
// Make newform 6552.2.a.f in Magma, downloaded from the LMFDB on 29 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6552_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_6552_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_6552_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_6552_a();"
function MakeCharacter_6552_a()
    N := 6552;
    order := 1;
    char_gens := [1639, 3277, 5825, 5617, 2017];
    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_6552_a_Hecke( : Kf := Kf)
    return MakeCharacter_6552_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], [0], [-2], [1], [-4], [-1], [-6], [4], [8], [6], [8], [2], [-2], [-4], [12], [6], [-12], [-6], [0], [-16], [-2], [0], [4], [-2], [-10], [-10], [4], [-8], [-14], [-2], [-8], [-12], [6], [-12], [10], [16], [-14], [-8], [-20], [-18], [0], [10], [8], [10], [-14], [-12], [4], [-8], [4], [-6], [14], [-24], [-10], [4], [10], [16], [14], [8], [6], [22], [28], [30], [-20], [-32], [-6], [-22], [0], [-30], [0], [-30], [-2], [0], [-12], [-34], [24], [-36], [-18], [-14], [-18], [14], [-12], [-22], [32], [-14], [-20], [0], [30], [34], [30], [-24], [-28], [36], [8], [-24], [-24], [40], [-2], [42], [-28], [42], [-12], [2], [-36], [-10], [-44], [-2], [-12], [22], [32], [18], [-28], [42], [-26], [-44], [-48], [-2], [-20], [24], [-18], [-40], [-22], [18], [6], [36], [28], [-2], [34], [-48], [-20], [-14], [16], [40], [-40], [38], [30], [22], [-18], [12], [-2], [30], [-4], [-22], [40], [-44], [-6], [36], [10], [-30], [44], [40], [2], [-30], [36], [-48], [4], [16], [24], [-10], [10], [22], [-36], [-18], [32], [-36], [-18], [-20], [-32], [2], [-22], [-42], [12], [-30], [8], [-30], [-28], [-30], [8], [-30], [-44], [10], [40], [-44], [38], [62], [24], [-42], [38], [-52], [-6], [-36], [2], [-4], [-4], [14], [40], [-54], [-22], [14], [-42], [8], [-22], [28], [58], [-46], [-20], [30], [16], [-8], [54], [20], [-46], [-26], [48], [20], [-36], [46], [-16], [-6], [-72], [-54], [38], [40], [14], [16], [-36], [54], [10], [-16], [52], [68], [30], [-4], [-24], [-42], [20], [4], [22], [-54], [-48], [-24], [-36], [-28], [-76], [34], [6], [48], [-16], [-52], [32], [40], [2], [30], [56], [-46], [30], [4], [62], [-36], [38], [-42], [32], [24], [42], [-70], [-2], [-20], [-6], [54], [-24], [6], [-46], [-56], [-18], [64], [-14], [-8], [12], [-30], [74], [60], [24], [-40], [48], [58], [68], [-24], [-46], [74], [24], [-6], [6], [-12], [-74], [44], [22], [30], [28], [-42], [76], [44], [-70], [66], [24], [0], [52], [2], [64], [-70], [-24], [-74], [48], [26], [-18], [36], [-64], [10], [-60], [-48], [14], [6], [-68], [58], [30], [-64], [54], [34], [80], [-84], [24], [-74], [-62], [-66], [52], [-60], [88], [-28], [42], [78], [86], [4], [2], [-90], [22], [-24], [-94], [8], [-38], [-48], [-12], [-10], [-4], [82], [34], [4], [-82], [-86], [-52], [-28], [38], [56], [2], [-6], [-48], [60], [-60], [10], [46], [24], [-86], [-56], [52], [64], [70], [76], [-26], [-84], [-40], [-66], [90], [10], [-18], [-26], [32], [-66], [-20], [-4], [0], [-30], [32], [-40], [-6], [-66], [-68], [76], [88], [90], [8], [-30], [-76], [-6], [50], [22], [-32], [6], [6], [20], [-22], [46], [-100], [-28], [-22], [14], [24], [4], [-6], [-82], [-32], [-4], [-18], [48], [46], [-70], [-72], [60], [-10], [6], [-64], [-10], [4], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6552_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6552_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_6552_2_a_f(:prec:=1)
    chi := MakeCharacter_6552_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_6552_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_6552_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6552_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![4, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;