
// Make newform 5775.2.a.t in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5775_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_5775_2_a_t();".
// 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_5775_2_a_t();".
// 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_5775_a();"
function MakeCharacter_5775_a()
    N := 5775;
    order := 1;
    char_gens := [3851, 4852, 4126, 3676];
    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_5775_a_Hecke(Kf)
    return MakeCharacter_5775_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 := [[1], [1], [0], [-1], [-1], [-6], [-2], [4], [0], [-2], [8], [-6], [10], [4], [8], [-6], [4], [-10], [12], [0], [-2], [16], [-4], [18], [-2], [6], [0], [12], [-2], [-18], [16], [4], [6], [-20], [-10], [8], [2], [12], [8], [-14], [12], [-10], [-24], [-2], [26], [16], [-12], [24], [12], [-26], [-26], [-24], [26], [-12], [6], [0], [-10], [16], [-22], [10], [4], [10], [-20], [32], [6], [18], [28], [-2], [-4], [-10], [22], [0], [8], [-22], [28], [-8], [6], [-14], [34], [-14], [12], [6], [-24], [30], [8], [12], [34], [22], [-34], [32], [-12], [0], [-8], [20], [20], [24], [6], [2], [4], [-34], [12], [18], [12], [-22], [-20], [14], [-36], [14], [16], [-14], [0], [10], [6], [20], [-8], [18], [-28], [-32], [18], [12], [-10], [-18], [-6], [-36], [-20], [30], [38], [-40], [-32], [-6], [-44], [48], [-16], [10], [10], [-22], [-30], [12], [-6], [-38], [-4], [-42], [40], [44], [-50], [0], [34], [-10], [52], [40], [34], [42], [-36], [8], [20], [40], [24], [-54], [-50], [14], [36], [22], [-8], [-12], [30], [16], [-32], [-46], [50], [50], [-28], [-34], [0], [-26], [-40], [50], [28], [-58], [-8], [-42], [-64], [20], [-38], [38], [56], [14], [2], [-44], [-6], [-32], [-34], [12], [52], [6], [-28], [-50], [34], [34], [46], [-40], [14], [-16], [42], [-38], [-12], [26], [-24], [20], [58], [-12], [-66], [30], [-8], [-52], [-24], [10], [32], [50], [32], [34], [22], [56], [34], [32], [44], [-26], [-2], [24], [24], [52], [-14], [-76], [0], [-6], [-4], [0], [-14], [42], [-28], [48], [4], [-4], [0], [14], [78], [8], [-72], [12], [-4], [56], [-30], [-30], [32], [-6], [66], [-4], [22], [-12], [18], [-18], [32], [36], [-18], [-6], [-42], [44], [-66], [-46], [36], [-38], [78], [-52], [38], [-32], [-10], [-16], [28], [30], [-54], [20], [-32], [-8], [-72], [46], [20], [40], [-82], [10], [48], [-62], [30], [-28], [22], [12], [-46], [78], [-8], [-30], [-36], [28], [-50], [18], [-80], [52], [-20], [-2], [60], [30], [0], [-70], [-48], [-74], [-78], [44], [-48], [-62], [60], [-40], [-2], [-30], [20], [6], [14], [48], [6], [-62], [20], [44], [-88], [-38], [-18], [2], [16], [60], [44], [-20], [-2], [-66], [26], [48], [-54], [-42], [-42], [88], [34], [76], [-26], [-12], [80], [-78], [20], [54], [-34], [-64], [54], [-42], [56], [12], [-2], [0], [-6], [74], [72], [20], [20], [-10], [66], [72], [-38], [-20], [-12], [-24], [22], [8], [98], [60], [72], [38], [-30], [-66], [-58], [54], [24], [78], [-60], [-96], [-64], [-38], [-44], [56], [66], [18], [100], [-12], [-96], [54], [16], [18], [28], [22], [-10], [-2], [-32], [78], [54], [-56], [18], [34], [20], [12], [-26], [-22], [-20], [-76], [-26], [6], [56], [-16], [-58], [24], [62], [2], [-8], [-36], [86], [-70], [68], [-54], [52], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5775_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5775_2_a_t();".
// 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_5775_2_a_t(:prec:=1)
    chi := MakeCharacter_5775_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_5775_2_a_t();".
// 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_5775_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5775_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<13,R![6, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;