
// Make newform 2304.3.e.i in Magma, downloaded from the LMFDB on 24 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_i();".
// 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_i();".
// 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, 4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 5, 0, 1], [0, -9, 0, -3], [4, 0, 2, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    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_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, 0, 0], [1, 0, 0, 0], [-1, 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, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, -1], [0, 0, -2, 0], [0, 0, 0, 3], [0, 0, -3, 0], [4, 0, 0, 0], [0, 14, 0, 0], [0, -19, 0, 0], [0, 0, 0, 11], [0, 0, 0, -8], [0, 0, 7, 0], [-56, 0, 0, 0], [0, -26, 0, 0], [0, 29, 0, 0], [0, 0, -24, 0], [0, 0, 0, 20], [-28, 0, 0, 0], [0, 22, 0, 0], [-20, 0, 0, 0], [0, 0, 0, 21], [0, 0, -18, 0], [0, 0, 13, 0], [-8, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, -7], [0, 0, -24, 0], [0, 0, 0, -31], [0, 0, 1, 0], [0, 0, 0, -65], [0, 0, -52, 0], [0, 0, -47, 0], [-148, 0, 0, 0], [0, -33, 0, 0], [0, 0, 0, -1], [0, 0, 0, -20], [-272, 0, 0, 0], [0, -104, 0, 0], [0, 53, 0, 0], [0, 0, -68, 0], [0, 0, 0, 101], [0, 56, 0, 0], [86, 0, 0, 0], [0, -117, 0, 0], [0, 0, 0, 69], [-244, 0, 0, 0], [0, 0, 0, 31], [0, 0, -70, 0], [0, 0, 0, -37], [0, 0, -41, 0], [0, -4, 0, 0], [-196, 0, 0, 0], [0, 0, -38, 0], [0, 0, 59, 0], [0, 12, 0, 0], [0, -105, 0, 0], [0, 0, 0, -77], [0, 0, 0, -19], [0, 0, 65, 0], [-496, 0, 0, 0], [0, 77, 0, 0], [-220, 0, 0, 0], [0, 148, 0, 0], [554, 0, 0, 0], [0, -9, 0, 0], [-244, 0, 0, 0], [-188, 0, 0, 0], [0, 0, -6, 0], [0, 0, 0, 176], [0, 0, -63, 0], [0, 250, 0, 0], [0, 0, 0, 139], [0, 0, 0, -156], [268, 0, 0, 0], [0, -196, 0, 0], [0, 179, 0, 0], [0, 0, 0, -152], [0, 0, -99, 0], [640, 0, 0, 0], [0, 0, 86, 0], [0, 0, 0, -143], [0, -34, 0, 0], [106, 0, 0, 0], [0, 0, 0, -177], [0, 0, -50, 0], [0, 0, 43, 0], [-344, 0, 0, 0], [0, -45, 0, 0], [0, 0, 0, 33], [0, 0, 2, 0], [0, -318, 0, 0], [0, 0, 0, 237], [0, 0, 92, 0], [128, 0, 0, 0], [0, 74, 0, 0], [0, -179, 0, 0], [0, 0, -163, 0], [484, 0, 0, 0], [0, 0, 0, -151], [184, 0, 0, 0], [0, 375, 0, 0], [0, 0, 78, 0], [0, 0, 249, 0], [152, 0, 0, 0], [790, 0, 0, 0], [0, 0, 272, 0], [0, 0, -89, 0], [0, -130, 0, 0], [202, 0, 0, 0], [0, 0, 0, -113], [0, 0, 0, 232], [0, 0, -263, 0], [200, 0, 0, 0], [0, 0, 0, 193], [0, 0, 257, 0], [-416, 0, 0, 0], [0, -278, 0, 0], [0, 45, 0, 0], [0, 0, 104, 0], [0, 0, 0, -140], [334, 0, 0, 0], [0, -359, 0, 0], [0, 0, 134, 0], [832, 0, 0, 0], [0, 177, 0, 0], [0, 0, 0, -57], [0, -90, 0, 0], [0, 0, 0, 249], [0, 0, 0, -49], [872, 0, 0, 0], [0, -80, 0, 0], [0, 0, 0, 139], [0, 0, 0, 347], [0, 0, 219, 0], [218, 0, 0, 0], [0, 425, 0, 0], [-404, 0, 0, 0], [0, 617, 0, 0], [0, 0, -25, 0], [1132, 0, 0, 0], [0, 295, 0, 0], [0, 0, 0, 3], [0, 0, 308, 0], [0, 0, 0, 153], [0, 338, 0, 0], [0, 0, 