
// Make newform 7920.2.a.be in Magma, downloaded from the LMFDB on 16 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7920_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_7920_2_a_be();".
// 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_7920_2_a_be();".
// 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_7920_a();"
function MakeCharacter_7920_a()
    N := 7920;
    order := 1;
    char_gens := [991, 5941, 3521, 6337, 6481];
    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_7920_a_Hecke(Kf)
    return MakeCharacter_7920_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], [2], [-1], [-4], [4], [-8], [4], [-2], [-8], [-2], [2], [6], [12], [-6], [-12], [10], [-16], [0], [4], [-12], [-6], [-6], [14], [-14], [-16], [-6], [-10], [18], [-2], [0], [-6], [-12], [18], [-4], [-14], [12], [2], [8], [-12], [18], [16], [4], [-8], [-24], [0], [4], [-18], [-10], [-16], [-20], [-2], [12], [14], [-18], [18], [20], [20], [-30], [-10], [28], [-34], [0], [-22], [-10], [36], [-4], [18], [26], [-10], [4], [-12], [-32], [28], [8], [-6], [18], [-14], [10], [12], [-6], [-24], [14], [-32], [36], [-6], [-16], [-6], [4], [0], [20], [-44], [-20], [4], [-22], [-30], [-22], [10], [-10], [2], [8], [-2], [-18], [-4], [-10], [24], [12], [0], [14], [-18], [-32], [-6], [20], [-16], [2], [-4], [48], [-38], [-36], [14], [36], [12], [32], [-36], [-18], [-6], [-24], [8], [8], [20], [30], [16], [2], [14], [-2], [-46], [38], [-26], [42], [52], [-42], [24], [2], [-14], [40], [-40], [36], [-28], [0], [24], [-26], [-52], [-46], [-8], [0], [-16], [30], [32], [-46], [40], [-48], [6], [36], [58], [-8], [-32], [-32], [22], [34], [-60], [18], [-52], [8], [-8], [30], [36], [-6], [14], [-26], [32], [60], [2], [32], [36], [6], [-8], [-16], [42], [32], [-50], [2], [60], [42], [14], [46], [-58], [-42], [-32], [14], [38], [-40], [-10], [-10], [12], [-2], [-32], [-2], [2], [60], [-32], [54], [-8], [64], [12], [26], [26], [62], [-24], [66], [-38], [52], [38], [-16], [-6], [-34], [-30], [-48], [-2], [-24], [2], [52], [16], [-22], [-16], [-2], [46], [-60], [-36], [-8], [36], [0], [-16], [50], [-40], [36], [12], [60], [-4], [-66], [-20], [78], [48], [-38], [24], [60], [58], [-62], [46], [24], [-62], [22], [34], [-20], [-42], [-20], [30], [38], [-18], [-36], [-50], [-32], [62], [24], [-48], [36], [32], [82], [6], [0], [-34], [32], [-18], [-26], [-46], [-56], [54], [12], [40], [30], [-54], [-28], [72], [72], [44], [-30], [64], [-6], [4], [-10], [-40], [-24], [-32], [-8], [76], [-22], [72], [-26], [-24], [56], [34], [-30], [78], [32], [-46], [70], [-36], [-12], [14], [78], [-8], [46], [46], [60], [-16], [-34], [-44], [-84], [-26], [-8], [82], [-10], [-36], [-2], [-32], [-28], [-66], [8], [86], [2], [-6], [30], [-70], [-56], [26], [0], [-74], [52], [40], [34], [-20], [10], [14], [62], [-46], [-84], [0], [-64], [40], [72], [-28], [-82], [-76], [-8], [-4], [-74], [-40], [-14], [-90], [-20], [36], [-66], [-30], [-88], [58], [28], [12], [-40], [78], [-88], [82], [78], [38], [4], [88], [-80], [-8], [42], [-86], [24], [62], [-22], [-36], [8], [48], [16], [-88], [-74], [28], [46], [-50], [66], [10], [102], [-62], [52], [2], [-2], [48], [76], [-56], [76], [-60], [-44], [28], [-18], [-32], [16], [30], [-86], [-58], [76], [56], [96], [-14], [-78], [-36], [6], [28], [-60], [14], [-4], [52], [-44], [2], [-78], [-10], [-86], [104], [32], [-60], [6], [-82], [-16], [2], [-64], [10], [34], [-34], [-78], [-42], [72], [94], [30], [50], [22], [14], [-68], [-100], [-62], [52], [32], [0], [-30], [-10], [-28], [24], [-4], [-70], [-76], [-18], [-108], [48], [46], [68], [-20], [46], [24], [54], [6], [50], [46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7920_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7920_2_a_be();".
// 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_7920_2_a_be(:prec:=1)
    chi := MakeCharacter_7920_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(3449) - 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_7920_2_a_be();".
// 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_7920_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<13,R![4, 1]>,<17,R![-4, 1]>,<19,R![8, 1]>,<23,R![-4, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;