
// Make newform 2548.2.a.a in Magma, downloaded from the LMFDB on 10 March 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2548_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_2548_2_a_a();".
// 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_2548_2_a_a();".
// 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_2548_a();"
function MakeCharacter_2548_a()
    N := 2548;
    order := 1;
    char_gens := [1275, 885, 197];
    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_2548_a_Hecke(Kf)
    return MakeCharacter_2548_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], [-3], [-3], [0], [-5], [-1], [-1], [-7], [-1], [-2], [-5], [-11], [-6], [-12], [-7], [11], [3], [-5], [-1], [0], [-11], [-5], [4], [5], [2], [3], [3], [-5], [-15], [-2], [-12], [11], [21], [16], [1], [-9], [7], [1], [-16], [15], [-23], [10], [23], [13], [-6], [-11], [-20], [0], [3], [-11], [3], [-24], [-15], [-4], [19], [9], [-5], [-19], [11], [-10], [-3], [-18], [32], [-31], [-21], [5], [-15], [-18], [-23], [-10], [-15], [-19], [17], [7], [20], [21], [23], [9], [-31], [-3], [-36], [22], [39], [2], [7], [27], [-6], [17], [-18], [-24], [-21], [-25], [7], [-20], [1], [8], [-15], [7], [-1], [-3], [4], [9], [-43], [-21], [-23], [-7], [36], [-19], [-27], [14], [7], [-35], [42], [-9], [16], [15], [36], [33], [15], [-28], [13], [-30], [-33], [-9], [-3], [-6], [29], [15], [-40], [-19], [11], [0], [-5], [42], [-15], [14], [13], [35], [-30], [-45], [28], [-3], [9], [12], [27], [-16], [-34], [11], [-1], [9], [13], [-18], [-36], [-45], [-19], [52], [7], [-59], [22], [-51], [41], [-18], [-28], [27], [13], [-17], [29], [-45], [-6], [15], [19], [-42], [-21], [9], [49], [46], [-16], [-39], [32], [-9], [13], [-4], [26], [57], [39], [-49], [-37], [31], [-35], [63], [-49], [-48], [59], [-7], [-15], [-65], [-43], [-5], [-30], [-9], [21], [-24], [-43], [-49], [-12], [67], [-11], [-21], [54], [-67], [15], [10], [-16], [-43], [-17], [-55], [-31], [35], [41], [70], [-33], [-40], [17], [13], [-48], [2], [15], [9], [43], [-11], [-69], [-31], [16], [-5], [-12], [11], [41], [5], [48], [0], [-49], [7], [-31], [-35], [-54], [-41], [12], [-33], [47], [-36], [6], [25], [15], [46], [-13], [33], [-25], [-1], [42], [-75], [65], [-76], [-55], [30], [13], [-9], [-46], [2], [-48], [-21], [-17], [-5], [-47], [31], [2], [-27], [-43], [49], [-59], [-41], [-7], [-29], [-60], [-6], [9], [75], [-21], [-50], [55], [-30], [35], [53], [-39], [12], [10], [-45], [43], [2], [35], [28], [-65], [57], [9], [-12], [-9], [-62], [33], [-50], [29], [-65], [35], [-31], [11], [73], [84], [-53], [16], [-33], [-14], [54], [73], [-11], [58], [48], [13], [23], [-39], [-83], [-33], [90], [-55], [-41], [68], [-61], [-61], [-68], [-26], [-63], [58], [-37], [-11], [-18], [-66], [-64], [69], [36], [55], [-3], [36], [-13], [-43], [-63], [-34], [53], [-1], [6], [-3], [43], [-37], [72], [6], [-33], [53], [-35], [-79], [67], [94], [-89], [-74], [-15], [-5], [13], [66], [-79], [19], [-49], [24], [-67], [55], [-26], [41], [74], [-48], [-27], [-20], [39], [-89], [-85], [-59], [-16], [-82], [-39], [-81], [-41], [-15], [91], [47], [-10], [-40], [89], [73], [-21], [-23], [-3], [5], [-25], [-23], [62], [-69], [81], [-1], [47], [36], [99], [-34], [-41], [-15], [-35], [90], [-1], [-57], [93], [-16], [-12], [-74], [-79], [-33], [-26], [19], [105]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2548_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2548_2_a_a();".
// 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_2548_2_a_a(:prec:=1)
    chi := MakeCharacter_2548_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_2548_2_a_a();".
// 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_2548_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2548_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![3, 1]>],Snew);
    return Vf;
end function;