
// Make newform 7800.2.a.x in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7800_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_7800_2_a_x();".
// 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_7800_2_a_x();".
// 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_7800_a();"
function MakeCharacter_7800_a()
    N := 7800;
    order := 1;
    char_gens := [1951, 3901, 5201, 7177, 4201];
    v := [1, 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_7800_a_Hecke(Kf)
    return MakeCharacter_7800_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], [5], [1], [-1], [3], [6], [7], [6], [-2], [-1], [7], [-8], [-2], [-13], [-8], [-7], [-12], [1], [10], [3], [-8], [13], [-7], [4], [-4], [-9], [-16], [-2], [2], [6], [-6], [5], [21], [4], [-22], [25], [-16], [18], [6], [-11], [0], [-21], [8], [-20], [-4], [-8], [-18], [14], [1], [9], [-18], [-12], [22], [24], [0], [2], [10], [-18], [16], [-16], [-23], [12], [26], [8], [-36], [-8], [-13], [-32], [-14], [-24], [-12], [4], [38], [-26], [-8], [23], [6], [10], [30], [24], [24], [-36], [-15], [31], [15], [37], [-33], [-3], [23], [35], [-1], [28], [38], [12], [17], [-22], [-16], [-2], [-8], [-38], [11], [36], [-5], [29], [26], [-44], [-4], [-13], [-4], [1], [10], [6], [-16], [48], [37], [-11], [-18], [36], [-20], [-42], [27], [-24], [-6], [-28], [4], [-12], [30], [-13], [2], [-26], [29], [28], [34], [-16], [24], [-28], [-7], [-30], [56], [3], [16], [-54], [-34], [33], [25], [-39], [9], [-38], [-46], [-26], [-28], [57], [-46], [-12], [-37], [-11], [2], [-35], [-28], [-13], [0], [-60], [-8], [-4], [-47], [-40], [30], [-1], [44], [26], [28], [45], [-35], [4], [22], [-27], [36], [22], [-55], [24], [26], [34], [14], [-24], [-52], [19], [56], [23], [8], [6], [55], [-13], [36], [-9], [10], [-2], [48], [-57], [-14], [-17], [5], [-67], [-33], [29], [50], [-4], [69], [-15], [-4], [2], [28], [6], [12], [-48], [0], [30], [22], [-42], [-41], [-26], [-30], [-52], [51], [25], [-15], [-22], [22], [-51], [4], [-43], [8], [20], [18], [66], [4], [-34], [6], [64], [24], [-59], [16], [4], [8], [-44], [16], [-3], [8], [0], [-2], [10], [62], [-15], [-75], [-41], [-10], [53], [-54], [-31], [-30], [21], [32], [-52], [66], [-5], [-75], [45], [47], [-42], [22], [23], [61], [-32], [74], [-77], [32], [28], [-8], [-78], [63], [-40], [0], [-82], [-56], [0], [-24], [-72], [10], [16], [-40], [52], [64], [44], [-58], [-46], [-48], [-47], [-44], [-51], [-70], [48], [-5], [13], [40], [83], [69], [-29], [0], [16], [15], [-22], [10], [32], [-8], [45], [63], [-75], [-10], [-28], [68], [-30], [58], [-7], [-6], [7], [-30], [41], [-85], [82], [-10], [-75], [-40], [-52], [-72], [26], [58], [44], [-50], [-34], [9], [41], [-11], [-84], [30], [44], [-27], [65], [16], [33], [-22], [-61], [27], [-70], [71], [-86], [-15], [76], [-82], [-22], [62], [-12], [-44], [20], [42], [-16], [-88], [81], [-59], [-14], [12], [44], [-26], [49], [72], [19], [50], [77], [16], [-54], [-10], [-58], [-56], [34], [20], [-24], [-86], [20], [-32], [-9], [100], [-54], [21], [12], [41], [-86], [-2], [30], [25], [30], [-58], [33], [-1], [88], [47], [89], [58], [18], [59], [-45], [98], [-43], [-73], [-31], [42], [10], [-45], [38], [48], [61], [-26], [-85], [48], [-76], [82], [76], [-13], [-45], [10], [-66], [66], [-67], [87], [-6], [-63], [-70], [46], [66], [-28], [-84], [36], [17], [12], [-6], [-26], [6], [-66], [-66], [-54], [-26], [45], [-48], [-6], [-70], [-29], [-16], [110], [10], [-42], [-18], [-1], [76], [-84], [49], [-7], [-89], [1], [90], [18], [-88], [40], [54], [37], [112]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7800_2_a_x();".
// 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_7800_2_a_x(:prec:=1)
    chi := MakeCharacter_7800_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(3361) - 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_7800_2_a_x();".
// 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_7800_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-5, 1]>,<11,R![-1, 1]>,<17,R![-3, 1]>,<19,R![-6, 1]>],Snew);
    return Vf;
end function;