_Hecke();"
function MakeCharacter_147_h_Hecke(Kf)
    N := 147;
    order := 6;
    char_gens := [50, 52];
    char_values := [[-1, 0, 0, 0, 0, 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, 0, -1, 0, 0, 0, 1, 0], [-1, 0, -1, -2, -1, -2, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, -2, 0], [-9, 0, 0, 0, 0, 0, 0, -1], [0, -2, -2, 0, 0, 0, 2, 0], [-3, 0, 5, 0, -3, 0, 0, -5], [-2, 0, -2, -4, -2, 4, 0, 2], [2, -6, 0, 4, -2, 6, -2, -2], [0, 0, 2, 0, 34, 0, 0, 0], [-4, 0, -14, 0, -4, 0, 0, 14], [-8, -6, 0, -16, -22, 6, -22, 8], [-40, 0, 0, 0, 0, 0, 0, 6], [4, 0, 4, 8, 4, 16, 0, -4], [0, 22, 16, 0, 6, 0, -10, 0], [0, 28, 1, 0, 27, 0, 26, 0], [39, 0, -7, 0, 39, 0, 0, 7], [0, 0, -8, 0, -6, 0, 0, 0], [-6, 30, 0, -12, 18, -30, 18, 6], [0, 0, 26, 0, -8, 0, 0, 0], [-32, 0, 36, 0, -32, 0, 0, -36], [9, -36, 0, 18, -18, 36, -18, -9], [-16, 0, -16, -32, -16, 10, 0, 16], [-2, 0, 0, 0, 0, 0, 0, -8], [0, -50, -7, 0, -43, 0, -36, 0], [88, 0, -12, 0, 88, 0, 0, 12], [-22, 0, -22, -44, -22, -14, 0, 22], [0, 0, 38, 0, 64, 0, 0, 0], [-22, 30, 0, -44, -14, -30, -14, 22], [-26, 0, 0, 0, 0, 0, 0, 4], [5, 0, 5, 10, 5, -44, 0, -5], [0, 16, 4, 0, 12, 0, 8, 0], [-65, 0, 0, 0, 0, 0, 0, 49], [40, 0, 40, 80, 40, 10, 0, -40], [0, 0, 12, 0, 134, 0, 0, 0], [0, 0, -77, 0, -99, 0, 0, 0], [108, 0, 14, 0, 108, 0, 0, -14], [14, 36, 0, 28, 64, -36, 64, -14], [3, 0, 3, 6, 3, -26, 0, -3], [0, -50, -58, 0, 8, 0, 66, 0], [-9, 0, 0, 0, 0, 0, 0, -85], [-22, 0, -22, -44, -22, 46, 0, 22], [0, 0, -42, 0, -34, 0, 0, 0], [20, 6, 0, 40, 46, -6, 46, -20], [0, 0, -4, 0, 76, 0, 0, 0], [30, 0, 0, 0, 0, 0, 0, -4], [138, 0, 0, 0, 0, 0, 0, 14], [0, -128, -47, 0, -81, 0, -34, 0], [-17, 0, 9, 0, -17, 0, 0, -9], [4, 0, 4, 8, 4, -104, 0, -4], [-6, 24, 0, -12, 12, -24, 12, 6], [0, 0, -36, 0, 230, 0, 0, 0], [53, -12, 0, 106, 94, 12, 94, -53], [-42, 0, -42, -84, -42, -26, 0, 42], [0, 82, 30, 0, 52, 0, 22, 0], [0, 82, 49, 0, 33, 0, -16, 0], [270, 0, -94, 0, 270, 0, 0, 94], [0, 0, -20, 0, -174, 0, 0, 0], [-56, -24, 0, -112, -136, 24, -136, 56], [0, 0, 145, 0, -15, 0, 0, 0], [-77, 126, 0, -154, -28, -126, -28, 77], [327, 0, 0, 0, 0, 0, 0, -91], [0, 40, -2, 0, 42, 0, 44, 0], [-80, 0, 90, 0, -80, 0, 0, -90], [40, 0, 40, 80, 40, -38, 0, -40], [52, 0, 78, 0, 52, 0, 0, -78], [366, 0, 0, 0, 0, 0, 0, 14], [0, -62, 44, 0, -106, 0, -150, 0], [-219, 0, 0, 0, 0, 0, 0, -55], [0, -74, -30, 0, -44, 0, -14, 0], [-6, 0, -6, -12, -6, -116, 0, 6], [0, 0, 74, 0, 34, 0, 0, 0], [-410, 0, -12, 0, -410, 0, 0, 12], [-236, 0, 0, 0, 0, 0, 0, 70], [-80, 0, -80, -160, -80, -164, 0, 80], [0, -230, -14, 0, -216, 0, -202, 0], [-157, 0, -95, 0, -157, 0, 0, 95], [-2, 0, -2, -4, -2, 190, 0, 2], [0, 0, -126, 0, -316, 0, 0, 0], [-9, -24, 0, -18, -42, 24, -42, 9], [170, 0, 0, 0, 0, 0, 0, 96], [0, 232, 82, 0, 150, 0, 68, 0], [138, 0, 0, 0, 0, 0, 0, 8], [200, 0, 124, 0, 200, 0, 0, -124], [-50, 0, -50, -100, -50, -26, 0, 50], [-84, 96, 0, -168, -72, -96, -72, 84], [318, 0, 74, 0, 318, 0, 0, -74], [67, -282, 0, 134, -148, 282, -148, -67], [72, 0, 0, 0, 0, 0, 0, -268], [-65, 0, -65, -130, -65, 112, 0, 65], [0, 244, 80, 0, 164, 0, 84, 0], [0, 0, -92, 0, 330, 0, 0, 0], [-98, 210, 0, -196, 14, -210, 