0, -292], [0, 0, 127, 0], [-1160, 0, 0, 0], [0, -160, 0, 0], [0, 0, 0, 152], [0, 0, -61, 0], [520, 0, 0, 0], [0, -12, 0, 0], [1288, 0, 0, 0], [0, 16, 0, 0], [0, 0, 0, -453], [0, 0, -235, 0], [-874, 0, 0, 0], [0, -565, 0, 0], [0, 0, 160, 0], [0, 0, 389, 0], [0, 0, 0, 157], [0, 0, -106, 0], [0, 0, 19, 0], [0, 708, 0, 0], [0, 0, 0, 67], [0, 0, 0, -84], [-848, 0, 0, 0], [0, 573, 0, 0], [0, 0, -52, 0], [0, 0, 0, 144], [0, 476, 0, 0], [-760, 0, 0, 0], [0, 0, 0, 441], [0, 0, 201, 0], [680, 0, 0, 0], [0, -193, 0, 0], [0, 0, 0, -357], [0, 0, 0, 292], [0, 0, 0, -335], [0, 0, 58, 0], [0, 0, 0, -9], [0, 0, -439, 0], [0, -324, 0, 0], [0, 717, 0, 0], [0, 0, 0, 155], [968, 0, 0, 0], [908, 0, 0, 0], [0, -812, 0, 0], [392, 0, 0, 0], [0, 0, -356, 0], [1640, 0, 0, 0], [0, -133, 0, 0], [0, 0, -236, 0], [0, 0, 151, 0], [646, 0, 0, 0], [0, 0, 0, 15], [0, 0, 369, 0], [0, -152, 0, 0], [0, -343, 0, 0], [0, 0, 0, -81], [0, 0, 0, 381], [124, 0, 0, 0], [0, 0, -118, 0], [0, 139, 0, 0], [0, 0, 0, 573], [0, 0, -150, 0], [0, 0, -353, 0], [-1160, 0, 0, 0], [218, 0, 0, 0], [0, -223, 0, 0], [0, 0, 0, 559], [0, 0, -116, 0], [0, 834, 0, 0], [-86, 0, 0, 0], [0, 0, 0, -731], [0, 0, -401, 0], [0, -420, 0, 0], [0, 713, 0, 0], [0, 0, 0, 189], [0, 0, 0, 279], [0, 0, 369, 0], [0, 0, 0, 113], [0, 0, -510, 0], [0, 0, 0, 432], [0, 0, -53, 0], [0, 198, 0, 0], [0, 0, 0, 319], [0, 0, 88, 0], [0, 0, 0, -427], [2320, 0, 0, 0], [0, 0, 0, -11], [0, 0, -473, 0], [-608, 0, 0, 0], [0, 782, 0, 0], [520, 0, 0, 0], [0, 727, 0, 0], [0, 0, -230, 0], [0, -510, 0, 0], [0, 0, -586, 0], [668, 0, 0, 0], [0, 0, 0, 101], [0, 0, 0, 796], [0, 0, 151, 0], [0, 710, 0, 0], [0, 0, 0, 9], [0, 0, 236, 0], [-2332, 0, 0, 0], [0, 588, 0, 0], [0, 0, 0, -855], [0, 0, 39, 0], [0, -710, 0, 0], [-160, 0, 0, 0], [0, -981, 0, 0], [0, 0, 68, 0], [0, 0, 0, -191], [1184, 0, 0, 0], [0, -571, 0, 0], [1954, 0, 0, 0], [0, 0, 0, 117], [0, 0, -364, 0], [0, 0, 0, -763], [0, 0, 0, 52], [0, 0, -769, 0], [-1228, 0, 0, 0], [0, -47, 0, 0], [0, 0, -41, 0], [-3092, 0, 0, 0], [0, -145, 0, 0], [0, 0, 0, 508], [200, 0, 0, 0], [-832, 0, 0, 0], [0, 0, 0, -97], [1544, 0, 0, 0], [0, 0, 0, -649], [0, 0, 6, 0], [0, 0, 0, -296], [1222, 0, 0, 0], [0, 0, 72, 0], [0, -1338, 0, 0], [0, 0, 0, 35], [0, -270, 0, 0], [0, 0, 0, 728], [-2012, 0, 0, 0], [0, -1224, 0, 0], [-2164, 0, 0, 0], [0, -457, 0, 0], [0, 0, 0, -941], [0, 0, -559, 0], [0, -179, 0, 0], [0, 0, 536, 0], [0, 0, 181, 0], [0, 0, -52, 0], [0, 0, 0, 804], [0, -1125, 0, 0], [0, 0, 0, -633], [0, -153, 0, 0], [0, 0, -378, 0], [2524, 0, 0, 0], [-1402, 0, 0, 0], [0, -413, 0, 0], [0, 0, 0, 409]];
    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_i();".
// 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_i(:prec:=4)
    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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![6, 0, 1]>,<7,R![-12, 0, 1]>,<13,R![-108, 0, 1]>,<19,R![-4, 1]>,<31,R![-1452, 0, 1]>],Snew);
    return Vf;
end function;