
// Make newform 45.3.g.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_45_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_45_g_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_45_3_g_a();".
// 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_45_3_g_a();".
// 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 := [25, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 5, 5];
    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_45_g();"
function MakeCharacter_45_g()
    N := 45;
    order := 4;
    char_gens := [11, 37];
    v := [4, 3];
    // 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_45_g_Hecke();"
function MakeCharacter_45_g_Hecke(Kf)
    N := 45;
    order := 4;
    char_gens := [11, 37];
    char_values := [[1, 0, 0, 0], [0, 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, 0, 0], [0, 2, 0, 1], [-5, 0, -5, 0], [0, -5, 0, 5], [10, 0, -10, 0], [0, -2, 0, 0], [0, 0, 18, 0], [0, 0, 0, 2], [0, 15, 0, 15], [8, 0, 0, 0], [10, 0, 10, 0], [0, 10, 0, -10], [10, 0, -10, 0], [0, -26, 0, 0], [0, 0, 0, -16], [0, -15, 0, -15], [-58, 0, 0, 0], [70, 0, 70, 0], [0, -20, 0, 20], [55, 0, -55, 0], [0, 0, 12, 0], [0, 0, 0, -34], [0, 0, 0, 0], [-5, 0, -5, 0], [0, -5, 0, 5], [-35, 0, 35, 0], [0, -38, 0, 0], [0, 0, 162, 0], [0, 0, 0, 74], [55, 0, 55, 0], [0, 55, 0, -55], [0, 10, 0, 0], [0, 0, -102, 0], [0, -15, 0, -15], [-22, 0, 0, 0], [-200, 0, -200, 0], [100, 0, -100, 0], [0, 94, 0, 0], [0, 0, 0, -70], [0, -45, 0, -45], [218, 0, 0, 0], [0, -50, 0, 50], [-125, 0, 125, 0], [0, -92, 0, 0], [0, 0, -18, 0], [-298, 0, 0, 0], [-215, 0, 215, 0], [0, 172, 0, 0], [0, 0, 78, 0], [0, 0, 0, -70], [0, 120, 0, 120], [212, 0, 0, 0], [0, 115, 0, -115], [0, -194, 0, 0], [0, 0, 0, 182], [0, 45, 0, 45], [-178, 0, 0, 0], [-230, 0, -230, 0], [0, -50, 0, 50], [-350, 0, 350, 0], [0, 0, 0, 128], [190, 0, 190, 0], [0, -80, 0, 80], [145, 0, -145, 0], [0, 10, 0, 0], [482, 0, 0, 0], [-155, 0, -155, 0], [0, -104, 0, 0], [0, 0, 318, 0], [0, 0, 0, -142], [0, 90, 0, 90], [-185, 0, -185, 0], [100, 0, -100, 0], [0, 0, 558, 0], [0, 0, 0, -178], [0, -165, 0, -165], [-260, 0, -260, 0], [0, -80, 0, 80], [0, 0, 348, 0], [0, -195, 0, -195], [338, 0, 0, 0], [0, 70, 0, -70], [145, 0, -145, 0], [0, 0, -78, 0], [0, 0, 0, -124], [0, 90, 0, 90], [-365, 0, -365, 0], [0, 265, 0, -265], [-35, 0, 35, 0], [0, -74, 0, 0], [0, -120, 0, -120], [-125, 0, -125, 0], [0, 145, 0, -145], [0, 0, 222, 0], [0, 0, 0, 290], [0, 75, 0, 75], [0, 250, 0, -250], [370, 0, -370, 0], [362, 0, 0, 0], [490, 0, 490, 0], [0, 160, 0, 0], [0, 0, 0, 128], [0, 60, 0, 60], [-658, 0, 0, 0], [-35, 0, -35, 0], [0, -98, 0, 0], [0, 0, 0, -106], [0, -330, 0, -330], [-382, 0, 0, 0], [655, 0, 655, 0], [-620, 0, 620, 0], [0, 22, 0, 0], [0, 0, -258, 0], [812, 0, 0, 0], [0, -140, 0, 140], [820, 0, -820, 0], [0, 430, 0, 0], [0, 0, 0, 326], [0, 285, 0, 285], [-802, 0, 0, 0], [775, 0, -775, 0], [0, -314, 0, 0], [0, 0, 0, 38], [-562, 0, 0, 0], [0, 115, 0, -115], [0, 0, 498, 0], [0, 180, 0, 180], [-905, 0, -905, 0], [550, 0, -550, 0], [0, 0, 198, 0], [0, 0, 0, -610], [728, 0, 0, 0], [-170, 0, -170, 0], [0, 10, 0, -10], [0, 0, -318, 0], [0, 0, 0, -286], [250, 0, 250, 0], [0, 562, 0, 0], [0, -180, 0, -180], [-142, 0, 0, 0], [0, -275, 0, 275], [-1115, 0, 1115, 0], [0, 16, 0, 0], [0, 0, -1062, 0], [0, 150, 0, 150], [-80, 0, 80, 0], [0, 154, 0, 0], [0, 0, 1482, 0], [0, 0, 0, -610], [700, 0, 700, 0], [0, -380, 0, 380], [640, 0, -640, 0], [0, -86, 0, 0], [40, 0, 40, 0], [0, 130, 0, -130], [0, 0, -792, 0], [0, -390, 0, -390], [865, 0, 865, 0], [0, -305, 0, 305], [0, 532, 0, 0], [0, 0, 0, 722], [-1145, 0, -1145, 0], [0, -515, 0, 515], [0, -482, 0, 0], [0, 0, 0, -106], [2, 0, 0, 0], [310, 0, 310, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_45_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_45_3_g_a();".
// 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_45_3_g_a(:prec:=4)
    chi := MakeCharacter_45_g();
    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_45_3_g_a();".
// 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_45_3_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_45_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![25, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;