
// Make newform 6930.2.a.n in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6930_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_6930_2_a_n();".
// 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_6930_2_a_n();".
// 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_6930_a();"
function MakeCharacter_6930_a()
    N := 6930;
    order := 1;
    char_gens := [1541, 1387, 2971, 2521];
    v := [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_6930_a_Hecke(Kf)
    return MakeCharacter_6930_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 := [[-1], [0], [1], [1], [-1], [2], [-6], [-4], [-6], [6], [2], [2], [12], [8], [-12], [6], [-12], [-10], [-4], [12], [2], [-10], [-18], [-6], [8], [-12], [-4], [0], [2], [-18], [-4], [0], [6], [-4], [6], [-10], [8], [20], [6], [-6], [-12], [2], [-12], [-16], [-18], [26], [2], [-16], [-18], [14], [-6], [0], [2], [12], [-18], [12], [-30], [-28], [8], [-6], [-4], [-6], [-28], [24], [8], [-18], [8], [-4], [-24], [2], [30], [0], [8], [-4], [8], [12], [36], [-4], [0], [-10], [-12], [2], [0], [-28], [32], [18], [0], [-28], [0], [8], [-24], [-12], [-16], [12], [-4], [-30], [-30], [42], [-28], [26], [-4], [-18], [-18], [30], [-22], [-28], [12], [-42], [24], [26], [8], [-16], [18], [-34], [-40], [48], [44], [24], [-18], [-12], [50], [-28], [-18], [-30], [14], [-6], [-46], [0], [-28], [14], [-46], [-12], [-4], [2], [36], [14], [6], [-4], [-30], [-6], [-28], [42], [-16], [48], [26], [48], [-10], [18], [-10], [54], [32], [-30], [20], [6], [-28], [12], [-22], [-18], [-34], [0], [-42], [42], [32], [36], [-54], [0], [-40], [14], [50], [6], [-60], [-10], [24], [32], [-34], [-54], [-34], [48], [32], [-22], [-64], [0], [50], [6], [42], [6], [-28], [44], [62], [48], [-16], [48], [20], [42], [12], [18], [-10], [62], [-42], [6], [54], [-28], [-64], [-10], [36], [-30], [2], [0], [-6], [-22], [32], [42], [32], [-12], [-12], [-10], [-64], [-12], [-66], [6], [14], [68], [0], [56], [-42], [-34], [-42], [0], [-64], [36], [74], [-16], [-70], [-54], [-28], [54], [38], [54], [48], [-48], [6], [20], [44], [-46], [-54], [60], [32], [-12], [14], [-36], [-16], [-60], [30], [62], [6], [-60], [-22], [-28], [42], [2], [-76], [72], [50], [-10], [30], [-22], [24], [18], [-76], [66], [-34], [-4], [-40], [74], [2], [-16], [66], [-46], [-34], [-24], [42], [20], [-18], [-58], [-4], [-48], [50], [42], [-46], [0], [60], [48], [-18], [12], [8], [48], [62], [30], [24], [44], [62], [-30], [-58], [-54], [68], [2], [-18], [62], [-12], [44], [-66], [72], [90], [68], [-12], [-22], [36], [72], [8], [-66], [8], [50], [-60], [56], [-42], [-70], [20], [-28], [36], [-54], [-10], [-18], [-64], [66], [2], [-48], [-70], [42], [26], [8], [2], [-78], [36], [-64], [-66], [60], [-34], [68], [60], [42], [-40], [86], [24], [8], [50], [42], [-72], [-12], [-66], [90], [8], [96], [18], [12], [-52], [26], [-66], [-52], [62], [12], [62], [-60], [42], [8], [-34], [36], [0], [2], [-96], [-34], [-6], [54], [56], [-42], [44], [24], [-28], [-88], [-4], [-24], [62], [6], [-48], [92], [-72], [-40], [-64], [54], [68], [-42], [50], [6], [-4], [90], [-60], [-52], [-10], [78], [-52], [-36], [62], [42], [-66], [-46], [-28], [-18], [-48], [-16], [-102], [-30], [-96], [-22], [18], [-96], [-16], [-78], [66], [18], [62], [0], [-58], [108], [20], [-84], [8], [54], [2], [-22], [-4], [56], [24], [84], [86], [0], [62], [78], [-52], [54], [110], [-70], [80], [-96], [24], [0], [62], [-12], [-70], [-48], [-34], [30], [-16], [32], [-84], [50], [68], [92], [50], [-24], [-30], [-70], [-52], [18], [0], [2], [-72], [86], [-60], [-46], [-102], [54], [-64], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6930_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6930_2_a_n();".
// 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_6930_2_a_n(:prec:=1)
    chi := MakeCharacter_6930_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_6930_2_a_n();".
// 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_6930_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6930_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![4, 1]>,<23,R![6, 1]>,<29,R![-6, 1]>,<31,R![-2, 1]>],Snew);
    return Vf;
end function;