
// Make newform 567.3.r.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_567_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_567_r_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_567_3_r_c();".
// 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_567_3_r_c();".
// 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], [-164120, 174898, 37037, 11602, -13500, -3078, -1809, -62], [-43268, 43592, -4070, 2287, -2558, 121, -207, 56], [-43268, 51751, -15883, 469, -3276, 945, -63, 163], [-202912, 346799, -114503, -655, -20310, 7365, 255, 1223], [-78703, -53320, 50464, 14528, 4425, -2460, -582, -448], [19023, -45746, 16429, 986, 3373, -808, 29, -152], [-532271, 669787, -247522, 4597, -44553, 13605, -534, 2323]];
    Rf_basisdens := [1, 46998, 2611, 6714, 46998, 15666, 5222, 46998];
    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_567_r();"
function MakeCharacter_567_r()
    N := 567;
    order := 6;
    char_gens := [407, 325];
    v := [5, 6];
    // 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_567_r_Hecke();"
function MakeCharacter_567_r_Hecke(Kf)
    N := 567;
    order := 6;
    char_gens := [407, 325];
    char_values := [[1, 0, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1], [0, -1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 2, 0, 0, 0], [9, -1, 0, 9, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, -2, 0, 6], [0, 0, -2, 0, -4, 2, 0, 0], [-34, -2, 0, -34, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, -14, 0], [0, 0, 0, 0, 0, 8, 0, -14], [0, -6, 0, -40, 0, 0, -6, 0], [0, 0, 4, 0, 12, -4, 0, 0], [0, 0, 16, 0, 6, 0, 0, 6], [0, 0, 0, 0, 0, -1, 0, -27], [0, 7, 0, -39, 0, 0, 7, 0], [6, 8, 0, 6, 0, 0, 0, 0], [0, 0, -6, 0, -24, 0, 0, -24], [-8, 0, 0, 0, 0, 0, -26, 0], [0, -36, 0, 32, 0, 0, -36, 0], [0, 0, -9, 0, -27, 9, 0, 0], [0, 0, 16, 0, -26, 0, 0, -26], [0, 8, 0, -2, 0, 0, 8, 0], [0, 0, 7, 0, 43, -7, 0, 0], [88, -12, 0, 88, 0, 0, 0, 0], [0, 0, 22, 0, -8, 0, 0, -8], [64, 0, 0, 0, 0, 0, -38, 0], [0, 0, 0, 0, 0, 22, 0, 8], [-26, 0, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 5, 0, -49], [0, 0, -4, 0, -12, 4, 0, 0], [65, 49, 0, 65, 0, 0, 0, 0], [0, 0, 0, 0, 0, 40, 0, -30], [0, 12, 0, 134, 0, 0, 12, 0], [99, 77, 0, 99, 0, 0, 0, 0], [-108, 0, 0, 0, 0, 0, 14, 0], [0, 0, 0, 0, 0, -14, 0, 50], [0, 0, 3, 0, -29, -3, 0, 0], [0, 0, -58, 0, 8, 0, 0, 8], [-9, 0, 0, 0, 0, 0, -85, 0], [0, 0, -22, 0, 68, 22, 0, 0], [34, 42, 0, 34, 0, 0, 0, 0], [0, 0, 20, 0, -26, 0, 0, -26], [76, 0, 0, 0, 0, 0, 4, 0], [-30, -4, 0, -30, 0, 0, 0, 0], [0, -14, 0, 138, 0, 0, -14, 0], [0, 0, 47, 0, 81, -47, 0, 0], [-17, 9, 0, -17, 0, 0, 0, 0], [0, 0, -4, 0, 108, 0, 0, 108], [0, 0, 0, 0, 0, 6, 0, 18], [0, -36, 0, 230, 0, 0, -36, 0], [0, 0, 53, 0, -41, 0, 0, -41], [0, 0, 0, 0, 0, -42, 0, 16], [0, 0, -30, 0, -52, 30, 0, 0], [0, 0, 49, 0, 33, 0, 0, 33], [-270, 0, 0, 0, 0, 0, -94, 0], [0, -20, 0, -174, 0, 0, -20, 0], [0, 0, 56, 0, -80, -56, 0, 0], [15, -145, 0, 15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 77, 0, 49], [327, 0, 0, 0, 0, 0, -91, 0], [0, 0, 0, 0, 0, 2, 0, -42], [0, -90, 0, 80, 0, 0, -90, 0], [0, 0, 40, 0, -78, -40, 0, 0], [0, -78, 0, -52, 0, 0, -78, 0], [-366, 14, 0, -366, 0, 0, 0, 0], [0, 0, 0, 0, 0, -44, 0, 106], [0, 55, 0, -219, 0, 0, 55, 0], [0, 0, 30, 0, 44, -30, 0, 0], [0, 0, 6, 0, 110, 0, 0, 110], [0, 74, 0, 34, 0, 0, 74, 0], [-410, -12, 0, -410, 0, 0, 0, 0], [-236, 0, 0, 0, 0, 0, 70, 0], [0, 0, 0, 0, 0, -80, 0, -84], [0, 0, 14, 