
// Make newform 392.3.k.j in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_392_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_392_k_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_k_j();".
// 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_k_j();".
// 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 := [4, 24, 22, -104, 83, -40, 18, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-34, -100, 125, -52, 31, -6, 2, 0], [2464, 13100, -13942, 5009, -3150, 410, -176, -23], [1228, -1576, -2896, 2807, -1395, 719, -161, 46], [-2860, -4184, -614, 4276, -1317, 1285, -211, 101], [-610, 5836, -14939, 13684, -5895, 2974, -616, 176], [946, 1127, -9814, 9128, -4095, 2072, -434, 124], [1379, 3280, -19628, 18256, -8190, 4144, -868, 248]];
    Rf_basisdens := [1, 18, 1026, 1026, 1026, 1026, 513, 513];
    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_k();"
function MakeCharacter_392_k()
    N := 392;
    order := 6;
    char_gens := [295, 197, 297];
    v := [3, 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_392_k_Hecke();"
function MakeCharacter_392_k_Hecke(Kf)
    N := 392;
    order := 6;
    char_gens := [295, 197, 297];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 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, -1, -1, -1, 0, 0, 0], [2, 2, 0, 1, 0, 0, -1, 0], [0, 0, 2, 2, 2, -2, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-4, -4, 0, 6, 0, 0, -6, 0], [0, 0, 2, 1, 0, 0, 0, -2], [18, 18, 0, 4, 0, 0, -4, 0], [0, -2, 0, 0, 0, 0, -19, 0], [0, 0, 16, 16, 16, 2, -8, 0], [0, 0, 12, 6, 0, 0, 0, 0], [0, 0, 0, -4, -8, -12, 4, 12], [0, 0, -20, -20, -20, -8, 10, 0], [10, 0, 0, 12, 0, 0, 0, 0], [-20, 0, 0, 2, 0, 0, 0, 0], [0, 0, 8, 8, 8, 8, -4, 0], [0, 0, 0, -12, -24, -20, 12, 20], [46, 46, 0, 11, 0, 0, -11, 0], [0, 0, 6, 6, 6, 10, -3, 0], [56, 56, 0, -16, 0, 0, 16, 0], [0, 0, -32, -16, 0, 0, 0, -16], [58, 58, 0, 8, 0, 0, -8, 0], [0, 0, 32, 32, 32, -8, -16, 0], [-22, 0, 0, 13, 0, 0, 0, 0], [0, -78, 0, 0, 0, 0, -24, 0], [34, 0, 0, -92, 0, 0, 0, 0], [0, 0, 0, 1, 2, 6, -1, -6], [0, 0, -40, -40, -40, 12, 20, 0], [0, -24, 0, 0, 0, 0, -28, 0], [0, 0, 0, -18, -36, 24, 18, -24], [-76, 0, 0, 44, 0, 0, 0, 0], [0, 0, -64, -32, 0, 0, 0, -2], [0, 34, 0, 0, 0, 0, 95, 0], [-46, -46, 0, -8, 0, 0, 8, 0], [38, 0, 0, 103, 0, 0, 0, 0], [0, 0, -16, -16, -16, 84, 8, 0], [0, 0, 0, -16, -32, 66, 16, -66], [0, 0, 0, -27, -54, -42, 27, 42], [0, -68, 0, 0, 0, 0, 122, 0], [0, 0, -136, -68, 0, 0, 0, 16], [0, 0, 38, 38, 38, 42, -19, 0], [-176, -176, 0, 84, 0, 0, -84, 0], [0, 0, 98, 49, 0, 0, 0, 54], [0, 0, -16, -16, -16, 48, 8, 0], [52, 52, 0, -148, 0, 0, 148, 0], [0, 0, 72, 36, 0, 0, 0, -4], [0, 0, 0, -28, -56, 108, 28, -108], [0, 0, 0, 116, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4], [-54, -54, 0, 113, 0, 0, -113, 0], [0, 0, 42, 42, 42, -58, -21, 0], [0, -306, 0, 0, 0, 0, 80, 0], [0, 0, -48, -24, 0, 0, 0, 90], [-194, -194, 0, -188, 0, 0, 188, 0], [66, 0, 0, -135, 0, 0, 0, 0], [0, 54, 0, 0, 0, 0, -264, 0], [0, 0, 0, -16, -32, 120, 16, -120], [0, 0, 0, 25, 50, 46, -25, -46], [0, 0, 208, 208, 208, -4, -104, 0], [0, 0, 0, -36, -72, -12, 36, 12], [-286, 0, 0, -96, 0, 0, 0, 0], [-262, -262, 0, -59, 0, 0, 59, 0], [0, 0, -78, -39, 0, 0, 0, -138], [-270, 0, 0, 33, 0, 0, 0, 0], [0, 0, 0, -8, -16, 16, 8, -16], [0, -246, 0, 0, 0, 0, 116, 0], [0, 0, 64, 64, 64, 4, -32, 0], [0, 268, 0, 0, 0, 0, 38, 0], [0, 0, 0, 116, 0, 0, 0, 0], [220, 220, 0, -78, 0, 0, 78, 0], [0, 0, -194, -97, 0, 0, 0, -38], [-250, -250, 0, 232, 0, 0, -232, 0], [0, 0, 0, 0, 0, 138, 0, 0], [0, 0, 0, 40, 80, -140, -40, 140], [0, 0, -208, -208, -208, 52, 104, 0], [204, 0, 0, 58, 0, 0, 0, 0], [0, 0, 56, 56, 56, -80, -28, 