
// Make newform 294.3.h.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_294_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_294_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_294_3_h_d();".
// 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_294_3_h_d();".
// 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 := [81, 0, -72, 0, 55, 0, -8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-148, 0, 0, 0, 0, 0, -1, 0], [576, 0, -440, 0, 55, 0, -8, 0], [0, 533, 0, 0, 0, 0, 0, 1], [0, -203, 0, 0, 0, 0, 0, -1], [1656, 0, -1265, 0, 220, 0, -23, 0], [0, 81, 0, -341, 0, 55, 0, -8], [0, -5688, 0, 4345, 0, -605, 0, 79]];
    Rf_basisdens := [1, 55, 495, 165, 165, 495, 297, 1485];
    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_294_h();"
function MakeCharacter_294_h()
    N := 294;
    order := 6;
    char_gens := [197, 199];
    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_294_h_Hecke();"
function MakeCharacter_294_h_Hecke(Kf)
    N := 294;
    order := 6;
    char_gens := [197, 199];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 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, -1, 0, 0, 1, -1, -1, 0], [0, 0, 0, 1, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 5, 2], [10, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 2, 5], [0, 0, 16, 0, 0, 0, 0, 0], [0, 0, 0, -10, -1, 0, 0, 0], [0, 0, 0, 2, 0, 0, 20, 2], [32, 10, -32, 0, 0, 10, 0, 0], [0, 0, -20, 0, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, -30, 9], [20, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, -36, 0, 36, 0], [0, 0, 0, 0, 20, 0, -20, -8], [0, 0, 14, 0, 0, 20, 0, 0], [-60, -4, 60, 0, 0, -4, 0, 0], [0, 0, 0, -14, 0, 0, 25, -14], [-30, -16, 30, 0, 0, -16, 0, 0], [0, 0, 32, 0, 0, 20, 0, 0], [0, 0, 0, -20, 0, 0, -50, -20], [0, 0, 0, -3, -30, 0, 0, 0], [-90, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -70, 0, 70, -11], [0, 0, 40, 0, 0, -30, 0, 0], [0, 0, 0, 30, 3, 0, 0, 0], [18, -40, -18, 0, 0, -40, 0, 0], [0, 0, 0, 20, 0, 0, -4, 20], [-80, 20, 0, 0, 0, 0, 0, 0], [0, 0, 0, -20, -50, 0, 0, 0], [0, 0, 0, 0, 44, 0, -44, 10], [120, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, -36, -60, 0, 0, 0], [0, 80, 0, 0, 0, 80, 0, 0], [-50, -60, 50, 0, 0, -60, 0, 0], [0, 0, -140, 0, 0, 48, 0, 0], [0, 0, 0, 20, 0, 0, -58, 20], [0, 0, 0, 25, 70, 0, 0, 0], [0, 0, 0, 0, -15, 0, 15, 6], [-134, 20, 0, 0, 0, 0, 0, 0], [0, 0, 0, -54, 105, 0, 0, 0], [-80, 72, 80, 0, 0, 72, 0, 0], [0, 0, 0, -20, 0, 0, -8, -20], [88, 0, -88, 0, 0, 0, 0, 0], [-220, 60, 0, 0, 0, 0, 0, 0], [-40, 72, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 24, 0, -24, 60], [0, 0, 90, 0, 0, -20, 0, 0], [0, 0, 0, -10, 32, 0, 0, 0], [0, 0, 0, -6, 0, 0, -135, -6], [130, 0, -130, 0, 0, 0, 0, 0], [0, 0, 0, 36, 0, 0, 180, 36], [0, 0, 0, 15, 198, 0, 0, 0], [0, 0, 0, 0, 231, 0, -231, -30], [0, 0, 0, 0, -130, 0, 130, -53], [0, 0, -88, 0, 0, -10, 0, 0], [40, -72, -40, 0, 0, -72, 0, 0], [0, 0, 0, -50, 0, 0, -80, -50], [-120, 8, 120, 0, 0, 8, 0, 0], [0, 0, 0, -45, 0, 0, 114, -45], [0, 40, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 230, 0, -230, 16], [0, 0, -90, 0, 0, 64, 0, 0], [0, 0, 0, 60, -132, 0, 0, 0], [0, 0, 172, 0, 0, 120, 0, 0], [-50, -208, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 57, 0, -57, 30], [-222, -140, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 162, 0, -162, -105], [0, 0, 0, 22, 295, 0, 0, 0], [-200, 48, 200, 0, 0, 48, 0, 0], [0, 0, 390, 0, 0, -32, 0, 0], [-12, -80, 0, 0, 0, 0, 0, 0], [0, 0, 0, 140, -4, 0, 0, 0], [0, 0, 0, 0, -100, 0, 100, -26], [0, 0, -310, 0, 0, -4, 