
// Make newform 3360.2.a.p in Magma, downloaded from the LMFDB on 31 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3360_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_3360_2_a_p();".
// 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_3360_2_a_p();".
// 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_3360_a();"
function MakeCharacter_3360_a()
    N := 3360;
    order := 1;
    char_gens := [1471, 421, 1121, 2017, 1921];
    v := [1, 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_3360_a_Hecke( : Kf := Kf)
    return MakeCharacter_3360_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]];
    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], [1], [-1], [1], [-4], [-6], [6], [0], [4], [-6], [0], [2], [2], [-4], [-4], [-6], [-12], [-10], [-4], [-8], [-14], [-8], [12], [-6], [-14], [2], [0], [20], [-2], [-10], [-8], [20], [-2], [8], [-6], [-8], [-6], [-12], [20], [-14], [20], [-2], [-24], [10], [-6], [16], [8], [-16], [12], [-10], [6], [0], [-30], [20], [-18], [4], [26], [-8], [-14], [-14], [28], [-22], [-20], [0], [34], [18], [-32], [10], [-4], [22], [6], [8], [8], [-6], [32], [20], [-6], [2], [-6], [10], [12], [30], [0], [-22], [-16], [-28], [10], [-6], [18], [8], [-12], [24], [32], [36], [-24], [-12], [42], [-6], [-20], [-34], [-44], [-46], [20], [18], [-16], [34], [36], [-18], [24], [10], [-48], [-14], [22], [-40], [-16], [-14], [28], [-36], [-46], [-4], [46], [-38], [-6], [12], [8], [42], [14], [-48], [48], [-14], [-8], [-52], [48], [34], [10], [34], [-6], [-12], [34], [42], [16], [34], [56], [-4], [-10], [16], [42], [-18], [8], [-12], [-14], [-30], [-12], [-12], [-44], [0], [40], [-14], [-54], [-30], [-20], [46], [-24], [36], [-50], [4], [8], [50], [-14], [10], [4], [-2], [0], [-14], [24], [10], [-24], [26], [56], [-2], [-16], [-12], [-62], [-58], [-52], [42], [2], [-20], [10], [-32], [42], [4], [40], [-62], [-12], [38], [34], [18], [-2], [-28], [-30], [16], [50], [2], [20], [18], [40], [-60], [18], [-40], [-22], [42], [-32], [60], [56], [-6], [-24], [-30], [-28], [2], [-50], [0], [-62], [-56], [20], [-34], [-42], [40], [-32], [-20], [-62], [56], [-8], [-30], [-68], [52], [-46], [-54], [-12], [48], [4], [-24], [56], [14], [-10], [8], [-8], [-36], [8], [36], [10], [-22], [12], [-22], [18], [-36], [-10], [-44], [-6], [2], [64], [28], [54], [-78], [-34], [64], [-46], [50], [-44], [-70], [46], [-44], [34], [8], [26], [-48], [12], [46], [74], [-76], [-20], [8], [12], [-74], [-20], [72], [34], [42], [64], [-70], [34], [-60], [-74], [60], [42], [50], [-80], [26], [-76], [28], [50], [-46], [0], [76], [-16], [2], [-60], [-2], [0], [10], [12], [-30], [-86], [-68], [-12], [26], [-60], [-8], [58], [18], [-88], [-46], [42], [40], [-74], [2], [64], [84], [84], [-38], [38], [-62], [-8], [44], [8], [68], [54], [6], [26], [-8], [-86], [22], [-46], [40], [18], [-44], [-42], [-60], [-16], [58], [-64], [-6], [-70], [-16], [-10], [78], [-48], [12], [54], [-76], [10], [34], [-60], [-60], [-28], [58], [-78], [64], [-22], [-36], [88], [-84], [-30], [8], [74], [-44], [0], [-14], [-14], [26], [34], [78], [-8], [14], [-24], [28], [-96], [74], [-4], [-28], [-46], [-38], [-92], [-92], [16], [42], [-96], [-30], [32], [10], [-98], [86], [88], [-58], [-62], [-16], [10], [-22], [76], [28], [98], [-22], [-36], [64], [-30], [10], [88], [24], [30], [-36], [-78], [-14], [20], [28], [-62], [98], [-68], [10], [-56], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3360_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3360_2_a_p();".
// 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_3360_2_a_p(:prec:=1)
    chi := MakeCharacter_3360_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_3360_2_a_p();".
// 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_3360_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 1]>,<13,R![6, 1]>,<17,R![-6, 1]>,<19,R![0, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;