
// Make newform 150.3.b.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_150_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_150_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_150_3_b_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_150_3_b_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 := [1, 0, 0, 0, 7, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -16, 0, 0, 0, -2, 0], [0, 5, 0, -13, 0, 1, 0, -2], [-7, -3, -8, -21, -2, 0, -1, -3], [0, -16, -18, 16, 0, -2, -3, 2], [0, 10, 0, 26, 0, 2, 0, 4], [0, 2, -12, -14, 0, 0, -2, -2], [-14, 22, 0, 58, -4, 2, 0, 8]];
    Rf_basisdens := [1, 3, 3, 3, 3, 3, 1, 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_150_b();"
function MakeCharacter_150_b()
    N := 150;
    order := 2;
    char_gens := [101, 127];
    v := [1, 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_150_b_Hecke();"
function MakeCharacter_150_b_Hecke(Kf)
    N := 150;
    order := 2;
    char_gens := [101, 127];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 1, -2, 0, 0, 0, 1], [0, 0, 0, 0, 0, -3, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0], [0, 2, 14, -4, 0, 0, 0, -1], [-8, 0, 0, 0, 4, 1, -2, 0], [0, 2, -1, -4, 0, 0, 0, -1], [0, 0, 0, 0, 4, 4, 4, 0], [8, 0, 0, 0, 0, 0, 0, 0], [0, -15, -4, 8, 0, 0, 0, -4], [0, 0, 0, 0, 2, -10, 2, 0], [0, -10, -3, 6, 0, 0, 0, -3], [0, -10, 5, 20, 0, 0, 0, 5], [0, -2, -14, 4, 0, 0, 0, 1], [0, 0, 0, 0, -4, -22, -4, 0], [-16, 0, 0, 0, 8, 2, -4, 0], [0, -38, 3, -6, 0, 0, 0, 3], [0, 0, 0, 0, -4, -19, -4, 0], [0, -21, 4, -8, 0, 0, 0, 4], [28, 0, 0, 0, -4, -1, 2, 0], [0, 2, 11, -4, 0, 0, 0, -1], [0, 0, 0, 0, -4, 8, -4, 0], [0, 41, 4, -8, 0, 0, 0, 4], [0, 0, 0, 0, 8, 44, 8, 0], [0, -30, 7, -14, 0, 0, 0, 7], [0, 2, 23, -4, 0, 0, 0, -1], [-20, 0, 0, 0, 24, 6, -12, 0], [0, -14, 58, 28, 0, 0, 0, 7], [0, -50, 3, -6, 0, 0, 0, 3], [0, 0, 0, 0, -24, -3, -24, 0], [0, 6, -90, -12, 0, 0, 0, -3], [-56, 0, 0, 0, -28, -7, 14, 0], [0, 0, 0, 0, 26, -10, 26, 0], [56, 0, 0, 0, 8, 2, -4, 0], [0, -17, -12, 24, 0, 0, 0, -12], [0, -46, -9, 18, 0, 0, 0, -9], [0, -18, -111, 36, 0, 0, 0, 9], [0, 22, -74, -44, 0, 0, 0, -11], [0, 0, 0, 0, 28, 70, 28, 0], [38, 0, 0, 0, 48, 12, -24, 0], [0, 0, 0, 0, 20, 29, 20, 0], [0, 61, -4, 8, 0, 0, 0, -4], [0, -18, -54, 36, 0, 0, 0, 9], [-92, 0, 0, 0, -12, -3, 6, 0], [-172, 0, 0, 0, -24, -6, 12, 0], [0, 2, 9, -18, 0, 0, 0, 9], [0, 14, 101, -28, 0, 0, 0, -7], [-98, 0, 0, 0, 48, 12, -24, 0], [0, 10, -74, -20, 0, 0, 0, -5], [0, 0, 0, 0, 8, -118, 8, 0], [104, 0, 0, 0, -72, -18, 36, 0], [0, 0, 0, 0, -24, 3, -24, 0], [0, 2, 218, -4, 0, 0, 0, -1], [0, 18, -93, -36, 0, 0, 0, -9], [0, 0, 0, 0, -18, 126, -18, 0], [-400, 0, 0, 0, 16, 4, -8, 0], [0, 163, 12, -24, 0, 0, 0, 12], [0, 0, 0, 0, -34, -106, -34, 0], [0, -82, 3, -6, 0, 0, 0, 3], [0, 26, 266, -52, 0, 0, 0, -13], [0, 46, 9, -18, 0, 0, 0, 9], [0, 0, 0, 0, 28, -89, 28, 0], [0, 161, 24, -48, 0, 0, 0, 24], [0, -22, 110, 44, 0, 0, 0, 11], [-28, 0, 0, 0, 88, 22, -44, 0], [0, 25, -24, 48, 0, 0, 0, -24], [0, 22, -119, -44, 0, 0, 0, -11], [-206, 0, 0, 0, -64, -16, 32, 0], [0, -90, -66, 180, 0, 0, 0, 45], [0, 0, 0, 0, -8, 70, -8, 0], [0, -62, -51, 102, 0, 0, 0, -51], [0, 223, -4, 8, 0, 0, 0, -4], [88, 0, 0, 0, -36, -9, 18, 0], [0, 14, 317, -28, 0, 0, 0, -7], [0, 0, 0, 0, 86, 74, 86, 0], [0, -107, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -88, -136, -88, 0], [460, 0, 0, 0, -24, -6, 12, 0], [0, 0, 0, 0, -36, -156, -36, 