
// Make newform 8820.2.a.ba in Magma, downloaded from the LMFDB on 26 October 2025.
// 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_ba();".
// 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_ba();".
// 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 := 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]];
    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], [4], [2], [0], [4], [0], [8], [2], [-2], [4], [4], [-4], [4], [-8], [-4], [-12], [-4], [4], [-8], [14], [-12], [10], [8], [-12], [-2], [20], [0], [12], [-4], [-8], [-8], [-12], [-20], [-4], [-12], [18], [-4], [16], [-12], [18], [-12], [-16], [-16], [-16], [24], [8], [20], [-4], [8], [-20], [30], [-12], [14], [8], [2], [-8], [8], [-6], [8], [16], [-28], [12], [8], [-14], [28], [8], [18], [20], [16], [-18], [-16], [20], [24], [4], [-16], [32], [-36], [-22], [-12], [4], [-16], [-4], [0], [6], [18], [16], [8], [40], [-32], [12], [16], [-12], [30], [18], [16], [2], [-20], [-36], [0], [-16], [-24], [20], [-16], [34], [-44], [-40], [-32], [34], [-12], [-24], [-20], [-24], [24], [28], [-44], [4], [-40], [46], [26], [12], [-8], [32], [6], [-24], [-8], [36], [0], [-36], [28], [-2], [30], [-40], [-6], [0], [2], [40], [-56], [48], [56], [52], [16], [0], [4], [6], [-32], [-12], [42], [26], [52], [-20], [-20], [20], [-4], [6], [12], [-34], [-12], [-44], [8], [-20], [-12], [44], [-60], [36], [-2], [10], [-12], [16], [36], [42], [32], [18], [-40], [40], [-8], [32], [-32], [-12], [46], [-6], [4], [-6], [-22], [32], [-6], [24], [-60], [-12], [24], [-30], [-28], [-54], [-50], [-26], [46], [-60], [-48], [64], [-20], [-40], [-12], [-30], [-16], [-12], [56], [32], [46], [6], [40], [-40], [-40], [8], [-40], [-22], [-60], [52], [6], [-56], [-24], [-56], [-56], [10], [-42], [60], [40], [-36], [54], [16], [-28], [-56], [-56], [52], [-16], [-12], [-36], [8], [-36], [-40], [0], [-34], [-2], [-64], [-72], [-76], [40], [36], [-54], [-6], [44], [32], [-14], [20], [-42], [-72], [58], [-12], [72], [-52], [-56], [-36], [-62], [72], [-16], [66], [60], [-60], [-32], [-44], [-44], [20], [-28], [24], [52], [-50], [10], [-68], [-36], [-60], [28], [-32], [64], [-28], [46], [-52], [8], [38], [80], [-72], [68], [-44], [70], [-78], [0], [54], [-44], [16], [44], [-44], [44], [36], [32], [2], [12], [-24], [-60], [-34], [-44], [32], [-32], [60], [-12], [-8], [-36], [28], [52], [-16], [80], [38], [18], [8], [44], [8], [56], [-40], [44], [44], [50], [52], [-8], [-24], [-40], [-48], [18], [30], [8], [-88], [46], [36], [-26], [-12], [-44], [36], [40], [-52], [72], [10], [-48], [22], [-96], [32], [-74], [42], [-64], [-36], [-60], [92], [-66], [-18], [-84], [-44], [64], [-10], [78], [88], [-38], [-36], [48], [-52], [-72], [-88], [90], [-68], [-20], [68], [10], [28], [2], [-28], [56], [-20], [40], [60], [-28], [20], [20], [-12], [82], [-26], [-44], [-88], [-12], [42], [-16], [34], [-40], [64], [-96], [36], [-16], [-10], [22], [48], [42], [64], [-80], [36], [76], [-20], [-36], [-8], [-22], [-2], [28], [-24], [2], [20], [-24], [68], [28], [-84], [-60], [-46], [-20], [-32], [-88], [96], [80], [84], [0], [-12], [-20], [22], [-38], [42], [-84], [-32], [-96], [-56], [-22], [68], [-48], [44], [-16], [44], [24], [-8], [-4], [0], [36], [-14], [-2], [0], [30], [60], [-100], [4], [64], [44], [60], [-26], [0], [-46], [-4], [64], [-72], [-56], [32], [-76], [88], [-34], [-68], [-20], [64], [-40], [60], [-20], [76], [-30], [60], [10], [-104], [20], [-18], [12], [-80], [-12], [60], [68], [-54], [14], [100], [-24], [56], [84], [24], [-80], [-104], [40], [-20], [-104], [22], [98], [68], [8], [62], [-24], [68], [48], [84], [84], [-32], [-62], [38], [32], [60], [80], [-50], [8], [-56], [-20], [48], [-20], [44], [118], [-108], [-30], [-48], [52], [-32], [60], [-76], [44], [-28], [46], [-2], [36], [56], [20], [64], [-48], [8], [-32], [-16], [-40], [8], [22], [0], [72], [84], [-68], [68], [64], [-92]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8820_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8820_2_a_ba();".
// 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_ba(: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_ba();".
// 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_ba( : 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![-4, 1]>,<17,R![-2, 1]>,<31,R![-8, 1]>],Snew);
    return Vf;
end function;