
// Make newform 23.7.b.b in Magma, downloaded from the LMFDB on 03 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_23_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_23_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_23_7_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_23_7_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 := [-17, -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_23_b();"
function MakeCharacter_23_b()
    N := 23;
    order := 2;
    char_gens := [5];
    v := [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_23_b_Hecke();"
function MakeCharacter_23_b_Hecke(Kf)
    N := 23;
    order := 2;
    char_gens := [5];
    char_values := [[-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 := 7;
    raw_aps := [[2, 3], [23, -8], [0, 0], [0, 0], [0, 0], [-97, -888], [0, 0], [0, 0], [-12167, 0], [-19321, 7896], [-30409, 2064], [0, 0], [-35449, 27264], [0, 0], [105887, -6432], [0, 0], [-253942, 0], [0, 0], [0, 0], [-306529, -54096], [-333097, -58848], [0, 0], [0, 0], [0, 0], [0, 0], [505802, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1673233, 641568], [-1344769, -631896], [0, 0], [2461079, 279336], [0, 0], [-3409369, 629184], [0, 0], [3645263, -1305528], [-6074926, 0], [1964666, 0], [3083519, -2247864], [0, 0], [0, 0], [-3435937, 2880192], [-7171513, -638232], [0, 0], [-12696838, 0], [1246466, 0], [0, 0], [0, 0], [-1396177, -5017488], [-13057081, -869424], [0, 0], [0, 0], [14175047, -5172432], [0, 0], [18047279, 1831656], [39618722, 0], [21296207, -3462792], [0, 0], [0, 0], [0, 0], [50707514, 0], [7271111, 11325504], [0, 0], [-61369126, 0], [-36300409, 72264], [0, 0], [708554, 0], [4058039, -17616264], [33541103, 9154272], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-43651753, -15021672], [0, 0], [47684639, 16446576], [0, 0], [0, 0], [0, 0], [0, 0], [45212879, 25668336], [-81749977, 20678952], [158037698, 0], [0, 0], [99457031, -12741576], [-62083294, 0], [0, 0], [0, 0], [-60228673, -35527632], [-109995649, 29235864], [-26287081, 51816216], [0, 0], [118058159, -35765304], [0, 0], [0, 0], [160353191, -4897896], [-32437753, -62844072], [0, 0], [0, 0], [0, 0], [0, 0], [-83979673, 77878128], [206306687, -12703272], [278321186, 0], [162397298, 0], [216027191, 2002944], [-369151486, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-89821033, -97095552], [93448703, -115151928], [0, 0], [0, 0], [-292640137, 66903312], [0, 0], [-324694537, 36000552], [-623541958, 0], [0, 0], [0, 0], [696357218, 0], [0, 0], [0, 0], [-146127841, 166242216], [0, 0], [0, 0], [0, 0], [-262578769, -124264176], [0, 0], [0, 0], [0, 0], [0, 0], [-283550542, 0], [511596191, 32314584], [-508385878, 0], [322501943, 160682592], [0, 0], [-1114784422, 0], [0, 0], [-636632854, 0], [136414847, 239742768], [-630146761, -6948024], [-73050937, -254678928], [-186924166, 0], [0, 0], [-1368751174, 0], [682343327, 25621968], [0, 0], [0, 0], [0, 0], [-563523001, 284482416], [0, 0], [0, 0], [554508407, 220438488], [0, 0], [815170007, 105667968], [0, 0], [0, 0], [0, 0], [-1863164158, 0], [1640782154, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_23_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_23_7_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_23_7_b_b(:prec:=2)
    chi := MakeCharacter_23_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    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_23_7_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_23_7_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_23_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([<2,R![-143, -7, 1]>],Snew);
    return Vf;
end function;