
// Make newform 1152.5.b.i in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1152_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1152_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_1152_5_b_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_1152_5_b_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 := [9, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 8, 0], [0, 12, 0, -4], [0, 96, 0, 32]];
    Rf_basisdens := [1, 3, 3, 3];
    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_1152_b();"
function MakeCharacter_1152_b()
    N := 1152;
    order := 2;
    char_gens := [127, 901, 641];
    v := [1, 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_1152_b_Hecke();"
function MakeCharacter_1152_b_Hecke(Kf)
    N := 1152;
    order := 2;
    char_gens := [127, 901, 641];
    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 := 5;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, -11, 0], [0, -27, 0, 0], [162, 0, 0, 0], [0, 0, -45, 0], [0, 0, 0, 9], [0, 163, 0, 0], [0, 0, 0, -8], [0, -189, 0, 0], [-1890, 0, 0, 0], [0, 0, -297, 0], [0, 0, 0, -18], [0, -247, 0, 0], [0, 0, -231, 0], [0, 297, 0, 0], [0, 0, 171, 0], [0, 0, 0, 99], [2750, 0, 0, 0], [0, 0, 0, -102], [0, 0, 953, 0], [-2430, 0, 0, 0], [7454, 0, 0, 0], [0, -187, 0, 0], [0, 0, 0, 33], [0, 0, 109, 0], [0, -1863, 0, 0], [702, 0, 0, 0], [0, 0, 0, -44], [0, 0, -979, 0], [32670, 0, 0, 0], [0, 0, 135, 0], [0, 109, 0, 0], [0, 0, 0, -121], [0, -3591, 0, 0], [0, 0, -3177, 0], [0, 0, 0, -693], [0, -3157, 0, 0], [0, 0, 913, 0], [0, -4509, 0, 0], [0, 0, 0, -396], [41374, 0, 0, 0], [0, -8455, 0, 0], [0, 0, 0, -319], [0, 0, -2673, 0], [0, 0, 0, -264], [0, 0, 7953, 0], [0, -1809, 0, 0], [2754, 0, 0, 0], [0, 0, 0, 1170], [-97570, 0, 0, 0], [0, 0, 6921, 0], [-80514, 0, 0, 0], [0, 0, 0, -189], [0, -5609, 0, 0], [0, 0, 0, 1166], [0, -11313, 0, 0], [-23166, 0, 0, 0], [0, 0, 7227, 0], [0, 4721, 0, 0], [0, 0, -3933, 0], [0, 0, 0, 1089], [93602, 0, 0, 0], [0, -6149, 0, 0], [0, 0, 1683, 0], [-93758, 0, 0, 0], [0, 0, -22495, 0], [0, -6075, 0, 0], [-108162, 0, 0, 0], [0, 0, 0, 639], [0, 0, 0, -782], [0, 6831, 0, 0], [0, 0, -10593, 0], [0, 0, 0, -1584], [0, -16139, 0, 0], [0, 29025, 0, 0], [-151902, 0, 0, 0], [28450, 0, 0, 0], [0, 0, -29739, 0], [0, -14985, 0, 0], [0, 0, 0, 2178], [215518, 0, 0, 0], [0, 0, 0, 959], [0, 0, 18909, 0], [-177822, 0, 0, 0], [-203294, 0, 0, 0], [0, 35831, 0, 0], [0, 0, 0, -4246], [0, 0, -5995, 0], [0, 0, 0, 2376], [0, 0, 0, -4635], [0, 0, -10975, 0], [0, 0, 39303, 0], [0, 0, 0, -2691], [0, 21439, 0, 0], [184734, 0, 0, 0], [0, 0, 42867, 0], [0, -28863, 0, 0], [0, 0, 4059, 0], [0, -2893, 0, 0], [0, 0, 50125, 0], [127710, 0, 0, 0], [0, 0, 3987, 0], [-270718, 0, 0, 0], [0, 0, -27899, 0], [62910, 0, 0, 0], [0, 0, 0, -3339], [203902, 0, 0, 0], [0, 0, 0, -4716], [0, 27783, 0, 0], [-363582, 0, 0, 0], [0, 0, 32427, 0], [0, 0, 0, -4445], [-67230, 0, 0, 0], [0, 0, -7605, 0], [0, 0, 0, 4851], [0, -11161, 0, 0], [0, 0, -22775, 0], [0, 70011, 0, 0], [-810722, 0, 0, 0], [0, -81971, 0, 0], [0, 0, -19311, 0], [0, 0, -58113, 0], [0, 64903, 0, 0], [0, 114939, 0, 0], [0, 0, 0, -8370], [0, 0, 0, 6235], [0, 88317, 0, 0], [0, 0, 17631, 0], [0, 0, 0, 3195], [0, 0, 0, -6842], [0, 58779, 0, 0], [691038, 0, 0, 0], [304030, 0, 0, 0], [0, -66715, 0, 0], [0, 0, -119889, 0], [0, 69047, 0, 0], [-803682, 0, 0, 0], [0, 0, -7029, 0], [0, 108449, 0, 0], [0, 0, 0, -385], [0, 0, -88251, 0], [0, 56241, 0, 0], [0, 0, 0, 10215], [0, -15201, 