
// Make newform 6930.2.a.bk in Magma, downloaded from the LMFDB on 25 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_bk();".
// 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_bk();".
// 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], [-2], [6], [-6], [-4], [0], [8], [0], [4], [4], [12], [0], [2], [-8], [12], [-6], [10], [12], [-14], [4], [-10], [-8], [-20], [4], [6], [8], [6], [22], [-6], [-4], [-6], [-10], [-24], [0], [18], [4], [2], [-8], [2], [2], [16], [-20], [16], [24], [6], [26], [-6], [-8], [0], [28], [-8], [-14], [28], [-22], [2], [32], [6], [-12], [-16], [-12], [0], [12], [-2], [-20], [6], [-20], [6], [0], [-26], [4], [-16], [-18], [14], [34], [0], [16], [-26], [22], [16], [12], [-40], [-14], [22], [26], [-10], [-38], [-8], [-18], [-12], [-28], [-16], [-2], [10], [4], [-16], [20], [18], [24], [-26], [-40], [4], [2], [-26], [-44], [44], [16], [-26], [42], [-32], [-8], [30], [34], [0], [36], [-8], [42], [-42], [-38], [4], [12], [8], [-30], [-48], [-44], [-2], [-36], [-48], [4], [52], [24], [-28], [6], [-48], [-30], [-54], [-30], [8], [14], [-12], [-26], [40], [22], [30], [48], [-30], [14], [14], [28], [-48], [-4], [12], [26], [-30], [6], [-18], [52], [-18], [-56], [0], [54], [56], [16], [-42], [-10], [18], [-40], [2], [-6], [-10], [-8], [-18], [4], [-30], [-40], [-30], [-62], [-10], [-28], [-30], [-42], [-6], [-26], [-22], [34], [-48], [56], [4], [-4], [6], [-36], [-56], [38], [44], [34], [-24], [56], [64], [-14], [16], [-44], [14], [-40], [4], [-42], [-60], [18], [2], [22], [66], [-56], [30], [-28], [-64], [10], [36], [4], [-36], [-62], [10], [56], [-40], [64], [-20], [48], [32], [-8], [-46], [-30], [-18], [-10], [0], [-10], [6], [-12], [-72], [-36], [-38], [-36], [-10], [-62], [12], [32], [72], [-4], [-8], [58], [-60], [-10], [34], [46], [-36], [-22], [28], [70], [22], [-80], [12], [-46], [38], [60], [-80], [6], [42], [-44], [-10], [-62], [0], [0], [14], [-18], [68], [72], [-68], [26], [18], [-56], [60], [24], [82], [48], [72], [-18], [62], [32], [0], [-22], [-42], [10], [-78], [-54], [42], [48], [26], [-22], [68], [-66], [2], [22], [12], [-12], [50], [-44], [14], [-28], [-26], [48], [2], [-18], [4], [-84], [40], [28], [-54], [40], [-46], [-82], [10], [30], [38], [30], [-2], [52], [-34], [56], [-30], [-16], [-24], [-28], [-76], [32], [30], [2], [74], [-58], [56], [-28], [-66], [50], [8], [64], [-60], [86], [96], [-20], [30], [30], [78], [6], [88], [76], [46], [-56], [-66], [-42], [14], [2], [72], [-66], [-4], [-38], [78], [-10], [-60], [14], [84], [28], [-12], [-84], [-4], [24], [-28], [-50], [78], [-20], [-66], [18], [-6], [-20], [26], [54], [4], [0], [82], [52], [48], [-2], [-18], [44], [-18], [-48], [-22], [-28], [50], [-4], [80], [-78], [-78], [-88], [56], [66], [76], [-8], [82], [-22], [0], [22], [78], [-8], [72], [-26], [-18], [-18], [-24], [60], [0], [-6], [-38], [2], [-82], [-4], [-42], [104], [-30], [-72], [-80], [58], [-24], [-100], [12], [58], [-58], [-10], [-10], [92], [-16], [-54], [78], [96], [70], [-4], [-50], [-32], [-56], [-14], [-78], [-100], [48], [-44], [-36], [-62], [-6], [76], [-54], [-98], [86], [76], [-28], [104], [-2], [32], [18], [-70], [42], [6], [0], [-4], [-12], [-16], [106], [-36], [66], [-58], [-56], [-8], [-60], [36], [-64]];
    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_bk();".
// 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_bk(: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_bk();".
// 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_bk( : 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![2, 1]>,<19,R![-6, 1]>,<23,R![6, 1]>,<29,R![4, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;