
// Make newform 8820.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8820_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_8820_2_a_c();".
// 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_8820_2_a_c();".
// 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_8820_a();"
function MakeCharacter_8820_a()
    N := 8820;
    order := 1;
    char_gens := [4411, 7841, 7057, 1081];
    v := [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_8820_a_Hecke(Kf)
    return MakeCharacter_8820_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], [0], [-1], [0], [-4], [0], [2], [6], [-6], [0], [-2], [2], [2], [4], [8], [-10], [4], [2], [12], [-8], [-8], [-8], [-4], [10], [4], [-18], [-4], [-6], [-18], [10], [-8], [20], [-6], [2], [0], [8], [8], [20], [-12], [-22], [-24], [-22], [12], [2], [10], [10], [-8], [-4], [-16], [14], [-18], [20], [-10], [-12], [2], [-26], [-30], [-6], [-6], [16], [4], [-26], [-16], [0], [-32], [14], [-32], [10], [-22], [26], [-18], [4], [-28], [14], [-12], [24], [4], [20], [-28], [18], [12], [14], [-24], [40], [2], [26], [-24], [-26], [-26], [32], [32], [0], [-16], [24], [-12], [0], [10], [14], [-4], [30], [12], [-34], [-4], [-36], [0], [8], [-28], [30], [-36], [-30], [-4], [-6], [-2], [10], [-40], [-32], [32], [28], [-26], [8], [2], [-26], [-42], [-14], [38], [-16], [10], [0], [-24], [-16], [16], [-6], [-52], [-50], [-6], [-22], [-6], [-28], [42], [-12], [-38], [-12], [-32], [34], [-14], [-40], [52], [30], [-6], [50], [2], [38], [36], [8], [20], [-12], [20], [-14], [28], [-46], [22], [-54], [-8], [-36], [6], [20], [4], [52], [-2], [22], [28], [-50], [24], [-54], [-50], [22], [-20], [12], [64], [38], [8], [-12], [38], [2], [54], [6], [34], [52], [22], [0], [36], [26], [-20], [-58], [54], [-42], [10], [46], [18], [-52], [44], [10], [-12], [-66], [12], [26], [-26], [-6], [-60], [58], [-66], [-6], [-8], [-28], [48], [34], [64], [30], [38], [30], [-70], [-50], [-36], [-24], [32], [14], [-38], [16], [-28], [16], [-50], [30], [-56], [60], [56], [12], [-70], [-14], [64], [-8], [-34], [-38], [24], [-50], [54], [0], [-8], [36], [44], [-14], [-38], [30], [-18], [-54], [66], [68], [50], [-28], [42], [72], [-24], [-70], [-10], [40], [-66], [-2], [-36], [62], [4], [54], [6], [-60], [76], [36], [-68], [-80], [10], [26], [-42], [76], [76], [-16], [36], [-42], [-20], [60], [-74], [6], [46], [-6], [60], [68], [-34], [68], [30], [6], [38], [10], [12], [-44], [-36], [22], [52], [54], [-40], [18], [-26], [54], [-48], [38], [-84], [-24], [20], [44], [-30], [-50], [60], [48], [36], [76], [2], [-50], [14], [48], [6], [-2], [-40], [12], [38], [42], [-10], [54], [50], [60], [68], [12], [-26], [-78], [14], [88], [22], [-58], [18], [8], [6], [-40], [-66], [-20], [-72], [-42], [70], [54], [12], [-32], [-78], [6], [24], [-68], [18], [-78], [14], [-70], [58], [-16], [-52], [46], [-14], [-8], [22], [-48], [2], [50], [24], [-38], [-38], [-4], [-24], [64], [-42], [-48], [-74], [98], [-96], [54], [-30], [0], [36], [-32], [44], [40], [-82], [46], [80], [-48], [8], [10], [86], [70], [-48], [-24], [30], [66], [88], [6], [-38], [14], [26], [-48], [0], [-12], [28], [-30], [-6], [-56], [50], [-38], [60], [-24], [-102], [80], [-4], [-20], [-54], [-100], [24], [-50], [-50], [36], [-10], [80], [-22], [48], [12], [-48], [-64], [18], [14], [10], [-60], [-26], [-60], [24], [-70], [-8], [-14], [78], [-88], [52], [2], [-74], [-20], [48], [78], [-42], [110], [12], [-34], [-28], [-108], [-86], [-68], [4], [-108], [6], [8], [50], [-44], [24], [-88], [-66], [-64], [98], [-104], [-18], [-52], [100], [-48], [-6], [-104], [30], [-32], [6], [88], [-42], [-76], [30], [-98], [12], [14], [76], [40], [-12], [54], [30], [-116], [-38], [96], [-78], [26], [8], [-116], [-104], [46], [-64], [78], [34], [-38], [14], [86], [-76], [108], [64], [16], [-78], [60], [26], [-30], [22], [-52], [-88], [-2], [36], [-116], [-78], [-82], [28], [80], [-86], [66], [70], [96], [-50], [-24], [8], [-32], [-44], [-44], [114], [-50], [-36], [-56], [-30], [-46], [0], [52], [36], [32], [56], [-120], [-54], [16], [40], [-56], [-70], [60], [50], [28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8820_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8820_2_a_c();".
// 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_8820_2_a_c(:prec:=1)
    chi := MakeCharacter_8820_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(4027) - 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_8820_2_a_c();".
// 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_8820_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8820_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 1]>,<13,R![0, 1]>,<17,R![-2, 1]>,<31,R![2, 1]>],Snew);
    return Vf;
end function;