
// Make newform 245.3.i.c in Magma, downloaded from the LMFDB on 02 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_245_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_245_i_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_245_3_i_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_245_3_i_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 := [625, 0, 0, 0, -25, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0], [0, 125, 0, 0, 0, 0, 0, 1], [0, -25, 0, 10, 0, 1, 0, 0], [0, 250, 0, 0, 0, 0, 0, -1], [0, 0, 0, -25, 0, 5, 0, 1]];
    Rf_basisdens := [1, 5, 25, 125, 125, 25, 125, 125];
    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_245_i();"
function MakeCharacter_245_i()
    N := 245;
    order := 6;
    char_gens := [197, 101];
    v := [3, 5];
    // 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_245_i_Hecke();"
function MakeCharacter_245_i_Hecke(Kf)
    N := 245;
    order := 6;
    char_gens := [197, 101];
    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, -1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, 0, -3, 0, 6, 0], [0, 4, 0, -4, 0, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, 8, 8, 4], [0, -6, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, -2, 4, 0, 2], [0, 0, 0, 14, 0, 0, 0, 0], [0, 0, 0, 0, 14, 0, 0, 0], [0, 18, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 2, 2, 1], [0, 0, 0, 0, -7, 0, 14, 0], [0, -34, 0, 0, 0, 0, 0, 0], [-16, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -20], [0, 0, 76, 0, 0, 0, 0, 0], [0, 0, 0, 0, -23, 0, 0, 23], [0, 0, 0, 0, 6, 0, -12, 0], [0, 0, 0, 0, 22, 0, 0, -22], [0, 0, 0, 0, -22, 22, 22, 11], [0, 0, 0, 0, -28, 0, 0, 0], [0, -18, 0, 18, 0, 0, 0, 0], [-74, 0, 74, 0, 0, 0, 0, 0], [0, 0, 0, -26, 0, 0, 0, 0], [0, 0, 0, 24, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 10, 0], [0, 28, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, -26, 0, -13], [0, 0, 46, 0, 0, 0, 0, 0], [-14, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 37], [0, -18, 0, 18, 0, 0, 0, 0], [0, 0, 0, 0, -44, 0, 0, 44], [0, 0, 0, 0, -67, 0, 0, 0], [178, 0, -178, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 6, 0, 3], [0, 0, 148, 0, 0, 0, 0, 0], [0, 86, 0, 0, 0, 0, 0, 0], [0, 0, 0, 114, 0, 0, 0, 0], [0, 0, 0, 0, 12, -12, -12, -6], [242, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, 0, 0, 14], [0, 0, 0, 0, 0, 0, 0, -113], [0, 0, 0, 0, 25, 0, -50, 0], [0, 84, 0, -84, 0, 0, 0, 0], [242, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 56, -56, -56, -28], [0, 0, 0, 0, 51, -102, 0, -51], [0, 0, 0, 0, 26, 0, 0, 0], [0, 106, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 54, -54, -54, -27], [0, 0, 0, 0, -36, 0, 72, 0], [0, -2, 0, 0, 0, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 163], [0, 0, 0, 0, 145, 0, 0, -145], [0, 0, 0, 0, -29, 0, 0, 29], [0, 0, 0, 0, 76, -76, -76, -38], [0, 0, 0, 0, 176, 0, 0, 0], [0, -106, 0, 106, 0, 0, 0, 0], [0, 0, -374, 0, 0, 0, 0, 0], [0, 0, 0, -218, 0, 0, 0, 0], [0, -94, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 17, -34, 0, -17], [0, 0, 0, 0, 0, 0, 0, -194], [0, 0, -254, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 22], [0, -206, 0, 206, 0, 0, 0, 0], [-466, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 0], [-14, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, -25, 