
// Make newform 5808.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5808_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_5808_2_a_b();".
// 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_5808_2_a_b();".
// 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_5808_a();"
function MakeCharacter_5808_a()
    N := 5808;
    order := 1;
    char_gens := [3631, 4357, 1937, 2785];
    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_5808_a_Hecke(Kf)
    return MakeCharacter_5808_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], [-1], [-4], [-2], [0], [-4], [2], [0], [6], [-10], [8], [-2], [-2], [4], [2], [4], [0], [8], [12], [-2], [6], [10], [4], [10], [-2], [-2], [-4], [-12], [-20], [-6], [-22], [12], [-2], [0], [10], [2], [18], [-4], [-12], [6], [0], [2], [-22], [-14], [-18], [-20], [12], [16], [-12], [10], [6], [-20], [18], [8], [18], [-16], [-20], [22], [-8], [-22], [4], [-14], [8], [-2], [-6], [-32], [28], [22], [-12], [20], [-6], [-20], [-8], [-4], [20], [6], [-20], [18], [-18], [10], [0], [-18], [32], [-6], [10], [-24], [30], [2], [18], [-4], [-28], [40], [-28], [12], [20], [4], [-20], [-18], [44], [-12], [-32], [-38], [-36], [10], [32], [-2], [-8], [-14], [30], [-42], [-22], [16], [-22], [20], [48], [-18], [-44], [22], [24], [-20], [42], [-14], [22], [16], [-52], [18], [10], [-10], [-28], [-4], [20], [44], [8], [-42], [-22], [50], [4], [-52], [28], [-10], [12], [38], [-24], [-12], [30], [-30], [-24], [22], [20], [-54], [-28], [-38], [-44], [-12], [12], [-2], [-10], [30], [-58], [-42], [-28], [6], [-22], [-32], [38], [46], [8], [52], [10], [-56], [60], [-38], [12], [6], [40], [-30], [-8], [-48], [14], [-60], [32], [12], [54], [-58], [46], [0], [32], [-44], [10], [-48], [-26], [44], [-52], [12], [28], [34], [38], [-46], [-18], [-56], [10], [-18], [18], [-10], [40], [8], [-20], [4], [-30], [28], [-38], [-48], [16], [52], [60], [-58], [-22], [18], [22], [24], [-32], [-30], [-10], [-24], [68], [20], [-46], [-30], [18], [12], [14], [-20], [22], [18], [-44], [8], [50], [-14], [0], [18], [36], [-28], [16], [10], [-14], [20], [-28], [-12], [-40], [-36], [-8], [-2], [2], [-10], [-54], [-60], [22], [-52], [-72], [2], [-26], [68], [60], [-44], [-42], [60], [0], [42], [64], [46], [42], [-68], [-66], [-70], [-18], [36], [12], [-80], [-2], [-28], [-16], [-12], [-12], [28], [48], [-42], [-66], [-58], [-60], [-30], [12], [-8], [26], [-28], [36], [-30], [68], [-36], [20], [-52], [-54], [42], [30], [36], [-12], [58], [12], [-10], [-30], [36], [-76], [-40], [38], [-44], [-72], [-10], [-20], [72], [-46], [30], [-48], [38], [58], [36], [-54], [62], [-60], [-4], [68], [6], [-72], [-12], [70], [84], [52], [12], [-30], [66], [-38], [-22], [-46], [-82], [10], [88], [-36], [20], [-38], [-28], [-28], [-92], [-68], [18], [12], [-6], [-40], [-74], [-90], [92], [-18], [-14], [-48], [-22], [-18], [-60], [12], [-6], [2], [-46], [18], [-72], [20], [24], [-10], [2], [-82], [-60], [-28], [-34], [30], [2], [12], [14], [-62], [-38], [-60], [-54], [48], [-42], [44], [82], [-10], [24], [0], [-68], [18], [26], [-10], [70], [28], [-42], [-40], [54], [38], [78], [90], [-98], [-42], [38], [-4], [-20], [46], [2], [-44], [-48], [102], [-8], [-60], [-28], [-82], [44], [-40], [32], [-18], [-60], [54], [28], [-4], [-90], [28], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5808_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5808_2_a_b();".
// 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_5808_2_a_b(:prec:=1)
    chi := MakeCharacter_5808_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_5808_2_a_b();".
// 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_5808_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5808_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 1]>,<7,R![2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;