
// Make newform 7920.2.a.j in Magma, downloaded from the LMFDB on 18 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_j();".
// 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_j();".
// 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], [-6], [6], [0], [0], [6], [-8], [-2], [2], [4], [0], [-6], [4], [-10], [12], [0], [-10], [12], [0], [6], [10], [-10], [16], [-8], [-10], [-2], [-16], [-4], [-10], [0], [-18], [-12], [-18], [-4], [20], [18], [-20], [-26], [16], [-10], [-6], [-8], [-16], [16], [-8], [6], [-18], [0], [2], [12], [-2], [-4], [-6], [-28], [2], [-14], [4], [-6], [20], [24], [-22], [-30], [-12], [-18], [24], [-10], [14], [-32], [-16], [34], [-4], [16], [26], [14], [6], [18], [-20], [-26], [24], [-6], [-12], [-36], [-18], [14], [22], [-16], [-36], [0], [-16], [-28], [-36], [36], [-22], [30], [28], [22], [-28], [-22], [-32], [-14], [40], [10], [-12], [14], [-16], [-30], [0], [-46], [-42], [36], [-8], [-34], [28], [-8], [-30], [-44], [22], [-42], [34], [44], [-12], [-18], [-26], [-16], [40], [2], [-16], [20], [32], [-10], [18], [26], [-22], [12], [2], [42], [-32], [46], [-16], [-8], [-2], [-24], [18], [30], [-44], [32], [-22], [38], [4], [12], [28], [-48], [52], [38], [38], [-50], [4], [54], [24], [-4], [-2], [24], [24], [-14], [18], [-54], [28], [-34], [48], [46], [-32], [14], [-40], [50], [-24], [30], [0], [20], [22], [38], [16], [18], [-46], [36], [2], [-8], [26], [-24], [28], [18], [-32], [-42], [-38], [54], [46], [36], [30], [-44], [-50], [2], [36], [2], [-24], [48], [42], [-20], [-18], [-30], [-40], [-12], [-32], [-70], [-48], [34], [32], [-46], [-10], [20], [6], [-40], [-24], [-10], [38], [-16], [32], [12], [62], [-8], [-60], [10], [-44], [-36], [-14], [26], [44], [-48], [68], [-32], [40], [-50], [54], [-48], [40], [-44], [72], [-36], [10], [-46], [32], [-6], [-6], [60], [-42], [52], [26], [54], [64], [48], [22], [2], [14], [-20], [26], [-18], [-4], [34], [14], [4], [-46], [44], [38], [64], [76], [6], [-62], [-20], [-12], [64], [-12], [-10], [-36], [-24], [-70], [18], [64], [34], [50], [36], [-66], [-76], [-14], [54], [80], [42], [-12], [36], [22], [42], [36], [44], [-36], [-46], [-68], [14], [0], [50], [-52], [-54], [-14], [-4], [68], [74], [-12], [-72], [10], [-54], [32], [34], [-18], [-40], [-66], [-62], [-44], [92], [36], [42], [62], [34], [-4], [24], [-80], [28], [62], [6], [26], [-88], [22], [54], [30], [-24], [-14], [-44], [-90], [52], [-32], [-86], [40], [-86], [-70], [-16], [-50], [14], [56], [12], [62], [32], [26], [42], [-72], [60], [-44], [50], [-38], [16], [34], [12], [20], [-4], [6], [4], [-82], [-36], [48], [62], [-6], [46], [-30], [54], [-8], [-74], [-8], [16], [32], [-42], [44], [-56], [50], [10], [60], [-84], [-40], [14], [-12], [54], [-20], [-2], [46], [14], [-24], [-90], [-2], [20], [-18], [-78], [76], [68], [-18], [10], [36], [-48], [22], [-78], [8], [-48], [94], [36], [26], [58], [-72], [76], [26], [-78], [-84], [-22], [60], [88], [58], [60], [-60], [-32], [-2], [22], [10], [-10], [44], [12], [-84], [-10], [22], [16], [26], [-74], [-44], [60], [-110], [14], [-108], [-88], [-104], [-78], [50], [-6], [-26], [12], [-94], [70], [-84], [80], [-36], [70], [-68], [14], [-44], [68], [-6], [-52], [40], [52], [-72], [34], [-44], [90], [-102], [-112], [-20], [-54], [74], [-30]];
    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_j();".
// 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_j(: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_j();".
// 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_j( : 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![6, 1]>,<17,R![-6, 1]>,<19,R![0, 1]>,<23,R![0, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;