0], [224, 0, 0, 0, -104, -26, 52, 0], [0, 0, 0, 0, 36, 225, 36, 0], [0, -37, 36, -72, 0, 0, 0, 36], [-428, 0, 0, 0, -36, -9, 18, 0], [0, 38, 65, -76, 0, 0, 0, -19], [0, 0, 0, 0, -62, -2, -62, 0], [0, 7, 72, -144, 0, 0, 0, 72], [0, 0, 0, 0, -32, 88, -32, 0], [0, -114, -59, 118, 0, 0, 0, -59], [0, 34, -149, -68, 0, 0, 0, -17], [0, 0, 0, 0, -16, 206, -16, 0], [0, -98, -9, 18, 0, 0, 0, -9], [0, 0, 0, 0, 120, 87, 120, 0], [-80, 0, 0, 0, 12, 3, -6, 0], [0, 30, 201, -60, 0, 0, 0, -15], [0, 0, 0, 0, 52, 100, 52, 0], [0, 0, 0, 0, 40, 136, 40, 0], [0, -110, 65, -130, 0, 0, 0, 65], [-262, 0, 0, 0, -16, -4, 8, 0], [0, -126, 5, -10, 0, 0, 0, 5], [0, -50, 190, 100, 0, 0, 0, 25], [0, 66, -129, -132, 0, 0, 0, -33], [0, 0, 0, 0, 2, 62, 2, 0], [-220, 0, 0, 0, -16, -4, 8, 0], [0, -193, -80, 160, 0, 0, 0, -80], [0, 14, -187, -28, 0, 0, 0, -7], [0, 6, 222, -12, 0, 0, 0, -3], [0, 0, 0, 0, 24, -168, 24, 0], [680, 0, 0, 0, 72, 18, -36, 0], [0, -414, -1, 2, 0, 0, 0, -1], [0, -207, -56, 112, 0, 0, 0, -56], [0, -46, -586, 92, 0, 0, 0, 23], [-128, 0, 0, 0, -44, -11, 22, 0], [-304, 0, 0, 0, -168, -42, 84, 0], [0, 0, 0, 0, -30, -78, -30, 0], [0, 266, 9, -18, 0, 0, 0, 9], [0, -38, -185, 76, 0, 0, 0, 19], [0, -10, 338, 20, 0, 0, 0, 5], [0, 0, 0, 0, -36, -186, -36, 0], [-520, 0, 0, 0, -8, -2, 4, 0], [0, -345, -44, 88, 0, 0, 0, -44], [0, 126, 282, -252, 0, 0, 0, -63], [0, -98, -191, 196, 0, 0, 0, 49], [932, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 146, 14, 146, 0], [-110, 0, 0, 0, -80, -20, 40, 0], [0, 0, 0, 0, 32, -82, 32, 0], [0, -266, 51, -102, 0, 0, 0, 51], [0, 203, 36, -72, 0, 0, 0, 36], [880, 0, 0, 0, 12, 3, -6, 0], [0, -34, 125, 68, 0, 0, 0, 17], [920, 0, 0, 0, -56, -14, 28, 0], [0, -133, 52, -104, 0, 0, 0, 52], [0, 0, 0, 0, 80, 68, 80, 0], [514, 0, 0, 0, -48, -12, 24, 0], [0, 26, -538, -52, 0, 0, 0, -13], [0, 518, 61, -122, 0, 0, 0, 61], [0, 34, 262, -68, 0, 0, 0, -17], [0, 0, 0, 0, -124, -220, -124, 0], [-124, 0, 0, 0, 24, 6, -12, 0], [0, 0, 0, 0, -30, -78, -30, 0], [0, 170, 75, -150, 0, 0, 0, 75], [0, 34, 571, -68, 0, 0, 0, -17], [-68, 0, 0, 0, 56, 14, -28, 0], [0, 0, 0, 0, -80, -224, -80, 0], [0, 149, 120, -240, 0, 0, 0, 120], [0, -42, 654, 84, 0, 0, 0, 21], [376, 0, 0, 0, 20, 5, -10, 0], [0, -10, -259, 20, 0, 0, 0, 5], [0, 287, -44, 88, 0, 0, 0, -44], [0, 0, 0, 0, 66, -42, 66, 0], [0, -326, -133, 266, 0, 0, 0, -133], [0, 114, 291, -228, 0, 0, 0, -57], [0, 402, -61, 122, 0, 0, 0, -61], [0, 0, 0, 0, 52, -179, 52, 0], [268, 0, 0, 0, -108, -27, 54, 0], [0, 0, 0, 0, 158, 146, 158, 0], [0, -181, 4, -8, 0, 0, 0, 4], [0, 0, 0, 0, 72, 480, 72, 0], [0, -210, -351, 420, 0, 0, 0, 105], [0, -38, 634, 76, 0, 0, 0, 19], [0, -762, -43, 86, 0, 0, 0, -43], [0, 0, 0, 0, -8, 385, -8, 0], [0, 126, 42, -252, 0, 0, 0, -63], [0, -42, -567, 84, 0, 0, 0, 21], [-280, 0, 0, 0, 320, 80, -160, 0], [0, 175, -48, 96, 0, 0, 0, -48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_150_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_150_3_b_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_150_3_b_b(:prec:=8)
    chi := MakeCharacter_150_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(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_150_3_b_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_150_3_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_150_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![7396, 0, 188, 0, 1]>],Snew);
    return Vf;
end function;