0, 0, 0], [0, 0, -404, 0, 0, 0, 0, 0], [0, 0, 0, 0, -44, 44, 44, 22], [0, 0, 0, 0, 25, -50, 0, -25], [554, 0, 0, 0, 0, 0, 0, 0], [358, 0, -358, 0, 0, 0, 0, 0], [0, 0, 0, 0, 34, 0, 0, -34], [0, 0, 0, 0, 2, 0, -4, 0], [0, 122, 0, -122, 0, 0, 0, 0], [74, 0, 0, 0, 0, 0, 0, 0], [0, -46, 0, 46, 0, 0, 0, 0], [0, 0, 0, 0, -73, 146, 0, 73], [0, 0, 0, 202, 0, 0, 0, 0], [0, 0, 0, 0, -169, 0, 0, 0], [0, 0, 0, 0, -112, 112, 112, 56], [0, 36, 0, 0, 0, 0, 0, 0], [674, 0, 0, 0, 0, 0, 0, 0], [0, 0, -62, 0, 0, 0, 0, 0], [0, 0, 0, 0, -200, 0, 0, 200], [0, 0, 0, 0, 67, 0, -134, 0], [0, 0, 0, 0, -84, 84, 84, 42], [0, 0, 0, 0, 83, 0, 0, 0], [0, 0, 1006, 0, 0, 0, 0, 0], [0, 0, 0, -106, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 19], [0, 0, -464, 0, 0, 0, 0, 0], [-314, 0, 314, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 214], [0, 0, 0, 0, -185, 0, 0, 185], [0, 0, 0, 0, -142, 0, 0, 0], [136, 0, -136, 0, 0, 0, 0, 0], [0, 0, 0, 0, 38, -76, 0, -38], [0, 0, 0, 0, -262, 0, 0, 0], [0, 66, 0, 0, 0, 0, 0, 0], [0, 0, 0, 274, 0, 0, 0, 0], [0, 0, 0, 0, 182, -182, -182, -91], [-16, 0, 0, 0, 0, 0, 0, 0], [1156, 0, -1156, 0, 0, 0, 0, 0], [0, 0, 0, 0, -233, 0, 0, 233], [0, 0, 0, 0, 0, 0, 0, -236], [0, -278, 0, 278, 0, 0, 0, 0], [-958, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 22, -22, -22, -11], [0, 0, 0, 152, 0, 0, 0, 0], [0, 0, 0, 0, 77, 0, 0, 0], [0, -294, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 95, 0, -190, 0], [-358, 0, 0, 0, 0, 0, 0, 0], [0, 0, 238, 0, 0, 0, 0, 0], [0, 0, 0, 0, -60, 0, 120, 0], [0, 0, 0, 0, 328, 0, 0, -328], [0, 0, 0, 0, -199, 0, 0, 0], [466, 0, -466, 0, 0, 0, 0, 0], [0, 0, 0, -308, 0, 0, 0, 0], [0, 0, -1142, 0, 0, 0, 0, 0], [0, 0, 0, 54, 0, 0, 0, 0], [0, 0, 0, 0, -98, 0, 196, 0], [0, 0, 0, 0, 16, -32, 0, -16], [0, 0, 0, 0, 0, 0, 0, 403], [0, 0, 0, 0, 0, 0, 0, 73], [0, 0, 0, 0, -377, 0, 0, 377], [-1424, 0, 1424, 0, 0, 0, 0, 0], [0, 0, 0, 0, 117, -234, 0, -117], [0, 0, -662, 0, 0, 0, 0, 0], [0, -394, 0, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, -246, 246, 246, 123], [0, 0, 0, 0, -118, 236, 0, 118], [0, 0, 0, 0, 445, 0, 0, -445], [0, 0, 0, 0, 0, 0, 0, 40], [0, 0, 0, 0, 163, 0, -326, 0], [0, 54, 0, -54, 0, 0, 0, 0], [0, 142, 0, -142, 0, 0, 0, 0], [0, 0, 0, 0, 48, -96, 0, -48], [0, 0, 0, 114, 0, 0, 0, 0], [0, 0, 0, 0, 44, 0, 0, 0], [0, -482, 0, 0, 0, 0, 0, 0], [374, 0, 0, 0, 0, 0, 0, 0], [0, 0, -704, 0, 0, 0, 0, 0], [0, 0, 0, 0, 32, 0, -64, 0], [0, 0, 0, 0, 88, 0, 0, -88], [0, 0, 0, 0, 226, -226, -226, -113], [0, -118, 0, 118, 0, 0, 0, 0], [0, 0, 0, 204, 0, 0, 0, 0], [0, 0, 0, -548, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 18, 0], [0, 296, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -32], [-1244, 0, 1244, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -383]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_245_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_245_3_i_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_245_3_i_c(:prec:=8)
    chi := MakeCharacter_245_i();
    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_245_3_i_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_245_3_i_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_245_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![81, 0, -9, 0, 1]>,<3,R![100, 0, 10, 0, 1]>],Snew);
    return Vf;
end function;