0, 216, -14, 0, 0], [157, 0, 0, 0, 0, 0, -95, 0], [0, 0, 0, 0, 0, -2, 0, 192], [316, 126, 0, 316, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9, 0, -33], [0, -96, 0, 170, 0, 0, -96, 0], [0, 0, 82, 0, 150, 0, 0, 150], [138, 0, 0, 0, 0, 0, 8, 0], [0, -124, 0, -200, 0, 0, -124, 0], [0, 0, -50, 0, 24, 50, 0, 0], [0, 0, -84, 0, -12, 0, 0, -12], [0, -74, 0, -318, 0, 0, -74, 0], [0, 0, -67, 0, -215, 67, 0, 0], [-72, -268, 0, -72, 0, 0, 0, 0], [0, 0, 65, 0, -177, 0, 0, -177], [0, 0, -80, 0, -164, 80, 0, 0], [330, 0, 0, 0, 0, 0, 92, 0], [0, 0, 0, 0, 0, 98, 0, 112], [234, -232, 0, 234, 0, 0, 0, 0], [0, 0, -78, 0, 6, 0, 0, 6], [0, 0, 0, 0, 0, 135, 0, -53], [0, 0, -152, 0, 150, 0, 0, 150], [87, 0, 0, 0, 0, 0, 269, 0], [-514, 0, 0, 0, 0, 0, -84, 0], [0, 82, 0, -12, 0, 0, 82, 0], [0, 0, 24, 0, 274, 0, 0, 274], [0, 0, 0, 0, 0, -9, 0, 125], [0, 0, -94, 0, 104, 94, 0, 0], [-8, -42, 0, -8, 0, 0, 0, 0], [-666, 0, 0, 0, 0, 0, -4, 0], [0, 0, 115, 0, 29, -115, 0, 0], [0, 0, -70, 0, -312, 0, 0, -312], [0, 0, 0, 0, 0, 14, 0, -12], [0, 146, 0, -548, 0, 0, 146, 0], [-374, -210, 0, -374, 0, 0, 0, 0], [-300, 0, 0, 0, 0, 0, -226, 0], [0, 0, 0, 0, 0, -78, 0, -384], [0, -161, 0, 783, 0, 0, -161, 0], [44, 0, 0, 0, 0, 0, 324, 0], [0, 0, 122, 0, -72, -122, 0, 0], [-89, -21, 0, -89, 0, 0, 0, 0], [0, 0, 146, 0, -130, 0, 0, -130], [0, 0, 0, 0, 0, 34, 0, 132], [0, 0, -108, 0, -42, 108, 0, 0], [435, -199, 0, 435, 0, 0, 0, 0], [0, 328, 0, -222, 0, 0, 328, 0], [0, 0, 19, 0, -345, -19, 0, 0], [0, 0, -2, 0, 96, 0, 0, 96], [0, 255, 0, -109, 0, 0, 255, 0], [0, 0, 198, 0, -252, 0, 0, -252], [0, 62, 0, 928, 0, 0, 62, 0], [0, 0, 164, 0, -72, 0, 0, -72], [0, -72, 0, 488, 0, 0, -72, 0], [-31, -161, 0, -31, 0, 0, 0, 0], [-20, 0, 0, 0, 0, 0, 118, 0], [0, 0, 0, 0, 0, 30, 0, -90], [654, -16, 0, 654, 0, 0, 0, 0], [380, 0, 0, 0, 0, 0, 222, 0], [0, 0, 0, 0, 0, -44, 0, -54], [58, 372, 0, 58, 0, 0, 0, 0], [0, 0, 65, 0, -79, 0, 0, -79], [-69, -85, 0, -69, 0, 0, 0, 0], [0, 0, 0, 0, 0, 121, 0, -127], [0, 0, -38, 0, 36, 0, 0, 36], [257, 0, 0, 0, 0, 0, 303, 0], [0, 0, -216, 0, 94, 216, 0, 0], [-736, -200, 0, -736, 0, 0, 0, 0], [0, 0, -138, 0, 36, 0, 0, 36], [377, 0, 0, 0, 0, 0, 105, 0], [0, 0, -38, 0, -10, 38, 0, 0], [0, -45, 0, -1003, 0, 0, -45, 0], [0, 0, 156, 0, -58, -156, 0, 0], [1159, 195, 0, 1159, 0, 0, 0, 0], [0, 0, -202, 0, -72, 0, 0, -72], [276, 26, 0, 276, 0, 0, 0, 0], [0, 0, 150, 0, -300, 0, 0, -300], [-838, 0, 0, 0, 0, 0, -96, 0], [0, 0, 0, 0, 0, -30, 0, -62], [0, -396, 0, -454, 0, 0, -396, 0], [0, 0, -66, 0, -366, 66, 0, 0], [-304, 0, 0, 0, 0, 0, -144, 0], [0, 0, -246, 0, -104, 246, 0, 0], [-884, 0, 0, 0, 0, 0, -146, 0], [0, 0, 0, 0, 0, 43, 0, -17], [0, 0, 144, 0, -158, -144, 0, 0], [0, 0, -32, 0, 140, 0, 0, 140], [-534, -112, 0, -534, 0, 0, 0, 0], [0, 0, -103, 0, -453, 0, 0, -453], [0, 0, 0, 0, 0, -88, 0, 84], [0, 0, -94, 0, 378, 94, 0, 0], [848, 0, 0, 0, 0, 0, 252, 0], [0, -569, 0, 321, 0, 0, -569, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_567_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_567_3_r_c();".
// 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_567_3_r_c(:prec:=8)
    chi := MakeCharacter_567_r();
    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_567_3_r_c();".
// 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_567_3_r_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_567_r();
    // 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]>],Snew);
    return Vf;
end function;