
// Make newform 6930.2.a.l in Magma, downloaded from the LMFDB on 24 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_l();".
// 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_l();".
// 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], [-6], [6], [-4], [4], [-2], [0], [2], [2], [-4], [8], [-2], [-8], [2], [4], [-8], [-6], [12], [0], [-2], [-2], [6], [8], [12], [-14], [14], [8], [12], [6], [-12], [-18], [-4], [18], [-4], [0], [6], [20], [2], [24], [2], [10], [16], [4], [8], [-24], [2], [6], [4], [-10], [0], [10], [24], [-10], [-24], [14], [-18], [32], [6], [-32], [0], [-26], [-2], [-20], [34], [-12], [2], [26], [20], [24], [6], [20], [0], [-22], [-14], [30], [30], [24], [-10], [-36], [-26], [0], [4], [14], [42], [-2], [-36], [-20], [24], [28], [20], [28], [40], [-26], [6], [-8], [34], [12], [2], [32], [-42], [-20], [22], [12], [-34], [24], [-34], [-8], [14], [30], [8], [16], [-2], [-12], [-8], [-34], [28], [42], [-46], [14], [12], [-40], [6], [6], [-8], [16], [2], [-20], [48], [40], [-6], [50], [46], [6], [-8], [30], [-50], [-4], [22], [4], [-12], [50], [40], [-14], [-2], [-32], [-12], [6], [-18], [-36], [48], [52], [40], [20], [-2], [-30], [-50], [60], [-26], [-32], [-40], [-18], [8], [-24], [-38], [-6], [6], [-20], [58], [4], [-6], [0], [-42], [-4], [18], [-16], [-46], [-52], [52], [42], [-2], [20], [-18], [-18], [28], [10], [24], [-34], [44], [-12], [-58], [52], [-6], [-22], [2], [22], [0], [-66], [56], [42], [-2], [0], [-2], [-8], [12], [22], [8], [-30], [38], [12], [-12], [-24], [-22], [-24], [-70], [-52], [46], [-22], [-16], [-50], [-4], [-24], [34], [10], [72], [-32], [36], [-6], [-52], [28], [-10], [-52], [72], [2], [-14], [4], [-40], [-68], [-20], [24], [-66], [-26], [24], [-40], [-48], [36], [0], [70], [-22], [-44], [-6], [-18], [-52], [-42], [-16], [6], [-54], [-8], [-28], [-70], [-62], [-46], [8], [-6], [-18], [12], [-70], [-6], [44], [-42], [-4], [-38], [24], [60], [10], [-38], [36], [-48], [-72], [-24], [26], [64], [48], [74], [82], [-80], [34], [-30], [-60], [38], [-84], [38], [-18], [40], [-10], [-60], [24], [-54], [42], [-60], [12], [-60], [34], [68], [-78], [-64], [70], [-56], [34], [30], [36], [64], [70], [48], [-36], [46], [38], [-12], [-22], [-66], [-52], [6], [-14], [52], [12], [-40], [-30], [-30], [-74], [64], [-8], [20], [-36], [78], [-58], [66], [32], [50], [6], [-58], [64], [6], [-60], [82], [68], [-24], [-10], [68], [18], [-6], [-56], [18], [-90], [0], [84], [-18], [-36], [-50], [-22], [20], [20], [76], [-46], [54], [8], [2], [12], [-88], [88], [78], [48], [34], [-40], [-52], [74], [2], [-54], [-42], [-66], [16], [-34], [-12], [-72], [40], [74], [-92], [48], [-78], [-90], [76], [-36], [72], [26], [-80], [38], [20], [30], [98], [102], [0], [-62], [-90], [-56], [2], [70], [-28], [72], [-54], [-22], [-52], [44], [74], [-58], [36], [-8], [66], [-96], [42], [42], [-28], [60], [-26], [-50], [-28], [-66], [96], [64], [2], [-68], [-68], [-16], [-30], [54], [50], [-2], [-20], [16], [-36], [-34], [18], [20], [86], [30], [32], [16], [42], [-14], [-52], [8], [-108], [58], [-46], [10], [-94], [60], [58], [-10], [-36], [-16], [60], [-58], [40], [66], [-20], [60], [30], [40], [-16], [-44], [-24], [-94], [108], [42], [18], [80], [56], [110], [-26], [106]];
    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_l();".
// 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_l(: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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6930_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![6, 1]>,<17,R![-6, 1]>,<19,R![4, 1]>,<23,R![-4, 1]>,<29,R![2, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;