
// Make newform 1152.3.b.h in Magma, downloaded from the LMFDB on 19 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_3_b_h();".
// 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_3_b_h();".
// 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, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 4], [0, 8, 0, -4], [-4, 0, 8, 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_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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 0, 0, 0], [18, 0, 0, 0], [0, 0, 3, 0], [0, 0, 0, -6], [0, 1, 0, 0], [0, 0, 0, 7], [0, 18, 0, 0], [-18, 0, 0, 0], [0, 0, -9, 0], [0, 0, 0, -6], [0, 11, 0, 0], [0, 0, 9, 0], [0, 18, 0, 0], [0, 0, 3, 0], [0, 0, 0, 6], [-82, 0, 0, 0], [0, 0, 0, -9], [0, 0, -19, 0], [-126, 0, 0, 0], [110, 0, 0, 0], [0, 23, 0, 0], [0, 0, 0, -9], [0, 0, 13, 0], [0, 36, 0, 0], [126, 0, 0, 0], [0, 0, 0, -23], [0, 0, 29, 0], [-18, 0, 0, 0], [0, 0, -9, 0], [0, -35, 0, 0], [0, 0, 0, -19], [0, 54, 0, 0], [0, 0, -33, 0], [0, 0, 0, 12], [0, -13, 0, 0], [0, 0, -23, 0], [0, -36, 0, 0], [0, 0, 0, 12], [94, 0, 0, 0], [0, 47, 0, 0], [0, 0, 0, 23], [0, 0, -9, 0], [0, 0, 0, 45], [0, 0, 45, 0], [0, 0, 0, 0], [-126, 0, 0, 0], [0, 0, 0, -24], [158, 0, 0, 0], [0, 0, -27, 0], [126, 0, 0, 0], [0, 0, 0, -72], [0, 13, 0, 0], [0, 0, 0, 5], [0, -36, 0, 0], [-414, 0, 0, 0], [0, 0, 51, 0], [0, -133, 0, 0], [0, 0, 75, 0], [0, 0, 0, 12], [2, 0, 0, 0], [0, -131, 0, 0], [0, 0, 3, 0], [82, 0, 0, 0], [0, 0, -43, 0], [0, -54, 0, 0], [414, 0, 0, 0], [0, 0, 0, 78], [0, 0, 0, -17], [0, -90, 0, 0], [0, 0, -57, 0], [0, 0, 0, -60], [0, 13, 0, 0], [0, -54, 0, 0], [18, 0, 0, 0], [-14, 0, 0, 0], [0, 0, -63, 0], [0, -180, 0, 0], [0, 0, 0, 78], [334, 0, 0, 0], [0, 0, 0, -37], [0, 0, 81, 0], [306, 0, 0, 0], [466, 0, 0, 0], [0, -49, 0, 0], [0, 0, 0, 29], [0, 0, -31, 0], [0, 0, 0, 18], [0, 0, 0, -27], [0, 0, -7, 0], [0, 0, 87, 0], [0, 0, 0, -42], [0, -71, 0, 0], [846, 0, 0, 0], [0, 0, -21, 0], [0, 108, 0, 0], [0, 0, 147, 0], [0, 191, 0, 0], [0, 0, -119, 0], [-306, 0, 0, 0], [0, 0, -21, 0], [722, 0, 0, 0], [0, 0, 109, 0], [-1026, 0, 0, 0], [0, 0, 0, -42], [-146, 0, 0, 0], [0, 0, 0, 81], [0, -162, 0, 0], [1026, 0, 0, 0], [0, 0, -45, 0], [0, 0, 0, -59], [-558, 0, 0, 0], [0, 0, 3, 0], [0, 0, 0, -102], [0, -109, 0, 0], [0, 0, 73, 0], [0, 162, 0, 0], [-722, 0, 0, 0], [0, 1, 0, 0], [0, 0, -99, 0], [0, 0, -57, 0], [0, 313, 0, 0], [0, 288, 0, 0], [0, 0, 0, -162], [0, 0, 0, -71], [0, 72, 0, 0], [0, 0, 63, 0], [0, 0, 0, -96], [0, 0, 0, -173], [0, 144, 0, 0], [-18, 0, 0, 0], [670, 0, 0, 0], [0, 275, 0, 0], [0, 0, -129, 0], [0, -181, 0, 0], [558, 0, 0, 0], [0, 0, 195, 0], [0, -73, 0, 0], [0, 0, 0, 209], [0, 0, 117, 0], [0, 180, 0, 0], [0, 0, 0, 138], [0, 18, 0, 0], [-306, 0, 0, 0], [0, 0, 159, 0], [0, 0, 0, 156], [0, 90, 0, 0], [414, 0, 0, 0], [0, 0, -45, 0], [0, 0, 0, 72], [0, 0, 111, 0], [0, 0, 