
// Make newform 3380.2.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3380_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_3380_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_3380_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_3380_a();"
function MakeCharacter_3380_a()
    N := 3380;
    order := 1;
    char_gens := [1691, 677, 1861];
    v := [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_3380_a_Hecke(Kf)
    return MakeCharacter_3380_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 := [[0], [-3], [1], [3], [3], [0], [-7], [1], [-7], [-5], [-4], [-3], [7], [-9], [8], [-6], [5], [-5], [13], [-3], [-14], [-8], [12], [7], [-11], [-9], [-16], [-3], [-14], [13], [1], [4], [-3], [13], [3], [-8], [6], [11], [1], [-15], [19], [-14], [-3], [-15], [-23], [9], [-5], [-23], [-1], [26], [18], [-16], [-1], [5], [-19], [-7], [3], [23], [-23], [-10], [1], [9], [28], [-24], [-6], [-2], [13], [-18], [-13], [-25], [21], [-8], [9], [-27], [-9], [-13], [18], [33], [15], [-1], [-33], [34], [-9], [1], [3], [24], [-21], [-23], [11], [-28], [20], [-1], [-17], [23], [-16], [11], [15], [-34], [-23], [10], [-16], [-19], [-45], [19], [-40], [2], [37], [26], [8], [-13], [-9], [-31], [29], [12], [-15], [-21], [7], [17], [-11], [-15], [-17], [13], [22], [31], [-25], [2], [35], [1], [-28], [14], [39], [1], [-13], [1], [51], [-5], [25], [-1], [-7], [-25], [0], [15], [47], [12], [-13], [13], [10], [18], [44], [16], [5], [11], [-44], [-19], [17], [-12], [-43], [-53], [50], [42], [-55], [-31], [-48], [-43], [-3], [36], [-13], [13], [-30], [-42], [4], [19], [-3], [-19], [8], [-9], [-17], [50], [41], [63], [16], [36], [14], [-42], [-15], [-41], [-10], [-28], [19], [-47], [9], [-33], [-12], [-21], [35], [-35], [-10], [49], [62], [0], [-33], [-7], [17], [-58], [-45], [65], [40], [3], [-1], [1], [1], [-54], [21], [-5], [-11], [54], [-8], [27], [37], [-34], [59], [8], [14], [19], [5], [-30], [57], [69], [35], [20], [5], [-5], [31], [-18], [52], [64], [23], [49], [49], [-49], [69], [-41], [55], [-17], [45], [-16], [-29], [15], [5], [57], [-19], [-57], [-40], [-57], [-18], [-15], [-1], [-15], [-6], [33], [-8], [-75], [-38], [25], [33], [5], [31], [78], [31], [-15], [-18], [44], [1], [-55], [-19], [29], [15], [-70], [-45], [29], [-19], [51], [-24], [-13], [52], [16], [26], [-50], [65], [63], [-9], [3], [69], [-79], [-43], [-6], [-40], [-67], [15], [-23], [-11], [-10], [35], [20], [-23], [25], [-12], [-50], [15], [-2], [39], [11], [-30], [61], [27], [-17], [17], [72], [-51], [39], [-80], [82], [3], [17], [-46], [7], [52], [-53], [53], [-27], [-73], [-18], [7], [-21], [75], [24], [-49], [-19], [67], [56], [14], [-75], [54], [-73], [-27], [4], [18], [43], [-69], [9], [68], [-79], [-5], [-53], [-53], [-42], [21], [-11], [90], [-36], [-23], [15], [93], [39], [17], [-63], [13], [-61], [-86], [-51], [-7], [-56], [-54], [51], [-79], [-72], [25], [93], [75], [17], [58], [69], [12], [-54], [-35], [-27], [-91], [70], [76], [-9], [47], [25], [36], [-95], [41], [1], [-73], [94], [76], [35], [19], [-91], [-75], [98], [11], [49], [81], [75], [52], [-5], [-82], [73], [-65], [1], [1], [-30], [21], [60], [-107], [-69], [-17], [-106], [105], [-36], [-47], [-63], [48], [-10], [17], [-95]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3380_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3380_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_3380_2_a_b(:prec:=1)
    chi := MakeCharacter_3380_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(2999) - 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_3380_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_3380_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3380_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<7,R![-3, 1]>,<19,R![-1, 1]>],Snew);
    return Vf;
end function;