
// Make newform 5700.2.a.g in Magma, downloaded from the LMFDB on 10 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5700_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_5700_2_a_g();".
// 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_5700_2_a_g();".
// 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_5700_a();"
function MakeCharacter_5700_a()
    N := 5700;
    order := 1;
    char_gens := [2851, 1901, 3877, 4201];
    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_5700_a_Hecke( : Kf := Kf)
    return MakeCharacter_5700_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], [-1], [0], [3], [-2], [2], [-4], [-1], [-2], [3], [4], [-6], [-11], [-4], [0], [3], [-3], [-5], [-8], [-13], [15], [2], [0], [-11], [8], [-14], [14], [-1], [-6], [10], [-4], [6], [4], [-11], [-6], [12], [1], [23], [-9], [5], [11], [-12], [20], [2], [-4], [-15], [-4], [-12], [7], [-22], [-24], [6], [14], [4], [-13], [-16], [-14], [-1], [-17], [-18], [-4], [22], [-2], [-28], [6], [-35], [-14], [2], [-24], [17], [-4], [6], [0], [-22], [-2], [3], [26], [-18], [10], [0], [-12], [-28], [5], [4], [30], [10], [15], [9], [10], [20], [-10], [2], [-26], [0], [-27], [6], [-39], [-39], [28], [-10], [26], [-24], [-23], [-5], [-17], [18], [-28], [10], [40], [34], [22], [-23], [4], [-23], [-8], [18], [29], [12], [6], [12], [-8], [10], [-9], [51], [8], [6], [47], [28], [19], [-35], [-29], [-11], [-4], [41], [-36], [-25], [21], [38], [-39], [40], [16], [-46], [41], [12], [-20], [-9], [-37], [5], [45], [-1], [38], [-42], [29], [0], [28], [45], [-1], [18], [-35], [-42], [4], [39], [-23], [-15], [62], [-16], [-8], [14], [-6], [-56], [41], [0], [6], [-58], [54], [16], [15], [-14], [44], [-25], [24], [-11], [40], [-21], [0], [-4], [-8], [-2], [22], [-2], [36], [50], [-44], [-54], [-18], [-6], [-3], [-59], [24], [-42], [-6], [18], [20], [40], [46], [-16], [8], [-51], [60], [46], [-17], [44], [31], [-63], [25], [-60], [-31], [26], [28], [-26], [2], [58], [46], [-45], [-12], [15], [-37], [-15], [6], [-36], [69], [18], [48], [1], [-7], [54], [-17], [-54], [-30], [61], [-41], [29], [4], [-20], [46], [-36], [1], [52], [30], [-48], [5], [-30], [36], [26], [48], [-44], [11], [-66], [47], [6], [-20], [-39], [-75], [30], [-2], [-82], [27], [52], [32], [20], [24], [16], [-38], [31], [-16], [-64], [-36], [22], [74], [48], [-31], [-41], [-2], [-4], [73], [62], [42], [78], [53], [3], [-26], [-74], [51], [1], [-6], [30], [-44], [42], [-76], [20], [9], [34], [-16], [-1], [-76], [-16], [-76], [-10], [34], [-10], [-54], [6], [57], [26], [56], [0], [-86], [81], [-54], [86], [58], [-3], [43], [-54], [86], [-50], [18], [10], [0], [-52], [18], [7], [-15], [58], [-41], [-66], [-26], [57], [-79], [-17], [16], [-38], [-13], [70], [71], [-61], [17], [30], [-56], [36], [-38], [22], [28], [-46], [-16], [6], [78], [40], [-90], [8], [-55], [-8], [-51], [-28], [48], [-80], [24], [88], [68], [6], [18], [-86], [-41], [-92], [2], [-21], [-50], [-49], [90], [-68], [75], [34], [-29], [54], [-52], [-56], [0], [11], [27], [47], [-78], [6], [-12], [-37], [15], [72], [4], [75], [20], [22], [96], [94], [-46], [-69], [-39], [-40], [-62], [-9], [34], [-54], [26], [48], [68], [-35], [35], [-36], [-91], [10], [-70], [-21], [3], [-50], [98], [29], [-60], [33], [49], [-46], [-91], [-30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5700_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5700_2_a_g();".
// 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_5700_2_a_g(:prec:=1)
    chi := MakeCharacter_5700_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_5700_2_a_g();".
// 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_5700_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5700_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-3, 1]>,<11,R![2, 1]>,<13,R![-2, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;