
// Make newform 7920.2.a.bd in Magma, downloaded from the LMFDB on 11 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_bd();".
// 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_bd();".
// 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], [2], [-1], [-4], [-4], [0], [-4], [6], [8], [-2], [10], [-10], [-12], [-6], [-12], [10], [8], [0], [4], [-4], [2], [10], [-2], [-6], [-8], [-14], [6], [-14], [-2], [-16], [-22], [-20], [-6], [4], [2], [4], [-22], [-16], [-12], [-14], [0], [20], [0], [8], [8], [28], [6], [22], [24], [-20], [-2], [12], [14], [6], [-14], [-20], [20], [10], [-26], [-12], [-18], [16], [26], [6], [4], [-4], [10], [-22], [-26], [-12], [28], [-16], [-20], [0], [10], [-14], [18], [-22], [-20], [26], [-8], [-2], [8], [12], [42], [16], [-14], [-36], [-8], [-12], [-4], [28], [-28], [2], [2], [42], [26], [-42], [2], [0], [-10], [6], [4], [-10], [-16], [-44], [16], [-34], [14], [-16], [-6], [4], [-32], [18], [20], [-40], [42], [-36], [-50], [-44], [-12], [8], [28], [22], [26], [-24], [32], [8], [-36], [6], [0], [18], [-10], [30], [-30], [38], [22], [-14], [44], [-50], [32], [26], [-14], [-24], [-56], [28], [20], [24], [8], [22], [4], [42], [-16], [-48], [-8], [-18], [16], [-6], [0], [8], [6], [36], [42], [16], [16], [-16], [38], [-30], [36], [50], [60], [8], [40], [46], [-20], [-38], [-2], [-26], [24], [28], [-14], [24], [-36], [-42], [-40], [24], [10], [16], [14], [-54], [-20], [-6], [-42], [-18], [54], [-26], [56], [-10], [30], [-16], [38], [70], [44], [-50], [16], [22], [42], [44], [48], [-58], [-64], [56], [-36], [-70], [-22], [-10], [48], [18], [-22], [-4], [54], [-72], [2], [-2], [50], [-16], [14], [-24], [66], [-52], [8], [66], [24], [54], [46], [-36], [-20], [56], [12], [24], [56], [-14], [0], [-12], [52], [-36], [-60], [-26], [-52], [-58], [-8], [26], [16], [44], [26], [-14], [62], [8], [34], [62], [2], [-68], [38], [12], [-18], [54], [14], [4], [78], [-8], [30], [0], [16], [-36], [-40], [-30], [-10], [-48], [-42], [48], [6], [54], [-46], [-24], [38], [-44], [-56], [38], [-6], [-4], [48], [-72], [28], [-54], [-80], [74], [20], [38], [8], [64], [-40], [64], [28], [-6], [-32], [-58], [-40], [-88], [-6], [18], [-90], [-56], [-6], [-42], [-20], [-60], [-82], [6], [-32], [78], [-10], [-76], [88], [30], [-44], [36], [-18], [32], [34], [-10], [68], [-58], [56], [-36], [-66], [80], [-42], [-62], [58], [-50], [66], [-56], [-86], [-32], [-74], [28], [-24], [18], [36], [-86], [62], [14], [-62], [-92], [80], [0], [-32], [96], [-12], [6], [60], [24], [-92], [54], [80], [2], [86], [-36], [-76], [6], [-38], [48], [10], [60], [44], [-24], [6], [-88], [-94], [-18], [-90], [-68], [80], [8], [8], [74], [-102], [64], [-34], [-54], [-84], [64], [0], [-80], [-96], [-42], [60], [-26], [-50], [66], [-86], [38], [-54], [28], [-62], [-42], [24], [92], [-72], [4], [-20], [-100], [60], [-50], [48], [72], [14], [-30], [54], [60], [-32], [96], [18], [34], [-28], [-50], [28], [-28], [14], [12], [52], [76], [18], [18], [-58], [42], [32], [88], [-84], [22], [14], [-48], [82], [8], [42], [42], [-98], [-30], [38], [-8], [-26], [22], [34], [-42], [30], [-52], [28], [-78], [-12], [-64], [48], [98], [70], [36], [96], [36], [18], [84], [-114], [12], [64], [-50], [84], [44], [-34], [56], [-82], [22], [34], [-58]];
    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_bd();".
// 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_bd(: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_bd();".
// 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_bd( : 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![0, 1]>,<23,R![4, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;