
// Make newform 5775.2.a.f in Magma, downloaded from the LMFDB on 31 December 2025.
// 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_f();".
// 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_f();".
// 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 := 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]];
    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], [2], [-6], [4], [4], [10], [-4], [2], [10], [-12], [0], [-10], [-12], [-2], [-4], [8], [14], [4], [-8], [6], [10], [-2], [-8], [-4], [14], [18], [-8], [-4], [10], [-4], [18], [-4], [14], [20], [12], [-26], [-4], [-10], [16], [-14], [-26], [4], [-16], [24], [-8], [14], [26], [24], [2], [12], [2], [24], [10], [-8], [-2], [30], [28], [-18], [-12], [8], [-30], [-18], [-12], [-22], [28], [30], [18], [24], [8], [6], [4], [32], [22], [6], [18], [-22], [4], [6], [40], [2], [-8], [8], [18], [18], [30], [-16], [4], [-8], [8], [-12], [4], [20], [-6], [22], [4], [-18], [-44], [-18], [24], [-26], [8], [2], [-12], [2], [-32], [-6], [32], [-2], [18], [0], [8], [-30], [-36], [0], [30], [36], [38], [26], [6], [40], [-8], [-14], [6], [-48], [8], [-38], [0], [-24], [8], [18], [-6], [-30], [6], [-28], [46], [-18], [-52], [10], [32], [-4], [14], [0], [34], [-46], [48], [20], [-42], [14], [12], [-52], [-12], [-48], [20], [6], [38], [14], [24], [-38], [-8], [-44], [26], [32], [-16], [2], [-30], [-10], [-60], [22], [-48], [-6], [-28], [54], [-48], [-2], [0], [-34], [-56], [-4], [50], [42], [12], [-22], [46], [-4], [34], [-32], [42], [44], [-60], [-6], [-36], [58], [26], [-46], [-62], [12], [-6], [-8], [38], [-30], [44], [14], [28], [36], [-50], [8], [-14], [-62], [-40], [-28], [-32], [-62], [-16], [34], [-20], [14], [-50], [-16], [18], [-40], [24], [-18], [-6], [40], [0], [-20], [-6], [-68], [28], [30], [36], [-28], [10], [6], [12], [0], [-24], [28], [-72], [14], [-54], [-48], [-40], [60], [64], [64], [-10], [66], [-52], [-46], [-66], [-12], [54], [-52], [-26], [-34], [-8], [4], [-74], [58], [2], [0], [-42], [54], [12], [-26], [22], [-60], [-78], [12], [22], [16], [56], [-50], [-14], [44], [-36], [48], [12], [54], [4], [8], [30], [-74], [-44], [42], [-18], [20], [58], [-36], [-26], [-50], [-76], [6], [-76], [-52], [-10], [-18], [-44], [16], [76], [62], [16], [-42], [72], [22], [36], [30], [-10], [36], [8], [58], [-60], [-24], [26], [14], [-20], [54], [-42], [56], [-6], [66], [-28], [-28], [48], [-10], [-10], [-34], [-16], [-80], [-8], [60], [30], [-70], [74], [-32], [-46], [34], [78], [-8], [46], [68], [-18], [12], [-40], [-74], [-52], [14], [-58], [72], [22], [66], [24], [12], [66], [68], [-82], [-78], [-52], [-4], [-20], [-10], [-2], [-64], [34], [12], [-32], [16], [-6], [88], [2], [-4], [16], [86], [-6], [-42], [42], [42], [88], [66], [20], [-32], [40], [-10], [-4], [-72], [34], [-58], [-4], [-4], [16], [-22], [0], [30], [76], [34], [62], [-30], [56], [10], [-18], [0], [34], [-50], [-36], [-60], [14], [-74], [32], [96], [50], [82], [-64], [80], [2], [-12], [-74], [74], [60], [-12], [-94], [-18], [32], [30], [80], [-8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5775_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5775_2_a_f();".
// 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_f(: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_f();".
// 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_f( : 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![-2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;