
// Make newform 3654.2.a.f in Magma, downloaded from the LMFDB on 27 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3654_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_3654_2_a_f();".
// 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_3654_2_a_f();".
// 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_3654_a();"
function MakeCharacter_3654_a()
    N := 3654;
    order := 1;
    char_gens := [407, 2089, 379];
    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_3654_a_Hecke( : Kf := Kf)
    return MakeCharacter_3654_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 := [[-1], [0], [0], [-1], [4], [-4], [4], [8], [8], [-1], [2], [-6], [0], [-12], [-6], [6], [-6], [6], [8], [8], [-12], [8], [6], [8], [-8], [10], [4], [8], [-6], [-6], [20], [0], [-2], [22], [-10], [8], [18], [-12], [0], [16], [12], [-4], [-20], [18], [14], [4], [-12], [20], [-14], [26], [-26], [0], [10], [8], [-6], [4], [10], [-2], [-2], [18], [-14], [30], [-12], [22], [10], [26], [-36], [-10], [-28], [-24], [-26], [24], [22], [14], [-12], [-24], [-2], [20], [30], [-32], [22], [-22], [0], [4], [-16], [4], [34], [-26], [10], [-8], [-16], [-6], [40], [12], [4], [14], [28], [2], [14], [-2], [-8], [30], [0], [-34], [12], [-20], [42], [30], [12], [-28], [-34], [-38], [-18], [0], [48], [30], [6], [20], [-14], [-44], [-20], [-34], [14], [-36], [-18], [-22], [-30], [-28], [-18], [46], [-20], [-28], [-8], [-22], [-42], [40], [-34], [-30], [-18], [42], [-54], [-50], [-44], [28], [-2], [6], [46], [-34], [28], [-16], [10], [-4], [-48], [18], [-20], [-24], [-56], [6], [50], [48], [44], [10], [32], [24], [10], [30], [16], [-26], [22], [10], [-44], [32], [-48], [-18], [-40], [54], [16], [38], [58], [16], [-24], [36], [26], [10], [-40], [-24], [-38], [28], [14], [-60], [38], [-12], [20], [-30], [28], [2], [-50], [-6], [10], [-64], [-6], [-8], [18], [36], [48], [-60], [-42], [-24], [-54], [8], [14], [8], [8], [-44], [6], [-14], [8], [-24], [0], [-26], [-6], [-12], [-22], [-20], [-2], [14], [-24], [-16], [54], [-60], [-50], [-22], [-68], [42], [-74], [54], [-16], [6], [48], [-50], [76], [22], [-8], [18], [-58], [24], [76], [62], [40], [-40], [26], [-42], [-24], [32], [42], [-20], [14], [64], [-48], [-78], [-76], [20], [48], [10], [-48], [-8], [78], [12], [52], [18], [-24], [4], [-58], [-20], [-40], [-26], [76], [-34], [34], [22], [-24], [-8], [44], [0], [-4], [48], [-6], [-30], [-40], [74], [-26], [78], [-26], [-76], [-6], [0], [34], [-60], [74], [-36], [-40], [-46], [20], [36], [52], [34], [84], [-24], [-72], [14], [12], [42], [-34], [0], [-64], [-22], [-72], [-16], [54], [6], [-68], [54], [36], [-76], [14], [12], [-40], [-34], [60], [-14], [-38], [-30], [12], [-8], [-28], [82], [-38], [56], [-32], [48], [-86], [86], [-30], [80], [2], [20], [-10], [-20], [-42], [-26], [-46], [6], [50], [-44], [66], [68], [-90], [-10], [-54], [-80], [6], [34], [0], [48], [-8], [54], [-42], [-48], [10], [-68], [50], [-72], [66], [28], [38], [8], [12], [-36], [-94], [26], [-6], [6], [40], [-30], [86], [-44], [-92], [32], [-76], [58], [6], [16], [-20], [8], [-40], [-70], [-4], [0], [48], [-102], [4], [-54], [8], [38], [8], [56], [-2], [-46], [80], [70], [-50], [-78], [-24], [-4], [-82], [-54], [24], [16], [-88], [-18], [-46], [-66], [-92], [4], [38], [-80], [-16], [22], [-102], [6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3654_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3654_2_a_f();".
// 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_3654_2_a_f(:prec:=1)
    chi := MakeCharacter_3654_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_3654_2_a_f();".
// 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_3654_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3654_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-4, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;