
// Make newform 5445.2.a.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5445_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_5445_2_a_g();".
// 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_5445_2_a_g();".
// 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_5445_a();"
function MakeCharacter_5445_a()
    N := 5445;
    order := 1;
    char_gens := [3026, 4357, 3511];
    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_5445_a_Hecke(Kf)
    return MakeCharacter_5445_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], [0], [-2], [6], [7], [6], [0], [-1], [-7], [-6], [-8], [0], [6], [12], [1], [-7], [-6], [13], [-11], [0], [18], [-1], [-12], [-1], [12], [7], [12], [-17], [18], [12], [4], [6], [16], [17], [-1], [-6], [-6], [12], [17], [-12], [-5], [-24], [11], [-23], [29], [-12], [14], [24], [-12], [22], [-6], [6], [-6], [24], [-20], [-11], [-30], [1], [18], [13], [30], [-22], [6], [29], [13], [-30], [-35], [6], [6], [-4], [31], [8], [-6], [-30], [-7], [-18], [-5], [0], [-10], [-6], [5], [19], [18], [6], [10], [30], [32], [42], [24], [8], [42], [23], [-24], [-24], [0], [25], [-14], [-8], [-42], [36], [0], [-23], [-43], [-36], [-42], [-42], [-11], [-32], [-5], [6], [20], [8], [-30], [29], [24], [42], [6], [-1], [-29], [48], [30], [-13], [-6], [-34], [24], [8], [22], [19], [-36], [-37], [-25], [-6], [25], [-12], [28], [-18], [30], [-11], [-42], [-13], [12], [-7], [0], [-5], [54], [-49], [36], [-5], [0], [-25], [-6], [35], [12], [-11], [-30], [43], [30], [12], [-6], [-23], [-18], [18], [30], [-40], [-41], [-14], [0], [42], [-1], [-24], [-38], [-16], [-24], [37], [6], [28], [25], [35], [-54], [50], [24], [36], [-24], [-11], [17], [-26], [-66], [-61], [-36], [-25], [-36], [18], [6], [-14], [11], [-24], [42], [-24], [-17], [-13], [-29], [0], [66], [-7], [6], [36], [11], [43], [-24], [35], [0], [30], [-10], [-17], [54], [-54], [-24], [-5], [40], [-48], [53], [-54], [22], [-24], [-36], [40], [-30], [-46], [19], [-53], [-72], [-25], [-12], [-31], [54], [-6], [-30], [24], [28], [23], [11], [42], [6], [68], [42], [40], [-6], [-47], [48], [-24], [5], [30], [0], [14], [28], [18], [-53], [16], [6], [-53], [-29], [36], [41], [-42], [48], [-65], [-60], [26], [65], [-13], [-20], [49], [47], [-36], [-26], [-47], [12], [-72], [-25], [-18], [-50], [61], [24], [-61], [24], [-7], [54], [-12], [-36], [-54], [-42], [10], [-30], [-55], [60], [36], [7], [-41], [36], [49], [-6], [-76], [-25], [-18], [5], [-30], [79], [-48], [-72], [72], [8], [12], [7], [-42], [12], [47], [-60], [-11], [29], [-66], [-61], [-42], [-31], [20], [11], [30], [-30], [-5], [-84], [-59], [-18], [-20], [6], [47], [42], [-67], [-71], [38], [-12], [42], [-37], [78], [-54], [65], [17], [-66], [48], [-44], [-25], [-66], [64], [61], [-24], [12], [-84], [-42], [90], [19], [12], [12], [66], [29], [2], [-30], [-56], [-47], [-18], [47], [24], [-36], [40], [47], [-48], [12], [-53], [-42], [-71], [-18], [-36], [-41], [-54], [-5], [66], [8], [-94], [-92], [-18], [53], [-18], [48], [-73], [90], [-62], [-29], [96], [-103], [-30], [94], [48], [-104], [42], [54], [-20], [-73], [-24], [56], [54], [-26], [-48], [6], [31], [82], [36], [-84], [-73], [66], [96], [-24], [7], [42], [12], [-19], [84], [48], [48], [-43], [-6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5445_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5445_2_a_g();".
// 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_5445_2_a_g(:prec:=1)
    chi := MakeCharacter_5445_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_5445_2_a_g();".
// 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_5445_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5445_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![-1, 1]>,<23,R![-6, 1]>,<53,R![-6, 1]>],Snew);
    return Vf;
end function;