
// Make newform 7920.2.a.r in Magma, downloaded from the LMFDB on 19 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_r();".
// 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_r();".
// 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], [-2], [2], [8], [0], [-2], [8], [-10], [10], [0], [0], [-14], [-4], [14], [4], [8], [10], [-12], [4], [6], [-14], [6], [-8], [-4], [-10], [6], [12], [-12], [-18], [0], [-2], [-4], [-2], [12], [24], [22], [12], [6], [-16], [10], [-2], [0], [24], [-16], [4], [-10], [2], [-24], [26], [-12], [-18], [8], [10], [28], [-10], [-6], [16], [6], [0], [32], [-22], [18], [-12], [2], [-4], [-2], [14], [24], [0], [-34], [20], [24], [-22], [-18], [38], [-14], [-36], [-26], [24], [-14], [4], [-4], [14], [10], [-26], [-32], [20], [40], [-16], [20], [4], [24], [42], [-2], [32], [38], [-8], [-34], [-36], [18], [-8], [-14], [44], [42], [0], [10], [12], [-2], [30], [4], [32], [-34], [44], [16], [-22], [36], [-10], [10], [46], [-12], [-36], [6], [-10], [-16], [-16], [14], [-16], [-32], [-8], [-42], [-30], [34], [-14], [40], [34], [-30], [0], [-26], [24], [12], [-2], [-16], [14], [42], [-28], [-24], [6], [54], [4], [-48], [-28], [-16], [20], [6], [-14], [30], [-36], [-6], [-60], [-36], [54], [-32], [-40], [14], [26], [-30], [-20], [30], [-8], [-22], [-32], [-2], [8], [-30], [-28], [14], [-40], [-12], [-26], [-6], [0], [-30], [-50], [-4], [50], [-24], [2], [-20], [20], [-62], [-20], [-42], [-14], [30], [18], [0], [-26], [52], [-10], [-22], [-12], [-30], [16], [-36], [-46], [-36], [26], [-14], [16], [60], [24], [26], [-44], [-14], [8], [-54], [-50], [44], [-10], [32], [-36], [-58], [-50], [-24], [-52], [-28], [-50], [40], [52], [-46], [-52], [-8], [2], [38], [68], [0], [-28], [56], [-32], [14], [10], [-56], [-24], [-20], [-56], [-24], [-2], [-30], [72], [-70], [-66], [-4], [-10], [-8], [42], [66], [-12], [28], [-50], [-42], [14], [52], [74], [-18], [-24], [46], [46], [76], [-70], [-36], [-30], [-16], [20], [46], [42], [-12], [64], [16], [-48], [54], [32], [-40], [2], [-2], [64], [-78], [66], [-28], [34], [-20], [-26], [22], [-56], [-30], [4], [-64], [-62], [-66], [20], [36], [-68], [50], [-12], [46], [-32], [-2], [16], [-70], [-30], [-12], [72], [-14], [-12], [0], [2], [-22], [8], [-22], [-42], [-24], [-30], [50], [-28], [44], [-48], [-34], [-26], [-62], [-44], [52], [-40], [-84], [30], [-6], [-22], [-20], [54], [-58], [30], [-8], [-6], [-68], [22], [68], [48], [2], [-56], [74], [-54], [-4], [-26], [26], [-80], [-76], [2], [8], [62], [-54], [-88], [52], [4], [26], [30], [52], [42], [52], [-12], [-24], [22], [-28], [-2], [76], [8], [10], [-86], [26], [18], [-26], [68], [2], [-64], [-40], [-8], [-42], [80], [-48], [-30], [-46], [100], [-76], [-32], [-86], [68], [22], [100], [6], [14], [-10], [-20], [38], [-34], [44], [-50], [18], [-12], [28], [26], [6], [36], [-32], [26], [-78], [-8], [88], [14], [-8], [-6], [-82], [16], [84], [106], [42], [-36], [-54], [52], [0], [10], [-4], [-4], [-32], [-34], [38], [2], [-10], [20], [-68], [28], [102], [70], [0], [26], [-102], [56], [0], [-78], [6], [0], [96], [-44], [66], [34], [90], [-58], [-20], [22], [30], [92], [-24], [-60], [-26], [-56], [-110], [84], [84], [82], [92], [68], [-92], [-40], [-22], [60], [46], [10], [80], [-40], [74], [90], [-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_r();".
// 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_r(: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_r();".
// 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_r( : 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![2, 1]>,<17,R![-2, 1]>,<19,R![-8, 1]>,<23,R![0, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;