
// Make newform 1155.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1155_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_1155_2_a_d();".
// 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_1155_2_a_d();".
// 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_1155_a();"
function MakeCharacter_1155_a()
    N := 1155;
    order := 1;
    char_gens := [386, 232, 661, 211];
    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_1155_a_Hecke(Kf)
    return MakeCharacter_1155_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 := [[-1], [-1], [-1], [-1], [-1], [-2], [6], [4], [0], [-6], [0], [6], [-10], [4], [8], [-6], [-8], [2], [-8], [-8], [-14], [4], [-16], [-10], [-2], [-6], [0], [20], [-10], [-2], [16], [-20], [6], [4], [-14], [-12], [-6], [16], [-4], [-6], [-12], [-2], [0], [18], [-26], [-16], [16], [8], [24], [-10], [-6], [24], [22], [-16], [30], [-24], [18], [-16], [22], [30], [4], [-30], [4], [4], [-26], [-22], [-20], [-14], [12], [-14], [-18], [0], [-32], [-10], [20], [0], [-10], [-14], [2], [-2], [-16], [-26], [0], [-10], [0], [-12], [-14], [26], [-30], [4], [-28], [24], [4], [12], [20], [20], [-14], [14], [20], [14], [-20], [-18], [-8], [30], [16], [-42], [-12], [6], [0], [-34], [-8], [38], [30], [-20], [-8], [-14], [-36], [-24], [-30], [20], [-26], [2], [18], [-4], [28], [2], [-26], [-12], [32], [-26], [16], [8], [48], [-34], [22], [-18], [26], [12], [2], [6], [20], [-14], [20], [-36], [-2], [12], [22], [22], [44], [-48], [14], [54], [32], [-12], [16], [24], [52], [-34], [2], [-46], [36], [-46], [-56], [-16], [-42], [24], [32], [-58], [58], [58], [44], [22], [-32], [10], [-56], [62], [16], [14], [24], [-14], [4], [12], [30], [-18], [16], [-30], [62], [20], [26], [-24], [62], [60], [44], [42], [-60], [22], [10], [-42], [30], [36], [-14], [16], [-14], [-2], [-48], [66], [-24], [-20], [70], [4], [-46], [-54], [-4], [28], [0], [-14], [40], [-50], [8], [-30], [-66], [-32], [18], [-4], [48], [6], [-10], [-40], [-48], [-4], [-10], [-4], [20], [38], [28], [-44], [-30], [14], [-60], [12], [44], [52], [40], [46], [78], [-24], [56], [24], [0], [0], [-2], [-34], [-40], [-14], [10], [-52], [-10], [44], [10], [34], [24], [-12], [-22], [-74], [-18], [-28], [6], [30], [-36], [-58], [70], [-64], [-42], [60], [82], [64], [36], [22], [-6], [-76], [12], [-48], [-4], [26], [-12], [-48], [-38], [-18], [-24], [78], [14], [-52], [58], [-60], [-34], [-78], [-48], [-86], [28], [20], [50], [-26], [-36], [-20], [-68], [2], [28], [22], [0], [6], [-36], [-66], [-82], [32], [-48], [6], [-40], [80], [14], [-26], [-68], [18], [66], [-60], [66], [2], [-20], [-12], [-64], [-34], [-50], [10], [-56], [0], [48], [52], [-34], [14], [-54], [-8], [-50], [22], [26], [80], [2], [20], [38], [-64], [0], [90], [28], [2], [54], [-32], [-34], [30], [60], [60], [58], [-96], [54], [6], [24], [20], [-44], [34], [-62], [72], [-22], [-36], [68], [-32], [-54], [80], [78], [-96], [96], [-38], [-34], [-22], [58], [94], [-40], [2], [-28], [24], [56], [-78], [44], [-88], [-62], [90], [-76], [44], [24], [-54], [24], [30], [-92], [50], [26], [30], [-40], [70], [-38], [48], [30], [78], [-84], [8], [-6], [54], [12], [0], [90], [34], [24], [88], [-2], [92], [54], [78], [72], [-20], [-26], [98], [84], [22], [-108], [-88]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1155_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1155_2_a_d();".
// 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_1155_2_a_d(:prec:=1)
    chi := MakeCharacter_1155_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_1155_2_a_d();".
// 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_1155_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;