
// Make newform 810.3.j.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_810_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_810_j_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_810_3_j_a();".
// 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_810_3_j_a();".
// 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 := [961, 0, -434, 0, 165, 0, -14, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [721, 0, 0, 0, 0, 0, 1, 0], [0, 1711, 0, 0, 0, 0, 0, 1], [0, 721, 0, 0, 0, 0, 0, 1], [6076, 0, -2310, 0, 165, 0, -14, 0], [29078, 0, -11055, 0, 1155, 0, -67, 0], [0, 29078, 0, -11055, 0, 1155, 0, -67], [0, 12493, 0, -24915, 0, 2145, 0, -182]];
    Rf_basisdens := [1, 495, 495, 495, 5115, 15345, 15345, 15345];
    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_810_j();"
function MakeCharacter_810_j()
    N := 810;
    order := 6;
    char_gens := [731, 487];
    v := [1, 3];
    // 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_810_j_Hecke();"
function MakeCharacter_810_j_Hecke(Kf)
    N := 810;
    order := 6;
    char_gens := [731, 487];
    char_values := [[0, 0, 0, 0, 1, 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, -1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, -3, -1, 0, 0], [0, 0, -2, 0, 0, 0, -1, -2], [0, 0, -3, 0, 0, 0, 2, -3], [0, 0, -3, 0, 0, 0, 0, 0], [-3, 7, 0, 0, 0, 0, 0, 0], [-3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -15, -14, 0, 0], [0, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, 0, 17, 15, 0, 0], [0, 0, 0, 14, 0, 0, -14, 2], [0, 0, 6, 1, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 7, -4], [60, -14, 0, 0, -60, 14, 0, 0], [51, -9, 0, 0, 0, 0, 0, 0], [0, 0, -9, 4, 0, 0, 0, 0], [49, -24, 0, 0, -49, 24, 0, 0], [0, 0, -26, 8, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 6, -21], [0, 0, 0, 16, 0, 0, -16, 7], [-45, 63, 0, 0, 45, -63, 0, 0], [-27, 58, 0, 0, 27, -58, 0, 0], [0, 0, 0, -29, 0, 0, 29, 0], [0, 0, -22, 0, 0, 0, 10, -22], [0, 0, 6, 0, 0, 0, -8, 6], [0, 0, 2, 28, 0, 0, 0, 0], [-36, -30, 0, 0, 0, 0, 0, 0], [-19, -90, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -12, -30, 0, 0], [0, 0, 0, -34, 0, 0, 34, 14], [0, 0, -12, -18, 0, 0, 0, 0], [-27, -135, 0, 0, 27, 135, 0, 0], [0, 0, 0, 0, -38, 102, 0, 0], [0, 0, 12, -35, 0, 0, 0, 0], [20, -66, 0, 0, -20, 66, 0, 0], [0, 0, 19, -16, 0, 0, 0, 0], [0, 0, 0, -28, 0, 0, 28, -10], [0, 0, 0, 0, -21, 60, 0, 0], [-69, 29, 0, 0, 69, -29, 0, 0], [0, 0, 0, 20, 0, 0, -20, 12], [113, -42, 0, 0, 0, 0, 0, 0], [0, 0, 12, 0, 0, 0, 66, 12], [0, 0, -17, -34, 0, 0, 0, 0], [279, -1, 0, 0, 0, 0, 0, 0], [114, -30, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -171, -105, 0, 0], [0, 0, 2, 0, 0, 0, -74, 2], [87, -232, 0, 0, -87, 232, 0, 0], [0, 0, 0, 0, 59, -114, 0, 0], [126, 82, 0, 0, 0, 0, 0, 0], [0, 0, 48, 44, 0, 0, 0, 0], [191, 90, 0, 0, -191, -90, 0, 0], [0, 0, 0, -36, 0, 0, 36, -30], [0, 0, 0, 0, -87, 213, 0, 0], [-60, 118, 0, 0, 60, -118, 0, 0], [0, 0, 0, -18, 0, 0, 18, -72], [115, -147, 0, 0, 0, 0, 0, 0], [0, 0, 49, 0, 0, 0, -22, 49], [0, 0, 0, 0, 0, 0, -27, 0], [0, 0, 86, -53, 0, 0, 0, 0], [0, 0, 0, 0, 45, 129, 0, 0], [0, 0, 0, 0, 0, 0, 0, 60], [0, 0, 21, 114, 0, 0, 0, 0], [0, 0, 20, 0, 0, 0, 112, 20], [-57, -211, 0, 0, 57, 211, 0, 0], [266, 126, 0, 0, -266, -126, 0, 0], [0, 0, 95, -56, 0, 0, 0, 0], [0, 0, 0, 0, 150, 240, 0, 0], [37, -84, 0, 0, -37, 84, 0, 0], [-6, 182, 0, 0, 6, -182, 0, 0], [0, 0, 0, -18, 0, 0, 18, 84], [0, 0, 16, 0, 0, 0, 107, 16], [0, 0, -12, -18, 0, 0, 0, 0], [91, 165, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -465, -76, 