
// Make newform 8880.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8880_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8880_2_a_z();".
// 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_8880_2_a_z();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8880_a();"
function MakeCharacter_8880_a()
    N := 8880;
    order := 1;
    char_gens := [5551, 6661, 5921, 1777, 8401];
    v := [1, 1, 1, 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;

function MakeCharacter_8880_a_Hecke(Kf)
    return MakeCharacter_8880_a();
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 := 2;
    raw_aps := [[0], [1], [1], [-3], [5], [-2], [-7], [2], [-4], [-5], [7], [1], [1], [-9], [0], [3], [14], [-7], [4], [-8], [-12], [-4], [10], [-14], [1], [0], [-8], [-14], [-7], [-5], [-16], [6], [-22], [3], [-6], [-10], [-11], [15], [-20], [19], [24], [14], [7], [14], [10], [-4], [13], [-9], [7], [-28], [-20], [-17], [-2], [0], [14], [-11], [16], [-20], [16], [10], [12], [9], [-6], [-21], [-10], [-21], [0], [-22], [12], [-16], [17], [-34], [13], [22], [-16], [-4], [-31], [-34], [22], [34], [-20], [10], [-21], [-16], [33], [-42], [-12], [-39], [-15], [6], [-3], [24], [30], [-28], [-4], [-24], [-20], [3], [-16], [42], [7], [-18], [3], [-10], [-27], [10], [21], [46], [24], [-7], [14], [43], [6], [-5], [7], [41], [43], [-40], [36], [0], [31], [-8], [-42], [47], [41], [-34], [-27], [-34], [18], [-13], [-33], [13], [-20], [22], [-47], [2], [33], [4], [32], [-4], [-20], [-6], [32], [-21], [29], [-18], [6], [39], [6], [51], [-3], [-15], [49], [-15], [36], [40], [0], [-41], [32], [6], [-49], [-4], [4], [-53], [5], [-45], [-19], [6], [-35], [-30], [38], [-15], [-45], [-40], [-22], [-26], [18], [54], [9], [-4], [-12], [36], [64], [-25], [-57], [6], [7], [-49], [34], [40], [-62], [-16], [66], [-36], [42], [10], [-24], [-4], [-2], [12], [-50], [30], [17], [-19], [3], [-42], [14], [26], [-36], [5], [65], [-69], [34], [58], [33], [-33], [22], [-24], [-18], [57], [-4], [-46], [9], [-24], [-28], [46], [40], [68], [57], [54], [-7], [68], [6], [2], [62], [-11], [62], [-54], [66], [40], [29], [32], [16], [-65], [-30], [31], [16], [12], [17], [6], [20], [30], [3], [10], [-46], [27], [-22], [-66], [-33], [-17], [66], [48], [16], [41], [18], [-31], [36], [72], [68], [-2], [18], [41], [-74], [27], [18], [-25], [-42], [46], [25], [58], [0], [-51], [-67], [-24], [-20], [62], [49], [-21], [72], [-28], [-39], [-76], [26], [21], [29], [-14], [-82], [54], [-84], [68], [-47], [-78], [28], [-64], [-40], [-29], [67], [-58], [62], [80], [-32], [-60], [-3], [-2], [-62], [76], [71], [-12], [46], [-24], [-44], [30], [21], [23], [8], [8], [-67], [77], [66], [13], [-26], [-90], [-37], [-36], [-22], [26], [-26], [-58], [-30], [0], [74], [54], [30], [47], [30], [6], [20], [18], [-93], [-21], [-83], [52], [-29], [36], [10], [54], [57], [-14], [-2], [-4], [10], [6], [-58], [-58], [-42], [51], [-90], [-66], [6], [12], [-34], [-63], [66], [13], [37], [88], [60], [-28], [-60], [-12], [-65], [-6], [-56], [8], [70], [9], [78], [-34], [-19], [-40], [-21], [24], [-6], [9], [-12], [68], [40], [-21], [-23], [92], [-66], [-77], [-46], [-54], [66], [28], [84], [15], [32], [-63], [80], [69], [-87], [-22], [-29], [36], [38], [-33], [-24], [-66], [18], [-28], [84], [9], [-88], [-49], [-56], [-45], [-84], [-45], [50], [70], [34], [-96], [-86], [55], [-64], [-58], [-32], [57], [86], [-10], [56], [-68], [-80], [43], [-61], [-2], [-16], [-102], [33], [50], [30], [21], [21], [57], [-4], [-90], [-102], [26], [-95], [97], [-18], [-47], [1], [-19], [44], [-12], [30], [-36], [-4], [-108], [15], [-72], [27], [-62], [66], [29], [-1], [17], [-95], [-60], [16], [-9], [-16], [52], [88], [70], [-52], [100], [98], [63], [-6], [50], [6], [-50], [4], [-4], [50], [68], [31], [-2], [-66], [-4], [38], [-32], [101], [16], [58], [11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8880_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8880_2_a_z();".
// 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_8880_2_a_z(:prec:=1)
    chi := MakeCharacter_8880_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3643) - 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_8880_2_a_z();".
// 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_8880_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8880_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<11,R![-5, 1]>,<13,R![2, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;