
// Make newform 5800.2.a.c in Magma, downloaded from the LMFDB on 18 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5800_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_5800_2_a_c();".
// 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_5800_2_a_c();".
// 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_5800_a();"
function MakeCharacter_5800_a()
    N := 5800;
    order := 1;
    char_gens := [4351, 2901, 4177, 3801];
    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_5800_a_Hecke( : Kf := Kf)
    return MakeCharacter_5800_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 := [[0], [-2], [0], [2], [4], [4], [-4], [-4], [6], [-1], [0], [8], [10], [-2], [6], [4], [-4], [-2], [-2], [0], [12], [-8], [-14], [-10], [12], [6], [6], [-10], [10], [-12], [-10], [-12], [-12], [-12], [18], [-16], [24], [14], [2], [20], [12], [-10], [24], [4], [-12], [0], [-20], [14], [22], [10], [8], [-8], [-6], [20], [-8], [10], [18], [8], [-4], [18], [-6], [24], [26], [24], [16], [32], [-36], [-20], [30], [34], [-16], [-8], [-2], [-4], [-4], [-34], [26], [-4], [-30], [38], [4], [22], [-16], [-4], [-16], [-34], [14], [-8], [14], [22], [-22], [-8], [2], [12], [-20], [-30], [-30], [10], [2], [14], [-2], [-12], [46], [22], [-4], [-28], [30], [16], [16], [-38], [22], [-12], [-12], [44], [8], [-14], [-14], [42], [8], [28], [46], [-40], [48], [-30], [-52], [6], [26], [24], [22], [-48], [28], [34], [-16], [-8], [42], [30], [0], [38], [0], [-10], [-36], [46], [26], [18], [34], [0], [-24], [0], [-36], [14], [20], [-30], [-6], [-18], [-30], [-48], [-16], [22], [-8], [-18], [-6], [24], [-26], [-36], [-8], [-14], [-8], [8], [-34], [24], [-36], [-34], [-32], [-12], [8], [-26], [52], [54], [-6], [42], [6], [36], [-4], [0], [-2], [10], [-32], [-26], [-58], [-48], [-56], [14], [20], [-50], [2], [0], [-62], [20], [-24], [6], [34], [16], [-16], [14], [-60], [-52], [-16], [58], [-10], [68], [-4], [-66], [-54], [-54], [32], [50], [2], [66], [-6], [-24], [-26], [-56], [-18], [34], [6], [10], [4], [72], [30], [4], [-56], [-44], [-56], [-6], [2], [-18], [-2], [40], [36], [40], [-10], [-28], [-74], [-62], [16], [40], [10], [-20], [44], [-30], [56], [18], [-26], [-26], [0], [-20], [38], [42], [12], [-48], [2], [-62], [-38], [-24], [-52], [36], [-14], [58], [46], [68], [-18], [54], [32], [-24], [68], [26], [-54], [2], [-54], [-36], [70], [-40], [82], [38], [-46], [-16], [28], [-48], [-72], [86], [86], [-74], [48], [36], [-32], [-70], [0], [-12], [28], [74], [4], [-60], [-16], [-58], [-52], [72], [-78], [18], [0], [76], [6], [10], [82], [-62], [26], [-26], [-60], [-32], [-40], [46], [-20], [32], [22], [-62], [72], [2], [-20], [-46], [70], [-92], [-18], [12], [-48], [38], [-20], [78], [26], [36], [42], [10], [24], [-64], [-54], [-56], [-12], [-44], [-90], [50], [72], [-64], [28], [-8], [-66], [70], [-22], [-20], [48], [12], [68], [-66], [4], [-54], [78], [36], [82], [4], [64], [-50], [-70], [-44], [36], [-18], [42], [80], [52], [12], [8], [68], [-42], [-80], [-66], [96], [-10], [4], [-28], [22], [-88], [-28], [-58], [-42], [-50], [-60], [68], [-30], [32], [0], [0], [-42], [4], [38], [-70], [-52], [-50], [32], [-78], [-88], [-28], [66], [54], [-84], [-104], [60], [10], [92], [-4], [-50], [-72], [10], [-84], [-14], [-38], [88], [22], [-68], [56], [-12], [42], [-90], [52], [-56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5800_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5800_2_a_c();".
// 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_5800_2_a_c(:prec:=1)
    chi := MakeCharacter_5800_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_5800_2_a_c();".
// 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_5800_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![-2, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;