0], [0, 0, 0, -2, -4, 32, 2, -32], [0, 0, -326, -326, -326, -18, 163, 0], [0, 200, 0, 0, 0, 0, -84, 0], [298, 298, 0, -108, 0, 0, 108, 0], [-666, 0, 0, -29, 0, 0, 0, 0], [0, 0, -184, -92, 0, 0, 0, 84], [0, 0, 0, 40, 80, 166, -40, -166], [230, 0, 0, -260, 0, 0, 0, 0], [0, 0, 128, 128, 128, 76, -64, 0], [0, 56, 0, 0, 0, 0, 384, 0], [-98, 0, 0, 256, 0, 0, 0, 0], [0, -124, 0, 0, 0, 0, 276, 0], [0, 0, -258, -129, 0, 0, 0, 106], [0, 0, 160, 80, 0, 0, 0, 160], [0, 558, 0, 0, 0, 0, 149, 0], [0, 0, 0, -12, -24, 28, 12, -28], [0, 0, 0, -56, -112, -186, 56, 186], [-224, 0, 0, 116, 0, 0, 0, 0], [0, -56, 0, 0, 0, 0, 20, 0], [0, 0, 304, 152, 0, 0, 0, -60], [0, 0, 42, 42, 42, -210, -21, 0], [522, 522, 0, -220, 0, 0, 220, 0], [0, -694, 0, 0, 0, 0, -129, 0], [0, 0, 200, 200, 200, -12, -100, 0], [-524, 0, 0, 58, 0, 0, 0, 0], [0, 0, 0, -44, -88, -76, 44, 76], [310, 310, 0, 351, 0, 0, -351, 0], [0, -224, 0, 0, 0, 0, -316, 0], [52, 52, 0, -442, 0, 0, 442, 0], [102, 102, 0, -584, 0, 0, 584, 0], [150, 0, 0, -121, 0, 0, 0, 0], [0, 174, 0, 0, 0, 0, 32, 0], [0, 0, 0, 64, 128, 32, -64, -32], [258, 0, 0, 328, 0, 0, 0, 0], [0, 0, 400, 400, 400, -16, -200, 0], [0, 0, 0, 102, 204, -136, -102, 136], [-264, 0, 0, -308, 0, 0, 0, 0], [-222, -222, 0, 225, 0, 0, -225, 0], [0, 0, 144, 72, 0, 0, 0, 0], [-336, -336, 0, -212, 0, 0, 212, 0], [-774, 0, 0, -359, 0, 0, 0, 0], [0, 0, 0, 180, 360, -156, -180, 156], [0, 0, -28, -28, -28, 432, 14, 0], [-524, 0, 0, 114, 0, 0, 0, 0], [0, 0, 0, 117, 234, -210, -117, 210], [50, 0, 0, -40, 0, 0, 0, 0], [0, 0, 302, 302, 302, -166, -151, 0], [616, 616, 0, -44, 0, 0, 44, 0], [0, 706, 0, 0, 0, 0, -241, 0], [0, 0, 84, 42, 0, 0, 0, -296], [0, 0, 124, 124, 124, -312, -62, 0], [0, 0, -440, -440, -440, 80, 220, 0], [0, 0, -120, -60, 0, 0, 0, -272], [0, 0, -210, -210, -210, 66, 105, 0], [276, 276, 0, -190, 0, 0, 190, 0], [0, 0, -224, -112, 0, 0, 0, 118], [0, 0, 144, 144, 144, -150, -72, 0], [0, 0, 252, 126, 0, 0, 0, 88], [0, 194, 0, 0, 0, 0, -572, 0], [234, 0, 0, 292, 0, 0, 0, 0], [0, 0, 0, -9, -18, 226, 9, -226], [690, 690, 0, 165, 0, 0, -165, 0], [0, 0, 114, 57, 0, 0, 0, -2], [-844, -844, 0, 356, 0, 0, -356, 0], [790, 0, 0, 83, 0, 0, 0, 0], [0, 0, -480, -480, -480, 100, 240, 0], [0, 0, 0, 120, 240, -224, -120, 224], [912, 0, 0, 448, 0, 0, 0, 0], [0, 0, 0, -43, -86, 342, 43, -342], [0, 0, 56, 28, 0, 0, 0, -76], [0, 0, 198, 99, 0, 0, 0, 202], [506, 506, 0, -244, 0, 0, 244, 0], [0, 474, 0, 0, 0, 0, 499, 0], [0, 0, 432, 432, 432, 200, -216, 0], [0, 0, -564, -564, -564, -144, 282, 0], [594, 0, 0, -288, 0, 0, 0, 0], [1080, 0, 0, 140, 0, 0, 0, 0], [0, 0, -264, -264, -264, -184, 132, 0], [104, 104, 0, 392, 0, 0, -392, 0], [0, 0, -128, -64, 0, 0, 0, -14], [0, 0, 784, 784, 784, -48, -392, 0], [0, 502, 0, 0, 0, 0, 380, 0], [-746, 0, 0, 96, 0, 0, 0, 0], [0, 0, 0, 375, 750, 170, -375, -170], [0, -908, 0, 0, 0, 0, 654, 0], [938, 0, 0, -416, 0, 0, 0, 0], [0, 0, 208, 104, 0, 0, 0, -386], [0, -30, 0, 0, 0, 0, 443, 0], [-158, -158, 0, -8, 0, 0, 8, 0], [0, 0, 0, -116, -232, -100, 116, 100], [0, 0, 0, -168, -336, 400, 168, -400], [0, 0, 0, -225, -450, 338, 225, -338]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_392_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_392_3_k_j();".
// 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_k_j(:prec:=8)
    chi := MakeCharacter_392_k();
    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_k_j();".
// 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_k_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_392_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![4, -8, 14, -4, 1]>,<5,R![38416, 0, -16464, 0, 6860, 0, -84, 0, 1]>],Snew);
    return Vf;
end function;