
// Make newform 2304.3.e.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_e_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_2304_3_e_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_2304_3_e_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 := [2, 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_2304_e();"
function MakeCharacter_2304_e()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 2, 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_2304_e_Hecke();"
function MakeCharacter_2304_e_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    char_values := [[1, 0], [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, 0], [0, 0], [0, 1], [0, 0], [0, 0], [-10, 0], [0, -7], [0, 0], [0, 0], [0, 1], [0, 0], [24, 0], [0, 31], [0, 0], [0, 0], [0, 73], [0, 0], [-120, 0], [0, 0], [0, 0], [96, 0], [0, 0], [0, 0], [0, 41], [-144, 0], [0, 119], [0, 0], [0, 0], [-182, 0], [0, -127], [0, 0], [0, 0], [0, 17], [0, 0], [0, 191], [0, 0], [-264, 0], [0, 0], [0, 0], [0, 217], [0, 0], [-38, 0], [0, 0], [-190, 0], [0, 167], [0, 0], [0, 0], [0, 0], [0, 0], [-442, 0], [0, 103], [0, 0], [240, 0], [0, 0], [0, 223], [0, 0], [0, 191], [0, 0], [-230, 0], [0, -391], [0, 0], [0, 217], [0, 0], [0, 0], [-50, 0], [0, 383], [0, 0], [576, 0], [0, 0], [360, 0], [0, 497], [0, 0], [0, 0], [-504, 0], [0, 0], [0, 0], [0, -529], [456, 0], [0, -439], [-240, 0], [0, 0], [58, 0], [0, 0], [290, 0], [0, 0], [0, 0], [0, 631], [-336, 0], [0, 119], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -239], [0, 161], [0, 0], [682, 0], [0, 0], [0, -697], [0, 0], [0, -751], [0, 0], [-1150, 0], [0, 0], [0, -833], [0, 0], [1102, 0], [0, 0], [1224, 0], [0, 713], [0, 0], [0, 0], [0, 809], [0, 0], [0, 0], [0, -887], [0, 0], [600, 0], [-770, 0], [0, -623], [0, 0], [0, 0], [0, -911], [518, 0], [0, 0], [0, 0], [-1450, 0], [0, 0], [0, 0], [0, 0], [1190, 0], [0, 799], [-962, 0], [0, 553], [0, 0], [0, -1127], [0, 1039], [0, 0], [0, -1129], [0, 0], [0, 0], [1258, 0], [0, 0], [1656, 0], [0, -593], [0, 0], [0, 0], [-696, 0], [0, -1169], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 1049], [430, 0], [0, -1321], [0, 0], [0, 113], [0, 0], [0, 0], [0, -1193], [0, 0], [0, 0], [744, 0], [1680, 0], [0, 1057], [0, 0], [-1320, 0], [0, 0], [384, 0], [0, 0], [0, -679], [0, 0], [0, -241], [0, 0], [1560, 0], [0, 0], [0, 0], [-2170, 0], [0, -343], [0, 0], [0, -959], [470, 0], [0, 0], [-2160, 0], [0, 0], [-1056, 0], [0, 0], [0, 0], [0, 791], [0, 0], [0, 1687], [98, 0], [-490, 0], [0, 1697], [0, 0], [0, 1081], [0, 0], [-2150, 0], [1920, 0], [0, 0], [0, 287], [0, 0], [0, 0], [0, -1721], [0, 0], [2590, 0], [0, 1249], [0, 0], [0, 0], [0, 0], [-2542, 0], [0, 0], [0, 679], [0, 0], [0, 1513], [-1862, 0], [0, 0], [0, 1241], [0, 0], [0, 0], [-2760, 0], [0, -713], [0, 0], [0, 0], [0, 0], [-2870, 0], [0, 0], [0, 0], [0, 151], [0, 0], [0, 0], [2640, 0], [0, 863], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [2520, 0], [0, -977], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1430, 0], [0, 1519], [0, 0], [-480, 0], [0, 287], [0, 0], [2842, 0], [0, 0], [0, 1897], [-1870, 0], [0, 0], [0, 0], [-2438, 0], [-2664, 0], [0, -1337], [0, 0], [0, 2281], [0, 2359], [0, 0], [0, 2447], [-3480, 0], [0, 0], [3456, 0], [0, 0], [-2496, 0], [0, 0], [0, 0], [840, 0], [1298, 0], [0, 0], [0, 0], [0, 0], [0, 0], [3720, 0], [0, 0], [0, 0], [-3696, 0], [0, -337], [0, 0], [0, -2671], [0, -1271], [0, 0], [0, 1097], [0, 0], [-2184, 0], [0, -889], [0, 0], [0, 2663], [0, 0], [0, 0], [3410, 0], [0, -1657], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_e_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_2304_3_e_b(:prec:=2)
    chi := MakeCharacter_2304_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2304_3_e_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_2304_3_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![2, 0, 1]>,<7,R![0, 1]>,<13,R![10, 1]>,<19,R![0, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;