
// Make newform 7800.2.a.q in Magma, downloaded from the LMFDB on 24 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_q();".
// 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_q();".
// 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], [0], [-4], [-1], [-8], [5], [4], [9], [-4], [3], [5], [-6], [5], [5], [-6], [4], [3], [7], [-4], [-1], [6], [-6], [4], [14], [-8], [17], [-1], [10], [-5], [5], [11], [20], [24], [20], [-6], [-16], [23], [-7], [20], [-20], [6], [-4], [-4], [21], [-16], [-14], [6], [13], [-24], [-24], [20], [15], [24], [-18], [-7], [-8], [-26], [-25], [-4], [0], [23], [10], [9], [-24], [28], [6], [-9], [-14], [-3], [7], [-1], [6], [12], [27], [-15], [3], [38], [-8], [-5], [34], [-27], [5], [9], [-19], [-29], [-8], [0], [34], [-9], [25], [2], [0], [19], [16], [0], [32], [-22], [22], [2], [-22], [-21], [32], [-4], [-4], [28], [27], [24], [37], [-27], [-34], [-39], [0], [20], [-30], [-31], [24], [-42], [-19], [-37], [-29], [2], [36], [-7], [10], [46], [36], [12], [-49], [33], [29], [-39], [-40], [-19], [-8], [0], [-8], [-22], [-50], [36], [-6], [-35], [48], [-26], [40], [-47], [-54], [-2], [-23], [41], [-8], [44], [28], [-54], [18], [-11], [-3], [34], [24], [12], [8], [24], [23], [46], [16], [-43], [58], [-30], [22], [-34], [-13], [2], [-4], [41], [-30], [-20], [-42], [-35], [-6], [-46], [12], [-34], [11], [-12], [-43], [4], [43], [20], [8], [25], [-8], [-56], [-30], [57], [60], [-28], [34], [-30], [26], [4], [-37], [34], [7], [12], [-15], [50], [-9], [-62], [-28], [13], [-31], [-48], [-12], [-13], [-6], [8], [54], [-16], [-6], [30], [20], [55], [-64], [49], [62], [-50], [-68], [-7], [6], [-40], [70], [20], [-38], [-52], [24], [58], [-10], [-57], [-62], [74], [22], [49], [-49], [14], [30], [56], [-28], [-15], [6], [25], [22], [-3], [1], [-25], [40], [-32], [-17], [-59], [48], [-55], [27], [-67], [-24], [42], [-56], [58], [-57], [4], [30], [18], [11], [-54], [11], [-7], [52], [-22], [-5], [48], [57], [24], [54], [-62], [2], [-25], [2], [-14], [16], [-20], [-20], [-58], [25], [39], [-42], [62], [46], [-32], [-26], [84], [-44], [65], [-42], [56], [-21], [20], [-56], [-3], [38], [1], [48], [36], [26], [-30], [-74], [-51], [-29], [-6], [40], [-66], [12], [-70], [20], [-45], [-16], [81], [-29], [-65], [-7], [-72], [-51], [-53], [-82], [-44], [-58], [-27], [-26], [70], [-35], [-22], [29], [-5], [24], [16], [16], [20], [-48], [-24], [-20], [-51], [-83], [85], [54], [6], [31], [52], [-66], [-29], [38], [66], [-63], [79], [-4], [8], [-66], [10], [54], [44], [76], [-31], [9], [-4], [-36], [-1], [3], [-28], [90], [0], [6], [18], [57], [46], [39], [-16], [66], [19], [29], [-20], [4], [-29], [78], [62], [-42], [-30], [-52], [40], [-23], [-52], [-58], [74], [-100], [5], [48], [18], [33], [-38], [8], [66], [94], [23], [-78], [-42], [-42], [84], [26], [97], [-65], [31], [63], [-78], [80], [-45], [-51], [37], [-45], [104], [58], [-92], [-43], [92], [0], [78], [-92], [18], [-48], [98], [-42], [-52], [73], [-62], [-38], [-50], [69], [-63], [62], [-37], [20], [-88], [-21], [65], [46], [-45], [29], [-4], [66], [-38], [-90], [18], [-39], [-40], [19], [2], [-92], [11], [-86], [-33], [-68], [-20], [8], [54], [-66], [-104], [-48], [72], [36]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<17,R![8, 1]>,<19,R![-5, 1]>],Snew);
    return Vf;
end function;