
// Make newform 9200.2.a.i in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9200_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_9200_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_9200_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_9200_a();"
function MakeCharacter_9200_a()
    N := 9200;
    order := 1;
    char_gens := [1151, 6901, 2577, 1201];
    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_9200_a_Hecke(Kf)
    return MakeCharacter_9200_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], [-2], [0], [1], [5], [-1], [4], [-7], [-1], [5], [-2], [2], [11], [1], [-8], [0], [14], [10], [-8], [10], [-7], [-7], [-15], [10], [4], [18], [11], [0], [2], [-4], [-22], [-2], [6], [-4], [-2], [-10], [4], [4], [-22], [9], [-16], [-12], [27], [-6], [21], [17], [-2], [-20], [12], [-22], [15], [16], [-16], [4], [6], [0], [17], [-8], [1], [-28], [28], [12], [16], [-2], [-22], [5], [18], [20], [-16], [21], [-11], [33], [-3], [-4], [20], [-19], [16], [2], [-6], [-11], [3], [6], [32], [10], [32], [30], [30], [8], [21], [8], [27], [35], [-20], [-6], [-4], [15], [14], [-10], [-1], [-19], [-32], [-42], [33], [12], [-4], [41], [2], [23], [-14], [-30], [36], [14], [-8], [8], [-13], [24], [5], [28], [-25], [3], [4], [23], [42], [6], [8], [-48], [26], [-38], [-40], [38], [22], [-39], [-41], [-10], [-23], [-10], [-6], [47], [54], [17], [56], [31], [36], [-41], [-11], [-21], [7], [58], [-30], [16], [-2], [14], [32], [24], [7], [-47], [-32], [27], [-48], [-20], [-38], [-38], [4], [55], [8], [-21], [-28], [-23], [-20], [-6], [-9], [5], [15], [26], [56], [0], [26], [27], [61], [56], [46], [51], [-19], [-47], [-9], [-42], [42], [-9], [-23], [-2], [-61], [-46], [-9], [30], [-19], [-22], [-10], [30], [18], [48], [-12], [22], [57], [14], [-35], [-43], [28], [-14], [19], [20], [-33], [31], [19], [57], [-12], [-26], [14], [2], [-3], [46], [55], [-45], [-6], [27], [-14], [3], [2], [48], [-8], [-36], [-2], [44], [-43], [-51], [20], [-73], [-70], [-10], [-52], [6], [-19], [22], [70], [2], [6], [6], [-64], [24], [-25], [-69], [-78], [22], [65], [-68], [-21], [-60], [38], [-73], [69], [2], [8], [-7], [-66], [-38], [6], [-21], [20], [-64], [17], [61], [5], [-20], [24], [-27], [-79], [-8], [18], [-37], [80], [-28], [-50], [16], [47], [58], [-12], [22], [-2], [-8], [24], [45], [-48], [-73], [-51], [-76], [22], [-44], [5], [87], [-44], [-40], [14], [48], [45], [-18], [28], [-73], [8], [-56], [32], [-73], [8], [-10], [2], [-24], [56], [14], [-12], [-42], [-66], [78], [52], [8], [27], [74], [66], [64], [-60], [78], [45], [-60], [70], [-61], [14], [-90], [-76], [-76], [-62], [-56], [-73], [-68], [10], [68], [39], [8], [76], [82], [2], [-8], [63], [-52], [-62], [18], [-63], [8], [-52], [5], [57], [53], [49], [6], [-20], [-59], [-30], [3], [-48], [74], [66], [56], [-76], [-44], [52], [-54], [62], [-55], [71], [62], [25], [-22], [-92], [7], [42], [78], [-54], [-78], [-77], [6], [-48], [-98], [-36], [8], [20], [49], [26], [-8], [-24], [76], [-13], [12], [-31], [66], [19], [90], [37], [14], [74], [-42], [-20], [-2], [-29], [12], [38], [-61], [81], [65], [-56], [-31], [-26], [-30], [40], [-21], [-72], [98], [74], [66], [-23], [-17], [19], [-65], [80], [100]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9200_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_9200_2_a_i(:prec:=1)
    chi := MakeCharacter_9200_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_9200_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_9200_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![-1, 1]>,<11,R![-5, 1]>],Snew);
    return Vf;
end function;