
// Make newform 6930.2.a.s 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_s();".
// 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_s();".
// 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], [0], [0], [0], [4], [-2], [-2], [-6], [-8], [-12], [6], [6], [10], [-4], [-8], [4], [-4], [-16], [0], [6], [14], [0], [-14], [-12], [-6], [-2], [8], [-4], [-6], [8], [-6], [-8], [6], [-8], [-16], [-24], [-12], [-6], [8], [-22], [-14], [-6], [-12], [-6], [-4], [2], [18], [0], [-16], [-10], [14], [24], [-10], [20], [-2], [2], [28], [4], [28], [6], [2], [2], [28], [14], [-28], [-20], [-18], [8], [-22], [14], [-16], [2], [-22], [-34], [6], [4], [26], [18], [-8], [14], [4], [20], [10], [26], [20], [4], [30], [-20], [20], [-36], [-40], [-12], [-18], [34], [28], [-10], [-44], [18], [-32], [-6], [20], [-26], [-2], [12], [-20], [-8], [24], [-6], [42], [-22], [-8], [-46], [34], [42], [-22], [20], [46], [14], [0], [-20], [6], [14], [30], [34], [14], [4], [12], [8], [52], [-26], [-12], [12], [-18], [48], [-6], [-18], [-44], [34], [32], [36], [-10], [10], [-8], [-12], [2], [32], [10], [26], [-44], [52], [48], [-28], [-8], [-30], [36], [60], [12], [6], [-48], [-26], [-42], [22], [-4], [-48], [-46], [54], [-44], [14], [24], [-34], [-54], [-22], [-20], [6], [20], [8], [32], [48], [54], [-24], [-16], [-18], [-42], [34], [14], [56], [38], [4], [64], [10], [-12], [6], [30], [-30], [44], [-44], [2], [0], [-62], [-60], [-66], [66], [10], [-44], [-42], [-10], [-2], [22], [-48], [-62], [36], [26], [72], [-52], [44], [-54], [-6], [-52], [-30], [-8], [52], [26], [34], [-48], [28], [-60], [-46], [-20], [56], [46], [58], [-20], [-2], [-22], [16], [26], [-48], [-16], [-42], [42], [0], [0], [-2], [2], [-20], [24], [-42], [-60], [60], [-14], [28], [20], [10], [80], [34], [-32], [56], [-12], [-52], [-64], [-42], [66], [-66], [-62], [52], [46], [70], [64], [-18], [48], [20], [2], [-4], [-42], [2], [0], [-16], [36], [52], [0], [16], [-28], [-50], [-22], [-62], [48], [6], [-58], [42], [32], [-34], [-12], [54], [-74], [-32], [-8], [28], [86], [-72], [-48], [88], [-66], [-12], [78], [80], [22], [-40], [10], [18], [-76], [64], [-4], [54], [-72], [-10], [-54], [-92], [-78], [60], [48], [30], [-14], [56], [-46], [32], [14], [-10], [78], [0], [-32], [-36], [78], [-62], [-8], [10], [-52], [-74], [26], [-88], [-40], [90], [36], [-6], [-8], [-72], [6], [40], [62], [2], [-20], [-26], [0], [78], [52], [46], [24], [22], [-88], [-48], [-60], [-30], [6], [-40], [-8], [-78], [56], [-50], [88], [-46], [-24], [-54], [-18], [-40], [-32], [-8], [24], [-78], [-78], [96], [-10], [-16], [70], [-80], [38], [12], [30], [-66], [-50], [-8], [10], [0], [54], [84], [-98], [-96], [-90], [36], [62], [-32], [78], [-80], [-64], [14], [78], [-18], [-30], [-44], [-42], [-76], [-52], [62], [18], [64], [-42], [-66], [-48], [-6], [-52], [12], [84], [58], [-10], [40], [10], [-74], [-36], [50], [-60], [-28], [86], [102], [-70], [6], [-74], [-64], [32], [-18], [-30], [-46], [32], [20], [22], [-88], [-56], [-26], [-44], [44], [90], [52], [-40], [-42], [42], [-14], [-60], [-76], [-10], [-68], [-12], [-76], [-10], [112], [-74], [-8], [14], [-22], [-86], [-56], [-64], [26], [-6], [60], [44], [-66], [90], [48], [-66], [-102], [-48]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6930_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![0, 1]>,<17,R![0, 1]>,<19,R![0, 1]>,<23,R![-4, 1]>,<29,R![2, 1]>,<31,R![2, 1]>],Snew);
    return Vf;
end function;