
// Make newform 5400.2.a.x in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5400_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_5400_2_a_x();".
// 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_5400_2_a_x();".
// 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_5400_a();"
function MakeCharacter_5400_a()
    N := 5400;
    order := 1;
    char_gens := [1351, 2701, 1001, 2377];
    v := [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_5400_a_Hecke( : Kf := Kf)
    return MakeCharacter_5400_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], [0], [0], [2], [0], [-3], [-1], [-3], [4], [-5], [-10], [6], [6], [-8], [-3], [0], [5], [2], [2], [-6], [-11], [-9], [10], [-8], [-12], [12], [4], [9], [6], [-22], [10], [7], [-20], [2], [-8], [-18], [2], [17], [19], [-24], [17], [-6], [20], [-5], [-12], [-3], [-2], [-17], [-13], [10], [-30], [-17], [-20], [23], [-12], [18], [3], [-22], [-28], [-2], [11], [-26], [20], [20], [-11], [8], [-8], [12], [-7], [10], [-6], [-16], [-4], [-23], [-3], [18], [4], [34], [-31], [-6], [-1], [8], [24], [7], [-29], [12], [16], [-34], [-14], [7], [10], [2], [42], [43], [-35], [28], [30], [28], [-18], [-18], [-18], [-28], [24], [-9], [46], [-23], [3], [-40], [17], [-10], [-16], [25], [-4], [-19], [0], [-40], [-39], [-11], [4], [6], [8], [-50], [23], [13], [-42], [26], [18], [44], [8], [-19], [16], [53], [50], [44], [21], [-3], [38], [23], [-18], [-52], [-34], [6], [21], [34], [-20], [-8], [1], [37], [-23], [-22], [-4], [-16], [33], [44], [-36], [-40], [6], [12], [10], [3], [26], [42], [-6], [50], [-11], [1], [18], [55], [39], [-20], [31], [0], [-44], [33], [-16], [-5], [-44], [34], [10], [10], [-28], [-10], [42], [35], [60], [44], [4], [5], [44], [-46], [57], [33], [-8], [15], [-37], [17], [28], [-2], [0], [-2], [-16], [2], [51], [12], [-50], [-8], [-45], [-36], [-68], [32], [-18], [-32], [-28], [-12], [9], [58], [56], [-63], [-39], [-22], [19], [-44], [-34], [19], [-45], [-21], [66], [26], [-12], [22], [35], [32], [58], [76], [3], [-43], [-29], [74], [46], [4], [55], [58], [17], [58], [-42], [-28], [72], [-44], [-29], [42], [42], [15], [37], [-71], [0], [5], [-8], [-7], [8], [-58], [9], [-41], [-12], [-9], [28], [24], [42], [-74], [-18], [-69], [-70], [-10], [67], [-46], [-8], [-39], [-31], [49], [6], [24], [23], [-3], [-49], [68], [-44], [-30], [6], [-39], [-40], [-40], [-31], [-42], [-30], [34], [0], [67], [-54], [48], [-4], [62], [9], [0], [-9], [13], [2], [76], [-61], [78], [58], [39], [-80], [76], [-40], [79], [-9], [-48], [-70], [-4], [-62], [57], [-36], [-8], [-44], [33], [-26], [-43], [-4], [-8], [15], [-22], [-90], [-40], [48], [4], [60], [-25], [-51], [42], [-66], [4], [53], [-24], [-75], [78], [-70], [-61], [0], [-24], [-37], [-12], [-26], [-20], [-44], [-50], [-59], [0], [-24], [46], [-19], [20], [-6], [-12], [46], [-32], [48], [41], [50], [-54], [34], [-97], [67], [-46], [43], [28], [42], [-30], [-2], [-54], [-26], [-8], [11], [28], [89], [8], [-64], [-55], [-32], [46], [-64], [-70], [-79], [50], [-82], [26], [-46], [-35], [-60], [-63], [-96], [10], [-74], [-88], [-54], [96], [-25], [-94], [90], [-56], [68], [26], [-37], [85], [43], [-54], [-16], [6], [-16], [-37], [-25], [-80], [-54], [47], [42], [14], [-69], [-79], [-26], [-31], [86]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5400_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5400_2_a_x();".
// 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_5400_2_a_x(:prec:=1)
    chi := MakeCharacter_5400_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_5400_2_a_x();".
// 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_5400_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-2, 1]>,<13,R![0, 1]>,<17,R![3, 1]>],Snew);
    return Vf;
end function;