
// Make newform 9200.2.a.bb in Magma, downloaded from the LMFDB on 28 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_bb();".
// 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_bb();".
// 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], [1], [0], [4], [-3], [2], [1], [1], [1], [0], [-8], [2], [1], [12], [6], [4], [-12], [0], [13], [-12], [17], [14], [5], [1], [2], [-4], [14], [19], [-10], [3], [-8], [12], [-15], [-1], [6], [4], [-10], [-1], [-6], [4], [15], [18], [4], [-9], [-8], [14], [5], [16], [-8], [10], [18], [6], [-5], [15], [-30], [4], [10], [-2], [12], [18], [17], [16], [-19], [-18], [-26], [-20], [19], [7], [-11], [-30], [18], [-4], [30], [-12], [-29], [14], [12], [4], [-25], [-19], [-3], [-6], [-36], [25], [22], [35], [-27], [13], [-24], [-2], [-12], [4], [-26], [-24], [20], [-24], [20], [-37], [-29], [32], [29], [2], [36], [19], [12], [1], [-13], [-11], [12], [-9], [14], [40], [-2], [20], [34], [-18], [16], [36], [-10], [27], [-8], [34], [34], [33], [-41], [-48], [16], [0], [4], [-26], [-4], [-26], [16], [16], [-55], [-1], [12], [-28], [-42], [26], [-12], [-32], [-30], [-23], [32], [-42], [28], [43], [15], [-16], [22], [2], [1], [36], [-4], [58], [-52], [-6], [51], [60], [-36], [-39], [-52], [49], [21], [30], [-50], [-10], [-21], [-36], [-17], [-20], [22], [-1], [-10], [21], [-51], [40], [4], [-8], [-20], [9], [10], [5], [44], [-10], [-30], [-33], [-51], [-14], [-7], [31], [27], [22], [-53], [-6], [13], [18], [-1], [54], [-42], [58], [48], [5], [-28], [12], [-4], [-24], [-42], [56], [38], [-14], [-48], [-17], [-12], [-65], [-70], [-51], [60], [-30], [40], [-36], [-3], [2], [-63], [64], [-9], [60], [56], [-67], [40], [-1], [38], [-49], [-44], [26], [-53], [-32], [61], [-56], [-21], [-55], [-2], [-28], [-3], [-28], [-56], [72], [-13], [48], [30], [-51], [-48], [-35], [-54], [-11], [12], [-31], [20], [7], [-16], [21], [-24], [-76], [-33], [-27], [-74], [-58], [79], [-76], [24], [33], [39], [6], [-38], [-10], [29], [-22], [2], [-23], [-30], [-10], [-6], [4], [15], [-60], [2], [-58], [42], [-30], [0], [52], [13], [-45], [-58], [78], [-50], [-58], [-84], [35], [-83], [-8], [36], [61], [11], [79], [68], [22], [-4], [-38], [70], [-42], [-45], [36], [50], [-14], [-53], [-52], [-73], [-53], [44], [-35], [-66], [-38], [42], [70], [3], [36], [-66], [54], [86], [36], [-58], [45], [-11], [48], [62], [-66], [-13], [-58], [52], [-15], [-44], [-44], [14], [8], [2], [43], [42], [18], [89], [-18], [-10], [-76], [52], [17], [-64], [37], [-50], [16], [52], [47], [56], [64], [15], [-91], [18], [32], [-37], [-84], [23], [-78], [96], [32], [-34], [-45], [-80], [-49], [-25], [-58], [16], [29], [78], [61], [100], [-84], [54], [4], [-16], [-70], [49], [44], [-95], [-85], [-60], [-59], [62], [-65], [-31], [-62], [80], [67], [-58], [-71], [-96], [-36], [-22], [-6], [-56], [5], [25], [0], [-27], [-68], [-13], [-36], [38], [96], [5], [94], [16], [86], [36], [-75], [19], [-62], [52], [39], [47], [-78]];
    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_bb();".
// 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_bb(: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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<7,R![-4, 1]>,<11,R![3, 1]>],Snew);
    return Vf;
end function;