0, 168], [0, 0, 0, -247], [-270, 0, 0, 0], [-674, 0, 0, 0], [0, -361, 0, 0], [0, 0, -107, 0], [-18, 0, 0, 0], [0, 0, 0, -245], [0, 0, -131, 0], [-1710, 0, 0, 0], [0, 0, 0, -216], [0, 0, 0, -13], [0, -162, 0, 0], [1358, 0, 0, 0], [0, -263, 0, 0], [0, 0, 53, 0], [0, -234, 0, 0], [0, 0, 0, 240], [1310, 0, 0, 0], [0, 0, 0, -137], [-990, 0, 0, 0], [0, 0, 111, 0], [0, 61, 0, 0], [0, 0, 0, -77], [0, 126, 0, 0], [0, 0, 0, -89], [0, 0, 53, 0], [0, -360, 0, 0], [450, 0, 0, 0], [0, 0, 0, -108], [0, -383, 0, 0], [0, 0, 0, 0], [0, 0, 183, 0], [-1022, 0, 0, 0], [0, 0, 0, 96], [-254, 0, 0, 0], [0, 0, -55, 0], [0, 0, -213, 0], [0, -193, 0, 0], [0, 0, -279, 0], [1026, 0, 0, 0], [626, 0, 0, 0], [0, -144, 0, 0], [1566, 0, 0, 0], [0, 0, 0, -348], [0, -397, 0, 0], [0, 0, 0, -71], [0, -324, 0, 0], [178, 0, 0, 0], [0, 0, 257, 0], [0, 301, 0, 0], [0, 0, 0, 153], [0, 0, 101, 0], [-126, 0, 0, 0], [0, 0, 123, 0], [766, 0, 0, 0], [0, -335, 0, 0], [0, 0, 0, 91], [0, 0, 125, 0], [0, 0, 0, -198], [2450, 0, 0, 0], [0, 0, 0, 27], [306, 0, 0, 0], [0, 0, 0, 180], [0, -349, 0, 0], [0, 36, 0, 0], [0, 0, 0, 63], [1170, 0, 0, 0], [0, 0, 0, -63], [0, 0, -223, 0], [0, 90, 0, 0], [-2430, 0, 0, 0], [0, 0, 0, -342], [0, 0, 0, 135], [0, 0, -63, 0], [0, -684, 0, 0], [0, 0, 291, 0], [0, 0, 0, -99], [-2430, 0, 0, 0], [0, 0, -321, 0], [0, 0, 0, -102], [-370, 0, 0, 0], [0, 85, 0, 0], [0, 0, -79, 0], [0, 0, 0, 90], [0, 0, 117, 0], [0, 0, 75, 0], [0, 0, 0, 97], [0, 342, 0, 0], [-2178, 0, 0, 0], [0, 0, 0, 210], [0, 0, 0, -401], [0, 0, -43, 0], [0, 0, -33, 0], [0, 0, 0, 324], [0, 468, 0, 0], [-738, 0, 0, 0], [0, 0, 0, -102], [542, 0, 0, 0], [0, 85, 0, 0], [0, 0, 41, 0], [0, -324, 0, 0], [0, 0, -129, 0], [0, -133, 0, 0], [-910, 0, 0, 0], [0, 0, 0, -163], [0, 0, 73, 0], [0, 324, 0, 0], [0, -666, 0, 0], [594, 0, 0, 0], [0, 0, 3, 0], [0, 131, 0, 0], [1314, 0, 0, 0], [0, 0, 63, 0], [0, -443, 0, 0], [0, 18, 0, 0], [0, 0, -69, 0], [2558, 0, 0, 0], [0, 0, 0, -169], [-158, 0, 0, 0], [0, 0, 0, 95], [0, 0, 25, 0], [0, 594, 0, 0], [-1970, 0, 0, 0], [0, 0, 405, 0], [0, 0, 0, -342], [0, 0, 0, -277], [0, 0, 0, 54], [0, -810, 0, 0], [0, 0, 63, 0], [0, 0, 0, 108], [2654, 0, 0, 0], [0, -35, 0, 0], [0, 0, 0, 225], [-162, 0, 0, 0], [0, 683, 0, 0], [0, 0, -139, 0], [2178, 0, 0, 0], [0, 0, -187, 0], [0, 486, 0, 0], [0, 767, 0, 0], [0, 0, 0, 397], [0, -481, 0, 0], [0, 0, -455, 0], [0, 0, 99, 0], [-1630, 0, 0, 0], [0, -529, 0, 0], [0, 0, 0, 167]];
    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_3_b_h();".
// 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_3_b_h(:prec:=4)
    chi := MakeCharacter_1152_b();
    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_1152_3_b_h();".
// 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_3_b_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![16, 0, 1]>,<7,R![48, 0, 1]>,<17,R![-18, 1]>],Snew);
    return Vf;
end function;