
// Make newform 48.7.g.a 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_a();".
// 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_a();".
// 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], [-9, 18]];
    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, -1], [-90, 0], [0, 12], [0, 108], [1762, 0], [-1638, 0], [0, 804], [0, 864], [-16002, 0], [0, -1020], [61130, 0], [-98550, 0], [0, -3156], [0, -11448], [-275346, 0], [0, 16308], [106634, 0], [0, 35388], [0, 4752], [100978, 0], [0, -4956], [0, 41364], [-819054, 0], [557026, 0], [1837998, 0], [0, -80604], [0, -29484], [-1460270, 0], [563634, 0], [0, 20076], [0, -45252], [2429370, 0], [0, 124068], [3463974, 0], [0, 323964], [1346618, 0], [0, -191868], [0, 410184], [1853910, 0], [0, -506196], [-5680478, 0], [0, -575424], [5391650, 0], [4248702, 0], [0, 192372], [0, -988212], [0, -219972], [0, 285876], [3522994, 0], [-13290030, 0], [0, 41472], [15655858, 0], [0, 926748], [-9470718, 0], [0, 820584], [31898142, 0], [0, -2265876], [-21702206, 0], [-5818302, 0], [0, 1544340], [21188430, 0], [0, -423828], [0, -2362392], [-211966, 0], [35921358, 0], [0, -1622028], [10902418, 0], [0, 5083452], [-12797206, 0], [49834386, 0], [0, 3459888], [0, 4338324], [55594586, 0], [0, 3614892], [0, 3425760], [-7972506, 0], [-83299318, 0], [-4705110, 0], [131798050, 0], [0, -4682988], [106634626, 0], [0, -7827624], [-80628046, 0], [0, 4691124], [0, -7459236], [-103224294, 0], [11704354, 0], [-53724330, 0], [0, -2677044], [0, 4831812], [0, 9899496], [0, 9651084], [0, -8072460], [0, 1565100], [0, 2065824], [-17033490, 0], [60856650, 0], [0, 10664316], [-184782062, 0], [0, 13256244], [214641126, 0], [0, 14725908], [221575914, 0], [0, -8395644], [-244187566, 0], [0, -2655180], [328345074, 0], [0, -4847472], [-26242990, 0], [0, 15123756], [-402536518, 0], [396287874, 0], [0, 2614788], [0, -25257828], [-273944358, 0], [0, 9139572], [0, 5098896], [116064198, 0], [0, 12582972], [494257322, 0], [402581810, 0], [-198483210, 0], [0, 11546172], [0, 26162772], [-88917138, 0], [-21617966, 0], [0, 27697032], [0, -4794948], [-125888222, 0], [0, 7501740], [0, -26453736], [0, -40723932], [-22274030, 0], [-594900486, 0], [713646146, 0], [-256623138, 0], [0, -9104940], [63099846, 0], [146427066, 0], [0, -40469748], [-133108578, 0], [0, 13561068], [0, -27495612], [-52830254, 0], [0, -30163968], [-392907094, 0], [-114312150, 0], [0, 15068748], [0, 7600608], [-623832982, 0], [-279302526, 0], [0, 13461108], [0, 33302232], [0, 39459612], [0, -16937856], [0, -38031732], [-283618710, 0], [51893570, 0], [1241483958, 0], [0, -91521684], [1008697770, 0], [0, -104909148], [0, 87868476], [-1267318710, 0], [0, 16530264], [0, 49704372], [-494303974, 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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_48_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([<5,R![90, 1]>],Snew);
    return Vf;
end function;