
// Make newform 392.3.g.h in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_392_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_392_g_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_392_3_g_h();".
// 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_392_3_g_h();".
// 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 := [16, 0, 6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 2, 0, 1], [3, 0, 1, 0]];
    Rf_basisdens := [1, 1, 4, 1];
    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_392_g();"
function MakeCharacter_392_g()
    N := 392;
    order := 2;
    char_gens := [295, 197, 297];
    v := [1, 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_392_g_Hecke();"
function MakeCharacter_392_g_Hecke(Kf)
    N := 392;
    order := 2;
    char_gens := [295, 197, 297];
    char_values := [[-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, -1, 0, 0], [-2, 0, -1, 0], [0, 2, 1, 2], [0, 0, 0, 0], [4, 0, -6, 0], [0, -2, -1, -2], [-18, 0, -4, 0], [-2, 0, 19, 0], [0, 16, 8, -2], [0, -12, -6, 0], [0, 8, 4, -12], [0, -20, -10, 8], [10, 0, 12, 0], [-20, 0, 2, 0], [0, 8, 4, -8], [0, 24, 12, -20], [-46, 0, -11, 0], [0, 6, 3, -10], [-56, 0, 16, 0], [0, 32, 16, -16], [-58, 0, -8, 0], [0, 32, 16, 8], [-22, 0, 13, 0], [-78, 0, 24, 0], [34, 0, -92, 0], [0, -2, -1, 6], [0, -40, -20, -12], [-24, 0, 28, 0], [0, 36, 18, 24], [-76, 0, 44, 0], [0, 64, 32, -2], [34, 0, -95, 0], [46, 0, 8, 0], [38, 0, 103, 0], [0, -16, -8, -84], [0, 32, 16, 66], [0, 54, 27, -42], [-68, 0, -122, 0], [0, 136, 68, 16], [0, 38, 19, -42], [176, 0, -84, 0], [0, -98, -49, 54], [0, -16, -8, -48], [-52, 0, 148, 0], [0, -72, -36, -4], [0, 56, 28, 108], [0, 0, 116, 0], [0, 0, 0, 4], [54, 0, -113, 0], [0, 42, 21, 58], [-306, 0, -80, 0], [0, 48, 24, 90], [194, 0, 188, 0], [66, 0, -135, 0], [54, 0, 264, 0], [0, 32, 16, 120], [0, -50, -25, 46], [0, 208, 104, 4], [0, 72, 36, -12], [-286, 0, -96, 0], [262, 0, 59, 0], [0, 78, 39, -138], [-270, 0, 33, 0], [0, 16, 8, 16], [-246, 0, -116, 0], [0, 64, 32, -4], [268, 0, -38, 0], [0, 0, 116, 0], [-220, 0, 78, 0], [0, 194, 97, -38], [250, 0, -232, 0], [0, 0, 0, -138], [0, -80, -40, -140], [0, -208, -104, -52], [204, 0, 58, 0], [0, 56, 28, 80], [0, 4, 2, 32], [0, -326, -163, 18], [200, 0, 84, 0], [-298, 0, 108, 0], [-666, 0, -29, 0], [0, 184, 92, 84], [0, -80, -40, 166], [230, 0, -260, 0], [0, 128, 64, -76], [56, 0, -384, 0], [-98, 0, 256, 0], [-124, 0, -276, 0], [0, 258, 129, 106], [0, -160, -80, 160], [558, 0, -149, 0], [0, 24, 12, 28], [0, 112, 56, -186], [-224, 0, 116, 0], [-56, 0, -20, 0], [0, -304, -152, -60], [0, 42, 21, 210], [-522, 0, 220, 0], [-694, 0, 129, 0], [0, 200, 100, 12], [-524, 0, 58, 0], [0, 88, 44, -76], [-310, 0, -351, 0], [-224, 0, 316, 0], [-52, 0, 442, 0], [-102, 0, 584, 0], [150, 0, -121, 0], [174, 0, -32, 0], [0, -128, -64, 32], [258, 0, 328, 0], [0, 400, 200, 16], [0, -204, -102, -136], [-264, 0, -308, 0], [222, 0, -225, 0], [0, -144, -72, 0], [336, 0, 212, 0], [-774, 0, -359, 0], [0, -360, -180, -156], [0, -28, -14, -432], [-524, 0, 114, 0], [0, -234, -117, -210], [50, 0, -40, 0], [0, 302, 151, 166], [-616, 0, 44, 0], [706, 0, 241, 0], [0, -84, -42, -296], [0, 124, 62, 312], [0, -440, -220, -80], [0, 120, 60, -272], [0, -210, -105, -66], [-276, 0, 190, 0], [0, 224, 112, 118], [0, 144, 72, 150], [0, -252, -126, 88], [194, 0, 572, 0], [234, 0, 292, 0], [0, 18, 9, 226], [-690, 0, -165, 0], [0, -114, -57, -2], [844, 0, -356, 0], [790, 0, 83, 0], [0, -480, -240, -100], [0, -240, -120, -224], [912, 0, 448, 0], [0, 86, 43, 342], [0, -56, -28, -76], [0, -198, -99, 202], [-506, 0, 244, 0], [474, 0, -499, 0], [0, 432, 216, -200], [0, -564, -282, 144], [594, 0, -288, 0], [1080, 0, 140, 0], [0, -264, -132, 184], [-104, 0, -392, 0], [0, 128, 64, -14], [0, 784, 392, 48], [502, 0, -380, 0], [-746, 0, 96, 0], [0, -750, -375, 170], [-908, 0, -654, 0], [938, 0, -416, 0], [0, -208, -104, -386], [-30, 0, -443, 0], [158, 0, 8, 0], [0, 232, 116, -100], [0, 336, 168, 400], [0, 450, 225, 338]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_392_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_392_3_g_h();".
// 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_392_3_g_h(:prec:=4)
    chi := MakeCharacter_392_g();
    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_392_3_g_h();".
// 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_392_3_g_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_392_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![2, 4, 1]>],Snew);
    return Vf;
end function;