14, 98], [234, 0, -232, 0, 234, 0, 0, 232], [-78, 72, 0, -156, -84, -72, -84, 78], [135, 0, 135, 270, 135, 82, 0, -135], [0, -2, -152, 0, 150, 0, 302, 0], [-87, 0, 269, 0, -87, 0, 0, -269], [514, 0, -84, 0, 514, 0, 0, 84], [-12, 0, 0, 0, 0, 0, 0, -82], [0, 298, 24, 0, 274, 0, 250, 0], [0, -116, 9, 0, -125, 0, -134, 0], [-94, 0, -94, -188, -94, 10, 0, 94], [0, 0, 42, 0, 8, 0, 0, 0], [0, 0, 4, 0, -666, 0, 0, 0], [-115, 144, 0, -230, -86, -144, -86, 115], [70, 0, 70, 140, 70, 382, 0, -70], [0, -2, -14, 0, 12, 0, 26, 0], [-548, 0, 0, 0, 0, 0, 0, -146], [-374, 0, -210, 0, -374, 0, 0, 210], [0, 0, 226, 0, -300, 0, 0, 0], [78, -462, 0, 156, -306, 462, -306, -78], [0, 0, -161, 0, 783, 0, 0, 0], [44, 0, 0, 0, 0, 0, 0, 324], [0, -50, -122, 0, 72, 0, 194, 0], [89, 0, 0, 0, 0, 0, 0, -21], [0, 16, 146, 0, -130, 0, -276, 0], [34, 0, 34, 68, 34, 166, 0, -34], [108, -150, 0, 216, 66, 150, 66, -108], [0, 0, 199, 0, -435, 0, 0, 0], [-222, 0, 0, 0, 0, 0, 0, -328], [19, 0, 19, 38, 19, -326, 0, -19], [0, 94, -2, 0, 96, 0, 98, 0], [109, 0, -255, 0, 109, 0, 0, 255], [198, 54, 0, 396, 450, -54, 450, -198], [-928, 0, -62, 0, -928, 0, 0, 62], [-164, 0, -164, -328, -164, -92, 0, 164], [488, 0, 0, 0, 0, 0, 0, 72], [-31, 0, -161, 0, -31, 0, 0, 161], [0, 0, -118, 0, -20, 0, 0, 0], [-30, -60, 0, -60, -120, 60, -120, 30], [654, 0, -16, 0, 654, 0, 0, 16], [380, 0, 0, 0, 0, 0, 0, 222], [-44, 0, -44, -88, -44, -98, 0, 44], [-58, 0, 0, 0, 0, 0, 0, 372], [0, -14, 65, 0, -79, 0, -144, 0], [0, 0, 85, 0, 69, 0, 0, 0], [-121, -6, 0, -242, -248, 6, -248, 121], [0, -2, -38, 0, 36, 0, 74, 0], [257, 0, 0, 0, 0, 0, 0, 303], [-216, 0, -216, -432, -216, -122, 0, 216], [0, 0, 200, 0, 736, 0, 0, 0], [-138, 102, 0, -276, -174, -102, -174, 138], [0, 0, -105, 0, 377, 0, 0, 0], [38, -48, 0, 76, 28, 48, 28, -38], [-1003, 0, 0, 0, 0, 0, 0, 45], [0, -98, -156, 0, 58, 0, 214, 0], [1159, 0, 195, 0, 1159, 0, 0, -195], [202, 0, 202, 404, 202, 274, 0, -202], [276, 0, 26, 0, 276, 0, 0, -26], [150, 150, 0, 300, 450, -150, 450, -150], [-838, 0, 0, 0, 0, 0, 0, -96], [-30, 0, -30, -60, -30, -92, 0, 30], [0, 0, -396, 0, -454, 0, 0, 0], [66, -432, 0, 132, -300, 432, -300, -66], [304, 0, -144, 0, 304, 0, 0, 144], [-246, 0, -246, -492, -246, -350, 0, 246], [-884, 0, 0, 0, 0, 0, 0, -146], [0, -26, -43, 0, 17, 0, 60, 0], [144, 0, 144, 288, 144, -14, 0, -144], [-32, -108, 0, -64, -172, 108, -172, 32], [534, 0, 0, 0, 0, 0, 0, -112], [103, 0, 103, 206, 103, 556, 0, -103], [0, 4, 88, 0, -84, 0, -172, 0], [0, -284, 94, 0, -378, 0, -472, 0], [0, 0, -252, 0, 848, 0, 0, 0], [0, 0, -569, 0, 321, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_147_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_147_3_h_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_147_3_h_e(:prec:=8)
    chi := MakeCharacter_147_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_147_3_h_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_147_3_h_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_147_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![441, 0, -294, 0, 175, 0, -14, 0, 1]>,<5,R![7056, 0, -4704, 0, 3052, 0, -56, 0, 1]>,<13,R![74, 18, 1]>],Snew);
    return Vf;
end function;