
// Make newform 2736.3.o.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_o_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_2736_3_o_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_2736_3_o_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 := [-14, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_2736_o();"
function MakeCharacter_2736_o()
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    v := [2, 2, 2, 1];
    // 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_2736_o_Hecke();"
function MakeCharacter_2736_o_Hecke(Kf)
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    char_values := [[1, 0], [1, 0], [1, 0], [-1, 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], [-4, -1], [-4, 3], [-4, 5], [0, 0], [4, 7], [19, 0], [-30, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-44, 3], [-44, 13], [0, 0], [0, 0], [-44, -15], [0, 0], [0, 0], [-4, 33], [0, 0], [90, 0], [0, 0], [0, 0], [102, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [124, -35], [116, 23], [-76, -45], [-116, 55], [0, 0], [10, 0], [-250, 0], [0, 0], [0, 0], [0, 0], [0, 0], [4, 85], [0, 0], [-90, 0], [76, 75], [0, 0], [0, 0], [0, 0], [-44, 105], [-236, 7], [244, -35], [0, 0], [44, -115], [0, 0], [164, 77], [0, 0], [142, 0], [-284, 33], [0, 0], [244, -93], [0, 0], [0, 0], [-284, -35], [-590, 0], [0, 0], [0, 0], [0, 0], [-356, 37], [-316, 105], [510, 0], [-44, -155], [-50, 0], [0, 0], [0, 0], [0, 0], [-164, 175], [404, -63], [0, 0], [0, 0], [762, 0], [0, 0], [0, 0], [0, 0], [0, 0], [124, -203], [0, 0], [236, 153], [316, -185], [316, 123], [-436, -43], [-942, 0], [0, 0], [-918, 0], [164, 195], [930, 0], [0, 0], [0, 0], [0, 0], [116, 225], [0, 0], [524, 47], [0, 0], [0, 0], [-458, 0], [556, 33], [524, 77], [30, 0], [0, 0], [0, 0], [0, 0], [-284, 273], [-604, 143], [662, 0], [-436, -165], [0, 0], [484, 147], [596, -187], [44, 287], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-236, 315], [-1098, 0], [-1318, 0], [-604, 245], [124, -333], [-1270, 0], [116, 315], [0, 0], [0, 0], [244, 297], [724, 55], [-404, -255], [0, 0], [0, 0], [0, 0], [-764, -65], [0, 0], [716, -265], [-724, -117], [0, 0], [0, 0], [0, 0], [-1030, 0], [0, 0], [-844, 195], [0, 0], [0, 0], [-76, -385], [596, -357], [0, 0], [0, 0], [0, 0], [-1762, 0], [-642, 0], [44, -423], [0, 0], [-1830, 0], [0, 0], [1790, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-956, -63], [0, 0], [-404, -377], [0, 0], [0, 0], [-1662, 0], [850, 0], [0, 0], [964, -305], [-202, 0], [1084, -65], [0, 0], [1076, -15], [556, -477], [0, 0], [0, 0], [0, 0], [-844, 413], [-724, 455], [0, 0], [0, 0], [0, 0], [402, 0], [0, 0], [-484, -427], [0, 0], [0, 0], [-1350, 0], [0, 0], [764, 345], [-310, 0], [2430, 0], [116, 533], [0, 0], [0, 0], [0, 0], [0, 0], [618, 0], [-164, 583], [-1076, 435], [0, 0], [596, 455], [0, 0], [-2270, 0], [-2298, 0], [1324, -213], [0, 0], [0, 0], [0, 0], [-1970, 0], [0, 0], [0, 0], [-1364, -17], [0, 0], [0, 0], [0, 0], [716, 483], [0, 0], [1444, -255], [0, 0], [0, 0], [0, 0], [-1396, 365], [484, -663], [0, 0], [0, 0], [0, 0], [-1084, -357], [-1516, 77], [956, -615], [1276, -497], [1098, 0], [0, 0], [0, 0], [-284, -645], [3070, 0], [0, 0], [0, 0], [404, -715], [764, -693], [0, 0], [0, 0], [916, 493], [2890, 0], [1556, -385], [-510, 0], [0, 0], [-1396, 527], [3162, 0], [164, 705], [0, 0], [0, 0], [-1550, 0], [0, 0], [0, 0], [836, -735], [0, 0], [916, -737], [0, 0], [0, 0], [-1756, 95], [0, 0], [524, -793], [0, 0], [0, 0], [1324, -663], [1724, -405], [0, 0], [-3490, 0], [1804, -323], [0, 0], [0, 0], [-1844, 85], [0, 0], [62, 0], [1364, 437], [0, 0], [1444, -693], [1796, -475], [-1124, -567], [0, 0], [-484, -765], [0, 0], [3798, 0], [-356, -803], [0, 0], [0, 0], [0, 0], [3702, 0], [0, 0], [-356, -833], [0, 0], [-1756, 627], [-836, 897], [0, 0], [-596, 915]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_3_o_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_2736_3_o_c(:prec:=2)
    chi := MakeCharacter_2736_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2736_3_o_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_2736_3_o_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![6, 9, 1]>,<7,R![-122, 5, 1]>],Snew);
    return Vf;
end function;