0, 0], [0, 0, 0, 6, 180, 0, 0, 0], [106, 240, -106, 0, 0, 240, 0, 0], [0, 0, 0, -24, 0, 0, 0, -24], [-248, 200, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -285, 0, 285, 42], [-670, 48, 0, 0, 0, 0, 0, 0], [0, 0, -448, 0, 0, -40, 0, 0], [0, 0, 0, 50, -415, 0, 0, 0], [0, 0, 0, 8, 0, 0, 500, 8], [0, 0, -750, 0, 0, -32, 0, 0], [0, 0, 0, 89, 0, 0, -10, 89], [360, 16, 0, 0, 0, 0, 0, 0], [0, 0, 0, -20, -440, 0, 0, 0], [0, 0, 0, 0, 10, 0, -10, -40], [40, 24, -40, 0, 0, 24, 0, 0], [0, 0, 0, -90, 0, 0, -165, -90], [0, 0, 276, 0, 0, -80, 0, 0], [0, 0, 0, 60, 0, 0, -228, 60], [0, 0, 0, 51, 90, 0, 0, 0], [0, 0, 0, 0, -30, 0, 30, 183], [0, 0, 160, 0, 0, -18, 0, 0], [0, 0, -496, 0, 0, 200, 0, 0], [-740, 68, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -460, 0, 460, -20], [0, 0, 0, 0, -92, 0, 92, -160], [0, 0, 0, -90, -120, 0, 0, 0], [20, -20, -20, 0, 0, -20, 0, 0], [-150, 152, 150, 0, 0, 152, 0, 0], [0, 0, 0, 100, 0, 0, -524, 100], [0, 0, 0, 45, -558, 0, 0, 0], [0, 0, 0, 0, 225, 0, -225, 78], [-22, 360, 0, 0, 0, 0, 0, 0], [0, 0, 440, 0, 0, 248, 0, 0], [290, -8, -290, 0, 0, -8, 0, 0], [0, 0, 0, -50, 0, 0, 508, -50], [-240, -70, 240, 0, 0, -70, 0, 0], [-136, -120, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 460, 0, -460, 50], [440, 144, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -546, 0, 546, 60], [0, 0, 0, 150, -144, 0, 0, 0], [0, 0, 0, -158, 0, 0, 325, -158], [198, 20, -198, 0, 0, 20, 0, 0], [260, -24, 0, 0, 0, 0, 0, 0], [0, 0, 0, 15, -78, 0, 0, 0], [0, 0, 0, 0, -609, 0, 609, -30], [0, 0, -872, 0, 0, -150, 0, 0], [0, 0, 0, -274, 0, 0, -160, -274], [0, 0, 334, 0, 0, 120, 0, 0], [0, 0, 0, -60, -780, 0, 0, 0], [120, -122, 0, 0, 0, 0, 0, 0], [0, 0, -190, 0, 0, -292, 0, 0], [-484, 280, 484, 0, 0, 280, 0, 0], [0, 0, 0, 270, 0, 0, -309, 270], [0, 0, 632, 0, 0, 20, 0, 0], [130, 392, 0, 0, 0, 0, 0, 0], [0, 0, 0, -301, 230, 0, 0, 0], [-666, 160, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 86, 0, -86, -305], [-40, -494, 40, 0, 0, -494, 0, 0], [0, 0, 0, 285, 0, 0, 102, 285], [0, 0, 0, 0, 140, 0, -140, 136], [384, -170, 0, 0, 0, 0, 0, 0], [0, 0, 0, -16, -220, 0, 0, 0], [-240, -76, 240, 0, 0, -76, 0, 0], [0, 0, 0, 30, 0, 0, 3, 30], [1146, -100, -1146, 0, 0, -100, 0, 0], [0, 0, 0, 220, 0, 0, 370, 220], [-490, -180, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 258, 0, -258, -165], [0, 0, -992, 0, 0, 120, 0, 0], [0, 0, 0, 70, 523, 0, 0, 0], [0, 0, -750, 0, 0, -296, 0, 0], [0, 0, 0, 309, 0, 0, -150, 309], [500, -168, 0, 0, 0, 0, 0, 0], [0, 0, 0, -180, 462, 0, 0, 0], [500, -272, -500, 0, 0, -272, 0, 0], [0, 0, 0, -102, 0, 0, -315, -102], [0, 0, 24, 0, 0, -380, 0, 0], [0, 0, 0, 147, 630, 0, 0, 0], [1010, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 170, 0, -170, 13], [0, 0, 0, -230, -23, 0, 0, 0], [0, 0, 0, 160, 0, 0, 496, 160], [-280, -36, 0, 0, 0, 0, 0, 0], [0, 0, 0, 336, -150, 0, 0, 0], [0, 0, 0, 0, 72, 0, -72, 150], [0, 0, 0, 0, 708, 0, -708, 240], [-536, -300, 536, 0, 0, -300, 0, 0], [-950, 108, 950, 0, 0, 108, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_294_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_294_3_h_d();".
// 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_294_3_h_d(:prec:=8)
    chi := MakeCharacter_294_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_294_3_h_d();".
// 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_294_3_h_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_294_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![1296, 0, -1584, 0, 1900, 0, -44, 0, 1]>,<13,R![-12, -20, 1]>],Snew);
    return Vf;
end function;