0, 0], [0, 0, -78, 0, 0, 0, -119, -78], [0, 0, 0, -106, 0, 0, 106, 23], [0, 0, -18, -106, 0, 0, 0, 0], [0, 0, 0, 0, 491, 30, 0, 0], [0, 0, -63, 50, 0, 0, 0, 0], [-391, -24, 0, 0, 391, 24, 0, 0], [0, 0, 0, 152, 0, 0, -152, -33], [0, 0, 0, -8, 0, 0, 8, 103], [-135, -189, 0, 0, 135, 189, 0, 0], [-447, 6, 0, 0, 447, -6, 0, 0], [0, 0, 0, 33, 0, 0, -33, 132], [0, 0, 82, 0, 0, 0, 98, 82], [0, 0, 30, 0, 0, 0, -78, 30], [0, 0, -96, 132, 0, 0, 0, 0], [-675, -48, 0, 0, 0, 0, 0, 0], [0, 0, -93, 0, 0, 0, 34, -93], [0, 0, 0, 51, 0, 0, -51, 30], [0, 0, 132, 18, 0, 0, 0, 0], [0, 0, 0, 0, 681, -57, 0, 0], [-357, -54, 0, 0, 0, 0, 0, 0], [0, 0, 192, -54, 0, 0, 0, 0], [0, 0, 0, -17, 0, 0, 17, -6], [0, 0, 0, 137, 0, 0, -137, 86], [-80, 210, 0, 0, 0, 0, 0, 0], [0, 0, 10, 0, 0, 0, -67, 10], [210, -66, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -384, -106, 0, 0], [0, 0, 12, 0, 0, 0, 19, 12], [0, 0, 0, 0, -127, 699, 0, 0], [0, 0, 0, 170, 0, 0, -170, 77], [309, 162, 0, 0, -309, -162, 0, 0], [-549, -237, 0, 0, 0, 0, 0, 0], [0, 0, -75, 162, 0, 0, 0, 0], [-437, 30, 0, 0, 437, -30, 0, 0], [0, 0, 16, 53, 0, 0, 0, 0], [0, 0, 0, -176, 0, 0, 176, -14], [0, 0, 0, 0, -315, 269, 0, 0], [266, -174, 0, 0, -266, 174, 0, 0], [-427, 153, 0, 0, 0, 0, 0, 0], [0, 0, 192, 0, 0, 0, 38, 192], [0, 0, 12, -45, 0, 0, 0, 0], [-339, -252, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 855, 49, 0, 0], [0, 0, 27, 0, 0, 0, -94, 27], [0, 0, 0, 0, -668, -258, 0, 0], [0, 0, 33, 0, 0, 0, -84, 33], [-444, -94, 0, 0, 444, 94, 0, 0], [-147, -268, 0, 0, 0, 0, 0, 0], [65, -345, 0, 0, -65, 345, 0, 0], [0, 0, 0, -100, 0, 0, 100, 114], [-636, 378, 0, 0, 636, -378, 0, 0], [0, 0, 0, 28, 0, 0, -28, 18], [0, 0, 108, 0, 0, 0, -36, 108], [0, 0, 92, 334, 0, 0, 0, 0], [827, -51, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -498, -576, 0, 0], [0, 0, 0, 0, -473, -537, 0, 0], [0, 0, 0, -166, 0, 0, 166, 2], [0, 0, 156, -153, 0, 0, 0, 0], [0, 0, 0, 0, 687, -102, 0, 0], [-459, -45, 0, 0, 0, 0, 0, 0], [0, 0, -82, -155, 0, 0, 0, 0], [0, 0, 0, 0, -1011, 281, 0, 0], [0, 0, 0, -15, 0, 0, 15, 90], [-692, 282, 0, 0, 0, 0, 0, 0], [0, 0, -192, 0, 0, 0, 100, -192], [0, 0, 114, -189, 0, 0, 0, 0], [-615, 424, 0, 0, 0, 0, 0, 0], [180, -930, 0, 0, 0, 0, 0, 0], [0, 0, -72, 0, 0, 0, 68, -72], [0, 0, 40, 0, 0, 0, -88, 40], [-945, 537, 0, 0, 945, -537, 0, 0], [0, 0, 0, 0, 653, 315, 0, 0], [75, -30, 0, 0, 0, 0, 0, 0], [0, 0, -83, -226, 0, 0, 0, 0], [0, 0, 0, -117, 0, 0, 117, -342], [0, 0, 0, -123, 0, 0, 123, -90], [0, 0, 0, 0, -531, -798, 0, 0], [0, 0, -102, 0, 0, 0, 318, -102], [0, 0, 111, 0, 0, 0, 348, 111], [-762, -366, 0, 0, 0, 0, 0, 0], [0, 0, -120, 0, 0, 0, -129, -120], [0, 0, 0, -14, 0, 0, 14, -176], [0, 0, 180, 4, 0, 0, 0, 0], [-399, 150, 0, 0, 399, -150, 0, 0], [-294, 814, 0, 0, 0, 0, 0, 0], [0, 0, 274, -136, 0, 0, 0, 0], [0, 0, 0, 342, 0, 0, -342, -90], [0, 0, 0, 0, 678, -402, 0, 0], [-243, -70, 0, 0, 243, 70, 0, 0], [433, -207, 0, 0, 0, 0, 0, 0], [0, 0, 159, 0, 0, 0, -120, 159]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_810_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_810_3_j_a();".
// 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_810_3_j_a(:prec:=8)
    chi := MakeCharacter_810_j();
    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_810_3_j_a();".
// 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_810_3_j_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_810_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![15376, 0, -23312, 0, 35220, 0, -188, 0, 1]>,<17,R![-89, 6, 1]>],Snew);
    return Vf;
end function;