0, 0], [1264734, 0, 0, 0], [0, 0, 98703, 0], [0, 0, 0, -5472], [0, 64881, 0, 0], [-434754, 0, 0, 0], [0, 0, 48411, 0], [0, 0, 0, -3807], [0, 0, -70785, 0], [0, 0, 0, -14454], [0, 0, 0, -6721], [371682, 0, 0, 0], [-532418, 0, 0, 0], [0, -76825, 0, 0], [0, 0, -59603, 0], [-604962, 0, 0, 0], [0, 0, 0, 17785], [0, 0, -67583, 0], [815778, 0, 0, 0], [0, 0, 0, 15741], [0, 0, 0, 5324], [0, 225315, 0, 0], [-1264162, 0, 0, 0], [0, -57299, 0, 0], [0, 0, -63379, 0], [0, -50787, 0, 0], [0, 0, 0, 4635], [-1293634, 0, 0, 0], [0, 0, 0, 10390], [-1212030, 0, 0, 0], [0, 0, -153945, 0], [0, -55067, 0, 0], [0, 0, 0, -17699], [0, -28539, 0, 0], [0, 0, 0, 15496], [0, 0, -150559, 0], [0, -226017, 0, 0], [-702, 0, 0, 0], [0, 0, 0, 2970], [0, -297155, 0, 0], [0, -131031, 0, 0], [0, 0, -130185, 0], [1183330, 0, 0, 0], [0, 0, 0, -4896], [211714, 0, 0, 0], [0, 0, 117833, 0], [0, 0, 84843, 0], [0, 24275, 0, 0], [0, 0, -3207, 0], [-2423358, 0, 0, 0], [613826, 0, 0, 0], [0, -116127, 0, 0], [-2095362, 0, 0, 0], [0, 0, 0, 30195], [0, 307835, 0, 0], [0, 0, 0, 5698], [0, -274185, 0, 0], [338434, 0, 0, 0], [0, 0, 27533, 0], [0, -371261, 0, 0], [0, 0, 0, -12144], [0, 0, -294055, 0], [-2772414, 0, 0, 0], [0, 0, -6597, 0], [539230, 0, 0, 0], [0, 232213, 0, 0], [0, 0, 0, 33187], [0, 0, 6173, 0], [0, 0, 0, -28701], [-249118, 0, 0, 0], [0, 0, 0, 37566], [442530, 0, 0, 0], [0, 0, 0, 25029], [0, 50243, 0, 0], [0, 418527, 0, 0], [0, 0, 0, -46005], [1619298, 0, 0, 0], [0, 0, 0, -28830], [0, 0, -168355, 0], [0, -288009, 0, 0], [-1205118, 0, 0, 0], [0, 0, 0, -20628], [0, 0, 0, -35415], [0, 0, 327993, 0], [0, -132543, 0, 0], [0, 0, -209781, 0], [0, 0, 0, -6072], [-2903742, 0, 0, 0], [0, 0, -56529, 0], [0, 0, 0, 2934], [861278, 0, 0, 0], [0, -356843, 0, 0], [0, 0, -420475, 0], [0, 0, 0, -20493], [0, 0, -172311, 0], [0, 0, -295317, 0], [0, 0, 0, -12155], [0, -73683, 0, 0], [674622, 0, 0, 0], [0, 0, 0, -37899], [0, 0, 0, -29288], [0, 0, -347875, 0], [0, 0, -149409, 0], [0, 0, 0, 14850], [0, -334179, 0, 0], [3344382, 0, 0, 0], [0, 0, 0, 26235], [-1145410, 0, 0, 0], [0, 75535, 0, 0], [0, 0, -300223, 0], [0, -17037, 0, 0], [0, 0, 101079, 0], [0, -787, 0, 0], [960674, 0, 0, 0], [0, 0, 0, -1312], [0, 0, 465073, 0], [0, 8451, 0, 0], [0, -15579, 0, 0], [-1549854, 0, 0, 0], [0, 0, -350181, 0], [0, 3575, 0, 0], [4044546, 0, 0, 0], [0, 0, 213111, 0], [0, 248413, 0, 0], [0, 350325, 0, 0], [0, 0, 384219, 0], [-284290, 0, 0, 0], [0, 0, 0, 51812], [3972034, 0, 0, 0], [0, 0, 0, 32219], [0, 0, -331595, 0], [0, 305505, 0, 0], [4675774, 0, 0, 0], [0, 0, 73917, 0], [0, 0, 0, 33264], [0, 0, 0, 25025], [0, 0, 0, -35451], [0, 623727, 0, 0], [0, 0, 139239, 0], [0, 0, 0, -67014], [5321822, 0, 0, 0], [0, 599185, 0, 0], [0, 0, 0, -13065], [-3629826, 0, 0, 0], [0, -171649, 0, 0], [0, 0, -154723, 0], [1210626, 0, 0, 0], [0, 0, -637483, 0], [0, 233145, 0, 0], [0, 434687, 0, 0], [0, 0, 0, 41416], [0, 918245, 0, 0], [0, 0, 164197, 0], [0, 0, -168237, 0], [-1682590, 0, 0, 0], [0, -795541, 0, 0], [0, 0, 0, -37066]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1152_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1152_5_b_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_1152_5_b_i(:prec:=4)
    chi := MakeCharacter_1152_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_1152_5_b_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_1152_5_b_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<5,R![64, 0, 1]>,<17,R![-162, 1]>],Snew);
    return Vf;
end function;