
// Make newform 8190.2.a.h in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8190_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_8190_2_a_h();".
// 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_8190_2_a_h();".
// 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_8190_a();"
function MakeCharacter_8190_a()
    N := 8190;
    order := 1;
    char_gens := [911, 3277, 2341, 6931];
    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_8190_a_Hecke(Kf)
    return MakeCharacter_8190_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], [0], [-1], [1], [0], [1], [-6], [-4], [0], [-6], [-4], [-10], [-6], [8], [0], [6], [12], [14], [-4], [0], [2], [8], [12], [-6], [2], [-6], [-4], [0], [2], [6], [8], [12], [6], [-4], [6], [8], [14], [-4], [0], [-6], [12], [-10], [0], [2], [18], [-16], [-4], [8], [-12], [-10], [6], [0], [-10], [-12], [-6], [24], [18], [20], [2], [6], [20], [-6], [20], [-24], [-34], [18], [-16], [26], [24], [-34], [6], [0], [-4], [-22], [-16], [24], [18], [14], [6], [38], [36], [26], [0], [38], [-16], [24], [30], [26], [-6], [8], [-36], [-12], [-16], [-12], [-16], [36], [-6], [6], [-4], [2], [8], [-6], [-12], [6], [-4], [-22], [12], [-42], [24], [26], [-28], [-34], [-42], [-4], [8], [30], [-4], [12], [6], [-12], [14], [2], [-30], [36], [-52], [-30], [-22], [0], [44], [14], [-40], [-24], [8], [2], [18], [14], [-6], [20], [42], [54], [44], [-42], [32], [-12], [14], [-12], [-10], [18], [20], [24], [14], [-42], [8], [-12], [56], [24], [-40], [-6], [38], [-30], [12], [54], [32], [60], [30], [0], [32], [62], [26], [18], [0], [14], [0], [26], [32], [54], [-64], [-18], [20], [62], [32], [12], [50], [-18], [48], [-54], [26], [-4], [-22], [-60], [38], [-60], [-52], [-54], [-24], [-54], [-22], [2], [-42], [-36], [30], [-40], [-10], [-46], [-36], [18], [-4], [24], [-18], [-4], [-46], [-54], [56], [12], [12], [-10], [32], [54], [0], [-54], [62], [-28], [30], [8], [-60], [62], [-6], [24], [-28], [0], [50], [-4], [-40], [-42], [20], [-48], [14], [-6], [-36], [0], [-12], [-76], [20], [26], [30], [-48], [32], [-12], [-16], [24], [14], [-6], [-24], [26], [-30], [0], [14], [-52], [-54], [2], [56], [-48], [14], [38], [-42], [-76], [30], [-30], [-4], [30], [14], [68], [74], [56], [-10], [-16], [-12], [-22], [74], [-60], [-60], [8], [-12], [-34], [-76], [24], [74], [18], [44], [78], [-30], [84], [6], [36], [-70], [18], [-40], [-30], [60], [-28], [-10], [66], [-16], [-48], [44], [2], [24], [-34], [-24], [-46], [12], [6], [-66], [8], [-48], [74], [-60], [-48], [26], [-18], [-28], [-22], [66], [-64], [-90], [-70], [80], [-76], [24], [-42], [50], [6], [8], [84], [80], [-12], [-46], [-42], [14], [-52], [14], [30], [66], [-64], [-30], [-36], [38], [-28], [-12], [42], [-4], [74], [-18], [-76], [38], [66], [-12], [84], [78], [96], [-82], [78], [-24], [24], [20], [-46], [18], [-16], [-70], [12], [-52], [-24], [-30], [56], [-46], [36], [-72], [-70], [-90], [-10], [90], [54], [56], [-18], [44], [-24], [32], [-82], [-52], [-36], [74], [-6], [72], [-52], [72], [-70], [-52], [54], [-76], [-18], [-10], [-18], [56], [78], [66], [-88], [-58], [78], [44], [12], [86], [78], [-48], [-28], [50], [66], [-72], [-28], [54], [36], [-30], [38], [-72], [12], [50], [-78], [48], [-90], [44], [-72], [38], [-48], [68], [-24], [-82], [54], [-46], [-94], [-40], [-4], [12], [30], [-22], [-24], [-22], [102], [68], [-72], [74], [14], [-52], [-36], [108], [90], [50], [90], [50], [-108], [14], [102], [-76], [-40], [-60], [14], [-52], [2], [80], [84], [30], [-52], [-40], [60], [60], [50], [60], [14], [114], [-28], [-84], [-6], [86], [-42], [38], [-6], [-64], [72], [50], [-12], [-76], [56], [-82], [-12], [74], [-54], [-36], [38], [20], [-78], [-40], [44], [-42], [-64], [6], [104], [74], [-6], [-48], [8], [26], [44], [-60], [108], [26], [-78], [-76], [50], [-54], [-34], [0], [92], [14], [32], [30], [24], [-46], [84], [110], [-78], [60], [18], [32], [-42], [8], [84], [-34], [0], [-58], [18], [50], [44], [-108], [-18], [20], [36], [54], [80], [80], [-12], [-112], [-30], [30], [44], [-60], [6], [-120], [-10], [104]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8190_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8190_2_a_h();".
// 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_8190_2_a_h(:prec:=1)
    chi := MakeCharacter_8190_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(4027) - 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_8190_2_a_h();".
// 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_8190_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8190_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<17,R![6, 1]>,<19,R![4, 1]>,<23,R![0, 1]>,<29,R![6, 1]>],Snew);
    return Vf;
end function;