
// Make newform 7920.2.a.o in Magma, downloaded from the LMFDB on 07 May 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_o();".
// 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_o();".
// 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+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], [4], [-1], [-4], [0], [4], [-6], [6], [-8], [2], [-6], [-8], [6], [6], [-12], [2], [10], [-12], [-16], [-8], [0], [-6], [14], [18], [-14], [-12], [-10], [-6], [16], [12], [-6], [4], [18], [-8], [-10], [-14], [0], [0], [0], [-22], [-24], [8], [0], [-8], [-20], [-2], [0], [14], [12], [0], [-10], [24], [-18], [-24], [30], [16], [20], [18], [-8], [-12], [4], [-12], [26], [-6], [28], [-16], [12], [26], [-18], [0], [10], [-16], [-20], [-18], [-18], [-10], [-30], [2], [-24], [-10], [-24], [14], [-32], [-18], [-18], [-28], [-30], [22], [-18], [0], [34], [12], [-20], [36], [-6], [-6], [4], [38], [-8], [-24], [-36], [-42], [-20], [14], [-42], [24], [-12], [38], [40], [-16], [18], [-8], [-20], [30], [-14], [18], [-18], [-36], [-22], [8], [12], [-30], [-8], [30], [26], [36], [-26], [32], [-20], [24], [4], [-10], [-42], [-10], [6], [4], [42], [-30], [28], [30], [34], [-12], [-34], [0], [56], [-24], [-32], [6], [8], [18], [10], [-48], [10], [24], [-8], [18], [-28], [18], [6], [12], [-32], [-36], [18], [6], [16], [8], [-34], [30], [36], [26], [-24], [44], [4], [30], [4], [-6], [52], [-34], [22], [-36], [14], [60], [-30], [54], [8], [22], [-22], [0], [-10], [24], [-32], [30], [24], [30], [14], [26], [36], [24], [-66], [-8], [50], [-10], [-24], [18], [-32], [-36], [-6], [28], [44], [-30], [-14], [-18], [24], [-10], [28], [-42], [66], [6], [2], [-32], [54], [34], [36], [-10], [-30], [-24], [-8], [12], [14], [52], [-8], [18], [-26], [-12], [50], [60], [12], [-24], [6], [-20], [10], [-46], [12], [-24], [34], [24], [76], [48], [68], [-42], [42], [-34], [-48], [60], [2], [-8], [42], [-52], [64], [-72], [-22], [-4], [-18], [28], [-18], [54], [-32], [-12], [14], [-38], [-82], [40], [-52], [-26], [-18], [-46], [62], [60], [48], [40], [-12], [-10], [64], [0], [-10], [60], [28], [6], [30], [42], [36], [60], [44], [-30], [-68], [54], [12], [-44], [-4], [24], [40], [-6], [76], [26], [-18], [-82], [60], [-88], [36], [-54], [-18], [22], [-24], [-10], [0], [-72], [-34], [-66], [28], [-10], [6], [-26], [-84], [2], [16], [34], [-60], [-72], [2], [-78], [16], [-12], [-20], [-30], [26], [12], [-70], [52], [-10], [-42], [-6], [40], [30], [36], [26], [22], [48], [78], [-68], [2], [-6], [-44], [26], [0], [48], [36], [60], [-66], [20], [54], [-18], [-60], [-2], [62], [24], [-56], [-82], [96], [40], [24], [-6], [-32], [62], [72], [-72], [68], [6], [-52], [-30], [78], [-56], [-48], [-68], [54], [-44], [86], [-92], [42], [50], [-66], [60], [58], [-84], [68], [-8], [-90], [16], [-42], [-58], [-30], [-68], [54], [-6], [16], [38], [-90], [94], [36], [-28], [-36], [54], [52], [-40], [54], [48], [-2], [102], [-60], [66], [44], [42], [-12], [-70], [-90], [-42], [78], [-80], [48], [-34], [-36], [76], [-30], [14], [54], [-82], [38], [82], [-104], [-90], [54], [14], [48], [-22], [36], [-20], [24], [14], [-46], [-92], [-36], [-36], [-90], [-22], [-6], [-82], [-12], [68], [-54], [-56], [4], [-84], [26], [-92], [-40], [-8], [54], [66], [-20], [-44], [90], [108], [-58], [96], [-4], [-66], [40], [12], [-6], [-46], [78]];
    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_o();".
// 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_o(: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_o();".
// 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_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<13,R![4, 1]>,<17,R![0, 1]>,<19,R![-4, 1]>,<23,R![6, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;