
// Make newform 48.7.g.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_48_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_48_g_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_48_7_g_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_48_7_g_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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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_48_g();"
function MakeCharacter_48_g()
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    v := [1, 2, 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_48_g_Hecke();"
function MakeCharacter_48_g_Hecke(Kf)
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    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 := 7;
    raw_aps := [[0, 0], [0, -9], [6, 0], [0, -116], [0, -372], [-2654, 0], [-7206, 0], [0, -2044], [0, -10080], [11550, 0], [0, 4932], [22346, 0], [103626, 0], [0, 73548], [0, -92856], [168462, 0], [0, -64428], [-260470, 0], [0, 183068], [0, -409008], [-395918, 0], [0, -321436], [0, -51468], [-251886, 0], [517474, 0], [1023822, 0], [0, 52516], [0, 1271796], [952594, 0], [-121038, 0], [0, -836820], [0, 365340], [-4478214, 0], [0, -1298492], [-1840506, 0], [0, 660860], [-5287750, 0], [0, -3728924], [0, 2296200], [5735478, 0], [0, -4716276], [3246562, 0], [0, 644928], [3123746, 0], [10853022, 0], [0, 5194612], [0, 2217772], [0, -2030724], [0, -9359724], [-14686094, 0], [-6101166, 0], [0, -15215232], [-2152142, 0], [0, 4617660], [-28615422, 0], [0, -863064], [-532290, 0], [0, -16660180], [-18247934, 0], [9520194, 0], [0, 22018356], [16834542, 0], [0, -25354548], [0, 29350824], [13072130, 0], [5077230, 0], [0, 32298004], [-25573742, 0], [0, 2863836], [-30656278, 0], [-53335278, 0], [0, 16390128], [0, -33236396], [-46832806, 0], [0, -45753652], [0, -11580192], [-47934522, 0], [-34378486, 0], [90923370, 0], [-115453790, 0], [0, -42741516], [114810562, 0], [0, -1977768], [108786098, 0], [0, 65471796], [0, -14114052], [90369882, 0], [-58204766, 0], [173271222, 0], [0, 72399628], [0, -84238428], [0, -4401816], [0, -98928116], [0, 1907988], [0, 72865548], [0, 122592], [-149791986, 0], [111421962, 0], [0, -155588068], [303425362, 0], [0, 11139220], [-171693114, 0], [0, 127820916], [256990890, 0], [0, 156605732], [-3415726, 0], [0, 134434644], [-88669326, 0], [0, -13717296], [256420946, 0], [0, 120200428], [259662266, 0], [140533890, 0], [0, -91240156], [0, -144281188], [-29409510, 0], [0, 147758292], [0, -235756464], [206095014, 0], [0, 45325212], [-55050454, 0], [-48281806, 0], [-402892074, 0], [0, -73292196], [0, 80376372], [445981326, 0], [159057298, 0], [0, -192863352], [0, 150665148], [-356906270, 0], [0, -49747572], [0, -15089832], [0, 334893284], [335601874, 0], [234765114, 0], [378738242, 0], [-134904066, 0], [0, 49575860], [882734502, 0], [31918074, 0], [0, 49617324], [771026622, 0], [0, -118165780], [0, -147463452], [-629211374, 0], [0, 129394368], [209381930, 0], [-332297238, 0], [0, -380427668], [0, 581359584], [731415914, 0], [-1166063742, 0], [0, 42864724], [0, -136195176], [0, -146667844], [0, -595875072], [0, 652428684], [-1400494038, 0], [355654466, 0], [104392470, 0], [0, 567867852], [313478250, 0], [0, -228161116], [0, -117972132], [-38798838, 0], [0, 438266136], [0, 293532148], [150555674, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_48_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_48_7_g_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_48_7_g_b(:prec:=2)
    chi := MakeCharacter_48_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    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_48_7_g_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_48_7_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_48_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([<5,R![-6, 1]>],Snew);
    return Vf;
end function;