
// Make newform 7920.2.a.k in Magma, downloaded from the LMFDB on 24 April 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_k();".
// 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_k();".
// 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], [0], [1], [0], [-2], [-2], [6], [-2], [0], [-6], [-2], [2], [-2], [-2], [8], [4], [4], [-2], [-10], [-4], [12], [0], [-18], [10], [8], [-4], [-4], [-8], [4], [-4], [12], [14], [-6], [-20], [-18], [-4], [-12], [-2], [-12], [2], [-6], [-2], [18], [-8], [-22], [0], [-20], [-14], [18], [16], [18], [-28], [-4], [0], [14], [-4], [12], [-18], [-30], [6], [2], [18], [6], [-2], [-32], [-26], [12], [24], [-4], [4], [8], [-12], [0], [-2], [-30], [22], [-32], [-14], [12], [-10], [-20], [-34], [-32], [16], [-12], [-6], [-18], [-16], [-28], [12], [24], [36], [28], [12], [-26], [8], [10], [4], [-26], [6], [12], [-30], [-30], [-10], [0], [34], [-30], [-30], [-28], [40], [-32], [-20], [24], [-12], [-16], [6], [34], [36], [26], [-34], [26], [20], [44], [-34], [26], [-2], [-32], [0], [-10], [24], [16], [2], [18], [-10], [14], [2], [42], [18], [50], [-54], [-32], [20], [-2], [6], [-32], [22], [-24], [-6], [20], [52], [36], [-40], [-24], [-42], [-4], [40], [2], [-14], [-24], [-18], [-12], [-24], [-60], [6], [8], [28], [-22], [18], [-20], [-30], [-60], [-26], [48], [44], [34], [-30], [24], [24], [64], [-36], [26], [-42], [34], [-46], [44], [56], [-38], [60], [42], [28], [44], [58], [-12], [36], [2], [46], [22], [52], [-14], [24], [6], [-10], [-28], [-34], [-8], [36], [-30], [-8], [-10], [-22], [-16], [12], [28], [-22], [-72], [-42], [-6], [-30], [48], [20], [8], [32], [4], [64], [16], [14], [-8], [28], [26], [26], [32], [-54], [-16], [52], [22], [-66], [4], [-66], [-4], [6], [-64], [34], [66], [-28], [40], [-72], [10], [64], [44], [-34], [46], [70], [18], [4], [38], [6], [-58], [46], [52], [20], [-52], [4], [-72], [-56], [-62], [-60], [10], [54], [10], [0], [34], [32], [14], [72], [-56], [28], [-38], [36], [-40], [-16], [16], [24], [46], [-46], [-2], [38], [-16], [6], [2], [72], [74], [-20], [-44], [18], [-48], [30], [-60], [38], [-18], [38], [56], [12], [-40], [34], [-16], [-10], [-18], [52], [20], [-2], [86], [-16], [16], [46], [-52], [64], [-18], [58], [22], [-58], [-70], [-56], [42], [50], [64], [-24], [-84], [6], [80], [-38], [40], [4], [-58], [60], [42], [82], [38], [76], [46], [-88], [70], [0], [54], [20], [82], [-92], [-28], [30], [2], [-34], [-58], [52], [-92], [30], [86], [20], [-50], [14], [-56], [-42], [-26], [44], [84], [-38], [-30], [-92], [22], [-40], [-40], [-36], [-70], [-56], [-6], [-24], [48], [-38], [-6], [-62], [-30], [-64], [-52], [-74], [78], [46], [40], [2], [10], [38], [-34], [-30], [60], [40], [70], [74], [56], [60], [-56], [30], [4], [36], [56], [0], [-90], [0], [2], [-50], [-36], [-48], [-86], [74], [24], [50], [-10], [14], [-72], [16], [-80], [84], [-54], [-32], [74], [36], [-26], [-66], [56], [70], [-64], [-52], [26], [-20], [32], [-18], [-82], [-48], [-26], [-14], [68], [4], [-36], [-92], [-80], [20], [6], [-30], [-58], [36], [26], [-36], [70], [-26], [108], [10], [34], [34], [-92], [44], [-16], [-8], [-8], [-32], [-12], [-82], [-66], [38], [-48], [0], [38], [-28], [-68], [-108], [6], [6], [52], [-76], [-110], [-56], [-96], [-6], [14], [50]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<13,R![0, 1]>,<17,R![2, 1]>,<19,R![2, 1]>,<23,R![-6, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;