
// Make newform 2800.2.a.i in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2800_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_2800_2_a_i();".
// 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_2800_2_a_i();".
// 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_2800_a();"
function MakeCharacter_2800_a()
    N := 2800;
    order := 1;
    char_gens := [351, 2101, 2577, 801];
    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_2800_a_Hecke(Kf)
    return MakeCharacter_2800_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], [0], [-1], [5], [-1], [-3], [6], [-6], [-9], [0], [-6], [8], [6], [3], [12], [-8], [-4], [-4], [-8], [-10], [3], [-12], [-16], [-7], [-14], [-9], [2], [-11], [6], [8], [6], [16], [-18], [-14], [19], [2], [6], [-9], [-19], [-4], [-20], [-11], [-8], [12], [0], [-13], [-25], [-13], [16], [-8], [7], [18], [-14], [-6], [18], [10], [-4], [-14], [-13], [29], [1], [-27], [14], [-29], [-30], [20], [26], [-10], [14], [33], [0], [-29], [0], [20], [12], [25], [29], [9], [14], [-36], [-19], [-23], [30], [34], [30], [-33], [28], [-24], [20], [33], [6], [26], [-33], [25], [-31], [18], [-6], [4], [-9], [20], [-24], [4], [34], [36], [-17], [8], [7], [33], [-34], [17], [22], [-10], [-2], [9], [34], [47], [0], [-30], [25], [8], [-32], [-33], [-20], [-20], [-21], [-41], [-50], [-16], [-5], [37], [-36], [35], [-8], [-46], [-10], [-1], [11], [-35], [-23], [-38], [-7], [32], [2], [16], [26], [26], [14], [28], [20], [10], [-16], [-16], [-8], [-30], [40], [-25], [36], [-7], [4], [-52], [-20], [22], [-28], [-18], [9], [16], [1], [37], [-18], [24], [10], [-27], [30], [-10], [36], [9], [-15], [60], [-28], [-8], [0], [6], [-6], [-40], [14], [16], [-25], [25], [12], [-66], [6], [-17], [-44], [-28], [15], [-1], [32], [61], [23], [-18], [10], [-10], [-2], [-6], [33], [10], [-32], [22], [12], [8], [26], [0], [3], [-6], [48], [-14], [40], [-46], [-54], [-62], [74], [45], [-18], [-53], [-66], [-55], [9], [-17], [28], [-24], [-42], [-72], [-63], [-37], [-73], [20], [40], [23], [6], [24], [32], [59], [54], [-9], [2], [32], [18], [19], [-66], [50], [-38], [52], [-16], [18], [9], [46], [4], [-33], [63], [-18], [78], [60], [-61], [-42], [-50], [1], [-14], [-36], [-38], [60], [8], [-55], [-28], [2], [-53], [-12], [-30], [-31], [-46], [-25], [-13], [32], [38], [3], [33], [-50], [18], [60], [-12], [33], [-47], [36], [16], [10], [-84], [26], [-49], [36], [-57], [53], [-62], [85], [-32], [15], [50], [-34], [-48], [48], [32], [0], [-21], [-34], [26], [78], [-56], [26], [-23], [37], [45], [-2], [-44], [26], [8], [-62], [20], [-81], [4], [-52], [32], [70], [16], [46], [29], [12], [-57], [13], [14], [66], [-77], [38], [-24], [-50], [-51], [-66], [-12], [38], [56], [14], [-3], [-88], [-4], [25], [11], [25], [10], [-18], [-78], [82], [86], [-28], [-72], [84], [-7], [-36], [-38], [13], [-70], [69], [65], [76], [24], [27], [-10], [-26], [-48], [5], [-9], [54], [-21], [-18], [66], [-68], [34], [44], [31], [25], [-17], [4], [-19], [-55], [-23], [51], [-68], [-23], [64], [16], [76], [13], [50], [0], [44], [-94], [-7], [-36], [26], [86], [15], [-76], [-52], [65], [50], [54], [36], [-58], [-24], [-63], [84], [-2], [15], [-67], [90], [-74], [12], [-69], [-19], [56], [66], [-28], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2800_2_a_i();".
// 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_2800_2_a_i(:prec:=1)
    chi := MakeCharacter_2800_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_2800_2_a_i();".
// 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_2800_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<11,R![-5, 1]>,<13,R![1, 1]>],Snew);
    return Vf;
end function;