
// Make newform 675.3.c.i in Magma, downloaded from the LMFDB on 15 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_675_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_675_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_675_3_c_i();".
// 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_675_3_c_i();".
// 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 := [5, 0, 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_675_c();"
function MakeCharacter_675_c()
    N := 675;
    order := 2;
    char_gens := [326, 352];
    v := [1, 2];
    // 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_675_c_Hecke();"
function MakeCharacter_675_c_Hecke(Kf)
    N := 675;
    order := 2;
    char_gens := [326, 352];
    char_values := [[-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, 1], [0, 0], [0, 0], [-12, 0], [0, -7], [11, 0], [0, -11], [16, 0], [0, 3], [0, -5], [9, 0], [-2, 0], [0, 10], [47, 0], [0, -41], [0, -38], [0, 2], [-26, 0], [-6, 0], [0, 16], [92, 0], [-69, 0], [0, -30], [0, -24], [58, 0], [0, -69], [-98, 0], [0, 72], [146, 0], [0, -5], [-226, 0], [0, -21], [0, 18], [-64, 0], [0, -59], [-221, 0], [133, 0], [61, 0], [0, -42], [0, 48], [0, -22], [92, 0], [0, -56], [-128, 0], [0, -84], [-287, 0], [362, 0], [-248, 0], [0, 8], [366, 0], [0, 54], [0, -148], [-53, 0], [0, 201], [0, -27], [0, -82], [0, -121], [-206, 0], [-42, 0], [0, 114], [342, 0], [0, 60], [-337, 0], [0, -126], [106, 0], [0, 230], [-36, 0], [-466, 0], [0, -98], [196, 0], [0, 291], [0, 78], [204, 0], [557, 0], [268, 0], [0, 9], [0, 81], [173, 0], [0, 258], [503, 0], [0, 157], [-446, 0], [0, -186], [242, 0], [286, 0], [0, 54], [0, -112], [134, 0], [0, -282], [-618, 0], [0, 158], [0, 6], [544, 0], [0, -2], [-434, 0], [0, 359], [0, -61], [0, 322], [47, 0], [324, 0], [623, 0], [0, 42], [0, 372], [0, 204], [854, 0], [-212, 0], [0, -246], [0, -517], [0, -224], [-151, 0], [38, 0], [857, 0], [0, -465], [-694, 0], [22, 0], [0, -276], [681, 0], [0, 166], [0, 398], [0, 362], [34, 0], [1002, 0], [0, 84], [0, 162], [952, 0], [0, 505], [-614, 0], [0, 402], [-602, 0], [406, 0], [-32, 0], [0, -161], [589, 0], [143, 0], [0, -354], [-1207, 0], [0, 312], [-1057, 0], [0, -340], [0, 430], [144, 0], [0, -438], [16, 0], [0, 40], [1056, 0], [0, 244], [-999, 0], [0, -14], [-1094, 0], [0, -355], [273, 0], [0, 110], [82, 0], [0, 609], [-87, 0], [0, 242], [-239, 0], [0, 742], [708, 0], [0, 13], [0, -210], [0, -299], [-512, 0], [0, -315], [0, 193], [0, -189], [1889, 0], [-321, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_675_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_675_3_c_i();".
// 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_675_3_c_i(:prec:=2)
    chi := MakeCharacter_675_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_675_3_c_i();".
// 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_675_3_c_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_675_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![5, 0, 1]>,<7,R![12, 1]>],Snew);
    return Vf;
end function;