
// Make newform 175.4.b.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_175_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_175_b_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_175_4_b_b();".
// 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_175_4_b_b();".
// 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 := [1, 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_175_b();"
function MakeCharacter_175_b()
    N := 175;
    order := 2;
    char_gens := [127, 101];
    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_175_b_Hecke();"
function MakeCharacter_175_b_Hecke(Kf)
    N := 175;
    order := 2;
    char_gens := [127, 101];
    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 := 4;
    raw_aps := [[0, 1], [0, -2], [0, 0], [0, 7], [-8, 0], [0, 28], [0, -54], [110, 0], [0, 48], [110, 0], [12, 0], [0, 246], [182, 0], [0, 128], [0, -324], [0, -162], [-810, 0], [-488, 0], [0, -244], [-768, 0], [0, -702], [-440, 0], [0, -1302], [-730, 0], [0, -294], [-688, 0], [0, 1388], [0, -244], [-90, 0], [0, 1318], [0, 1776], [-1118, 0], [0, -2274], [210, 0], [2010, 0], [1112, 0], [0, -124], [0, 2008], [0, -2884], [0, 2228], [820, 0], [3892, 0], [-5048, 0], [0, -2962], [0, -3334], [-1860, 0], [-4268, 0], [0, -5432], [0, 2046], [2980, 0], [0, 4458], [-4440, 0], [3302, 0], [1582, 0], [0, -2354], [0, -3872], [-180, 0], [2032, 0], [0, 5426], [842, 0], [0, -3782], [0, -4312], [0, -2674], [-3768, 0], [0, 2438], [0, 3186], [8672, 0], [0, -814], [0, -9344], [5180, 0], [0, 12178], [-440, 0], [0, 9816], [0, -442], [3960, 0], [0, 6708], [13350, 0], [0, 1356], [6222, 0], [-5150, 0], [-2310, 0], [1262, 0], [-4488, 0], [0, 17038], [-16200, 0], [0, -8772], [-2130, 0], [0, -10534], [-9268, 0], [0, -9392], [0, 10806], [-4940, 0], [0, 5216], [4412, 0], [-19060, 0], [0, 12768], [5500, 0], [-7338, 0], [0, -17582], [-1618, 0], [0, -16144], [0, -4654], [0, 10078], [5930, 0], [-19048, 0], [0, 14366], [0, 3626], [0, -1062], [10200, 0], [-25158, 0], [0, -25664], [0, 19018], [0, -17334], [-18730, 0], [-6928, 0], [16302, 0], [0, 4718], [0, 21436], [0, 4458], [26640, 0], [7432, 0], [0, 58], [0, 21516], [0, 18108], [-10078, 0], [18762, 0], [-6810, 0], [-4860, 0], [0, 13636], [0, 2088], [5160, 0], [0, -28152], [-16808, 0], [0, -21674], [7422, 0], [-13790, 0], [0, -6232], [0, 1766], [0, -1204], [7050, 0], [23282, 0], [10142, 0], [0, -9192], [0, 46716], [-11240, 0], [-700, 0], [0, -37492], [0, -28894], [2770, 0], [0, 17688], [0, 33566], [-16758, 0], [0, 11468], [0, 50356], [0, 8716], [7632, 0], [23080, 0], [-45110, 0], [0, -16674], [43832, 0], [0, 736], [0, 38138], [0, -26224], [18762, 0], [0, -38394], [0, 5388], [25472, 0], [0, 17096]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_175_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_175_4_b_b();".
// 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_175_4_b_b(:prec:=2)
    chi := MakeCharacter_175_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_175_4_b_b();".
// 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_175_4_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_175_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 0, 1]>,<3,R![4, 0, 1]>],Snew);
    return Vf;
end function;