
// Make newform 275.3.c.e in Magma, downloaded from the LMFDB on 10 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_275_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_275_c_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_275_3_c_e();".
// 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_275_3_c_e();".
// 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 := [105, 0, 1, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -1, 1, 0], [-10, 21, -3, 2], [-21, 40, 6, -4]];
    Rf_basisdens := [1, 1, 41, 41];
    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_275_c();"
function MakeCharacter_275_c()
    N := 275;
    order := 2;
    char_gens := [177, 101];
    v := [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_275_c_Hecke();"
function MakeCharacter_275_c_Hecke(Kf)
    N := 275;
    order := 2;
    char_gens := [177, 101];
    char_values := [[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]];
    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, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, -5, 0], [-4, 1, 0, 0], [0, 0, 4, 0], [0, 0, 7, 0], [0, -1, 0, 0], [0, 0, 0, 6], [0, 1, 0, 0], [-3, 0, 0, 0], [0, 0, 0, -1], [0, -2, 0, 0], [0, 0, 10, 0], [0, 0, 0, 14], [0, 0, 0, 1], [18, 0, 0, 0], [0, 7, 0, 0], [0, 0, 0, -6], [27, 0, 0, 0], [0, 0, 26, 0], [0, 6, 0, 0], [0, 0, -32, 0], [-37, 0, 0, 0], [0, 0, 0, -26], [0, 10, 0, 0], [0, 0, 0, -4], [0, 0, -70, 0], [0, 2, 0, 0], [0, 0, 0, 36], [0, 0, 14, 0], [0, 21, 0, 0], [0, 0, 0, 14], [0, -22, 0, 0], [0, 15, 0, 0], [0, -10, 0, 0], [0, 0, 0, -11], [0, 0, 0, -59], [0, 0, -133, 0], [0, 0, -64, 0], [-242, 0, 0, 0], [-258, 0, 0, 0], [-278, 0, 0, 0], [0, 0, -35, 0], [0, 0, 88, 0], [53, 0, 0, 0], [0, 17, 0, 0], [0, 0, 0, 86], [0, 0, 124, 0], [-292, 0, 0, 0], [0, 0, -77, 0], [0, -22, 0, 0], [0, -32, 0, 0], [282, 0, 0, 0], [0, 0, 0, 14], [0, 0, 89, 0], [-162, 0, 0, 0], [0, 24, 0, 0], [0, 0, -26, 0], [0, 16, 0, 0], [0, 0, -112, 0], [0, 0, -200, 0], [0, 0, -200, 0], [87, 0, 0, 0], [0, 0, 0, -64], [0, 0, 0, -11], [-138, 0, 0, 0], [0, 0, -109, 0], [0, 0, -2, 0], [0, -50, 0, 0], [0, 0, 0, -64], [0, 62, 0, 0], [0, 0, 0, 94], [0, 0, -244, 0], [428, 0, 0, 0], [0, 0, 0, 36], [228, 0, 0, 0], [0, 0, 0, -146], [107, 0, 0, 0], [0, 32, 0, 0], [-722, 0, 0, 0], [-408, 0, 0, 0], [0, 6, 0, 0], [0, 0, 0, -54], [0, -42, 0, 0], [0, 0, 0, 66], [-22, 0, 0, 0], [0, 0, -155, 0], [0, -33, 0, 0], [0, 0, 0, 136], [0, 0, 0, -121], [0, -4, 0, 0], [0, 0, 0, 74], [0, -57, 0, 0], [268, 0, 0, 0], [0, 0, 242, 0], [828, 0, 0, 0], [242, 0, 0, 0], [0, 0, -64, 0], [0, 43, 0, 0], [0, 0, 178, 0], [0, 0, -70, 0], [0, 0, 394, 0], [0, 14, 0, 0], [0, 59, 0, 0], [0, 0, 0, -36], [0, 0, 0, -191], [0, 0, -50, 0], [493, 0, 0, 0], [0, -10, 0, 0], [0, 0, 365, 0], [0, 0, 394, 0], [0, 0, 0, 64], [-972, 0, 0, 0], [-113, 0, 0, 0], [1007, 0, 0, 0], [0, 0, 0, -59], [0, 0, 0, 4], [0, 0, 0, 151], [0, 107, 0, 0], [862, 0, 0, 0], [0, 0, -319, 0], [0, 0, 14, 0], [0, 0, 0, 131], [872, 0, 0, 0], [0, -85, 0, 0], [768, 0, 0, 0], [263, 0, 0, 0], [0, 0, 0, 64], [0, 0, 98, 0], [0, 38, 0, 0], [0, 0, 305, 0], [-1003, 0, 0, 0], [0, 0, 0, -266], [0, -18, 0, 0], [0, 24, 0, 0], [0, 0, 0, 61], [0, 0, -334, 0], [0, 0, 0, 154], [0, -118, 0, 0], [0, 107, 0, 0], [0, 74, 0, 0], [0, 0, 0, -64], [0, 0, -266, 0], [-102, 0, 0, 0], [138, 0, 0, 0], [0, 0, -238, 0], [0, 0, 425, 0], [1068, 0, 0, 0], [0, 0, 0, 16], [0, 0, -76, 0], [-78, 0, 0, 0], [0, 0, 0, -339], [0, 0, -134, 0], [0, 0, 0, -151], [1037, 0, 0, 0], [0, 174, 0, 0], [633, 0, 0, 0], [0, 0, 326, 0], [0, 83, 0, 0], [0, 0, 0, 199], [0, 0, -553, 0], [0, 0, -193, 0], [-1048, 0, 0, 0], [0, 0, 0, 54], [0, 0, 0, 6], [1182, 0, 0, 0], [0, 0, 758, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_275_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_275_3_c_e();".
// 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_275_3_c_e(:prec:=4)
    chi := MakeCharacter_275_c();
    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_275_3_c_e();".
// 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_275_3_c_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_275_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![5, 0, 1]>,<3,R![-21, 0, 1]>],Snew);
    return Vf;
end function;