
// Make newform 6930.2.a.b 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_b();".
// 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_b();".
// 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], [4], [-4], [-6], [0], [-6], [-10], [4], [0], [2], [-4], [-10], [8], [0], [2], [-8], [4], [2], [2], [10], [0], [12], [2], [18], [8], [-4], [10], [4], [-6], [0], [-18], [-8], [-12], [-14], [-4], [-14], [0], [-14], [-6], [-8], [-16], [-16], [-12], [10], [22], [-4], [2], [-20], [-2], [-16], [10], [-8], [-2], [-6], [-16], [-14], [8], [-20], [-6], [10], [-20], [2], [-4], [22], [14], [-4], [-8], [-26], [-20], [0], [-6], [22], [14], [26], [-36], [-26], [12], [-14], [24], [-12], [-18], [-6], [-14], [-40], [20], [32], [-16], [-20], [-20], [36], [10], [18], [24], [-22], [20], [18], [12], [-14], [16], [18], [28], [-22], [-8], [26], [16], [-2], [-38], [-32], [-32], [-2], [4], [-24], [34], [-44], [-46], [-46], [26], [-12], [40], [10], [22], [-36], [-8], [2], [24], [24], [-40], [10], [-26], [50], [10], [-24], [2], [18], [44], [18], [8], [-20], [-6], [-36], [-38], [18], [40], [36], [-58], [-54], [-8], [12], [40], [-56], [-8], [-14], [10], [18], [-44], [-34], [16], [-12], [42], [-24], [8], [-14], [50], [10], [-12], [10], [-36], [-38], [-8], [-46], [-16], [-6], [0], [22], [40], [-60], [-22], [58], [4], [-6], [-2], [4], [26], [0], [34], [-60], [-4], [50], [28], [14], [-46], [-30], [58], [-52], [18], [40], [22], [50], [28], [18], [0], [-52], [-54], [8], [34], [-38], [8], [-36], [-48], [50], [0], [-34], [28], [-30], [42], [-8], [14], [-16], [12], [42], [-50], [72], [-8], [28], [26], [-4], [-8], [10], [4], [-20], [18], [-22], [-44], [12], [-36], [44], [-56], [46], [42], [72], [32], [12], [-40], [-24], [70], [62], [36], [2], [-38], [-36], [22], [8], [58], [18], [48], [60], [-66], [2], [38], [-48], [18], [-54], [-36], [58], [10], [56], [-54], [48], [-22], [-16], [12], [-22], [42], [-36], [-36], [-32], [-4], [46], [32], [-40], [66], [42], [-64], [22], [50], [4], [22], [76], [-74], [-30], [-8], [26], [60], [-64], [-14], [50], [24], [-36], [52], [-62], [-84], [-46], [-64], [54], [-12], [-38], [18], [80], [48], [42], [60], [12], [-46], [-62], [4], [-30], [50], [-24], [-82], [2], [12], [-84], [-32], [-38], [-70], [82], [-8], [60], [-88], [12], [-34], [-38], [-14], [-56], [82], [-22], [42], [-8], [42], [92], [-22], [-40], [-56], [42], [4], [-70], [66], [56], [-62], [66], [92], [-4], [22], [84], [78], [78], [-20], [-28], [52], [58], [-6], [56], [-62], [44], [-56], [-32], [66], [-56], [74], [28], [-4], [50], [-74], [26], [-14], [-54], [-32], [-34], [-52], [24], [64], [-42], [4], [-88], [18], [-70], [60], [-20], [16], [-54], [-24], [-30], [-52], [-22], [46], [42], [24], [-10], [74], [-48], [-54], [-22], [-60], [36], [66], [42], [4], [-24], [-6], [-22], [-68], [-8], [-42], [84], [2], [-6], [-36], [4], [74], [18], [-84], [-14], [-16], [0], [42], [-12], [-44], [0], [-2], [-54], [82], [-2], [88], [48], [20], [102], [26], [-44], [34], [22], [-104], [-12], [74], [38], [76], [-12], [-36], [30], [90], [-54], [82], [-12], [-14], [-22], [-12], [40], [-36], [6], [-8], [34], [40], [-108], [-54], [16], [16], [-12], [-36], [26], [12], [66], [-62], [72], [12], [-38], [42], [-66]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : 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![-4, 1]>,<23,R![4, 1]>,<29,R![6, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;