
// Make newform 5472.2.a.q in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5472_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_5472_2_a_q();".
// 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_5472_2_a_q();".
// 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_5472_a();"
function MakeCharacter_5472_a()
    N := 5472;
    order := 1;
    char_gens := [4447, 2053, 1217, 3745];
    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_5472_a_Hecke(Kf)
    return MakeCharacter_5472_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+1)]];
    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], [1], [1], [3], [-4], [3], [-1], [-8], [0], [-2], [-8], [0], [-11], [-7], [-2], [6], [-1], [10], [2], [5], [2], [0], [-6], [-12], [-14], [14], [10], [-2], [-12], [-16], [13], [-5], [5], [15], [18], [2], [4], [12], [-18], [-16], [6], [19], [-14], [6], [-7], [22], [-28], [-16], [-11], [29], [-5], [4], [-11], [20], [-11], [18], [-24], [-15], [2], [-15], [16], [30], [-15], [-26], [10], [26], [-8], [25], [-7], [-26], [27], [-4], [-24], [10], [-18], [-19], [-23], [30], [-22], [36], [30], [-28], [-28], [28], [9], [12], [-1], [-33], [-17], [33], [-32], [-38], [4], [-29], [-24], [-28], [-42], [26], [19], [4], [7], [4], [24], [-40], [15], [-3], [-6], [-30], [26], [-8], [9], [27], [-4], [25], [-28], [29], [27], [-25], [30], [48], [-10], [4], [16], [-45], [50], [-26], [-41], [7], [42], [-23], [6], [20], [-41], [-45], [-29], [46], [44], [22], [29], [-16], [-9], [9], [-30], [20], [22], [2], [26], [5], [-4], [-46], [7], [-7], [18], [30], [-54], [16], [10], [13], [-18], [-4], [-18], [-16], [-52], [2], [36], [8], [61], [-18], [-21], [12], [14], [28], [-50], [-28], [53], [20], [23], [8], [35], [-17], [12], [14], [46], [-41], [-5], [6], [-50], [60], [12], [32], [-21], [-10], [-44], [28], [24], [45], [-14], [-54], [29], [-12], [48], [58], [40], [20], [-45], [47], [-12], [-57], [62], [-26], [38], [29], [-24], [16], [8], [0], [-4], [42], [-39], [-62], [-40], [28], [19], [-18], [-17], [-46], [-54], [-30], [15], [11], [74], [-42], [18], [-33], [-57], [73], [45], [-36], [38], [-2], [-25], [-16], [38], [0], [-27], [-57], [18], [-40], [45], [-34], [29], [-12], [-6], [-3], [36], [-43], [8], [-12], [-38], [76], [6], [35], [-62], [33], [-46], [6], [-3], [-34], [21], [-50], [-52], [33], [45], [-82], [80], [-41], [14], [-70], [19], [74], [-48], [37], [2], [77], [-45], [19], [64], [41], [80], [2], [-13], [16], [-70], [-58], [30], [48], [81], [54], [35], [-39], [34], [-79], [70], [-25], [14], [-28], [76], [-49], [34], [59], [-66], [-44], [-16], [-40], [-2], [-63], [-34], [55], [39], [-40], [5], [-54], [22], [-58], [22], [-32], [4], [24], [-45], [-41], [-90], [-8], [-77], [-63], [13], [34], [-64], [-1], [-44], [-30], [-49], [36], [-10], [-54], [-44], [-78], [-60], [16], [-25], [88], [68], [21], [-54], [-66], [44], [-24], [-16], [-69], [48], [-55], [-22], [-12], [-30], [-21], [44], [-54], [94], [-70], [92], [-40], [81], [16], [29], [25], [-44], [0], [-69], [29], [-20], [-74], [5], [-33], [-11], [-28], [-80], [44], [6], [-69], [100], [46], [-70], [-68], [-65], [-58], [-18], [-34], [-44], [28], [-18], [-50], [73], [-48], [-74], [74], [31], [5], [32], [48], [-15], [-86], [-35], [-46], [47], [-6], [42], [100], [-10], [-42], [34], [46], [-8], [-63], [12], [-4], [84], [-48], [25], [-67], [101]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5472_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5472_2_a_q();".
// 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_5472_2_a_q(:prec:=1)
    chi := MakeCharacter_5472_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_5472_2_a_q();".
// 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_5472_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5472_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<7,R![-1, 1]>,<11,R![-3, 1]>,<13,R![4, 1]>,<23,R![8, 1]>],Snew);
    return Vf;
end function;