
// Make newform 8624.2.a.f in Magma, downloaded from the LMFDB on 16 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8624_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_8624_2_a_f();".
// 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_8624_2_a_f();".
// 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_8624_a();"
function MakeCharacter_8624_a()
    N := 8624;
    order := 1;
    char_gens := [5391, 6469, 7745, 3137];
    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_8624_a_Hecke(Kf)
    return MakeCharacter_8624_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], [-2], [0], [0], [1], [-2], [2], [4], [0], [-6], [2], [-2], [-6], [0], [6], [2], [-2], [2], [4], [0], [6], [-12], [4], [8], [-12], [14], [6], [-4], [2], [-14], [8], [0], [-6], [4], [6], [20], [8], [-12], [-24], [6], [12], [-12], [8], [2], [-10], [14], [-4], [-26], [0], [12], [-10], [0], [-14], [2], [24], [24], [-20], [-20], [-2], [-18], [-16], [2], [8], [-10], [8], [-2], [-12], [-26], [-8], [10], [-24], [-4], [-2], [-18], [4], [-30], [10], [16], [-18], [38], [30], [10], [-20], [20], [-28], [28], [-26], [18], [-30], [-16], [2], [-20], [8], [-16], [-20], [-20], [-12], [-4], [32], [-2], [-28], [22], [-44], [-18], [-12], [-20], [42], [-10], [-8], [-38], [-8], [2], [-38], [-46], [-40], [-22], [26], [18], [-46], [16], [-32], [6], [14], [-28], [-26], [2], [-38], [18], [42], [-22], [-4], [8], [0], [42], [-42], [-42], [52], [-44], [-40], [42], [8], [6], [-40], [24], [-40], [-14], [54], [-18], [10], [24], [-22], [-36], [-36], [12], [28], [24], [-40], [12], [6], [-6], [52], [2], [-20], [-22], [30], [-26], [-40], [38], [10], [12], [-60], [-56], [44], [14], [22], [60], [40], [30], [-28], [22], [-32], [-60], [42], [34], [24], [-20], [38], [2], [38], [24], [-12], [12], [20], [32], [24], [56], [-34], [-42], [-42], [12], [30], [-32], [20], [-42], [-6], [-24], [14], [20], [-22], [22], [-26], [12], [16], [-18], [28], [32], [-52], [-42], [72], [-30], [42], [-36], [30], [16], [-24], [50], [4], [-32], [44], [48], [-34], [-8], [32], [-54], [-54], [28], [52], [-66], [-4], [26], [52], [-52], [-14], [50], [-14], [24], [34], [30], [-60], [48], [38], [54], [-8], [0], [34], [36], [-26], [12], [16], [2], [4], [60], [66], [-34], [-48], [26], [-6], [20], [-8], [74], [-40], [4], [28], [-64], [-2], [30], [20], [-26], [-46], [-12], [-16], [64], [44], [-58], [12], [16], [-82], [-74], [6], [-54], [50], [2], [30], [44], [78], [-54], [42], [52], [28], [12], [-34], [18], [4], [36], [-44], [26], [-20], [-40], [72], [22], [88], [-6], [30], [-4], [36], [22], [-38], [0], [-4], [-2], [48], [-42], [-62], [80], [-62], [-88], [-36], [-46], [-36], [-6], [-42], [-50], [48], [-60], [-24], [-46], [-18], [-30], [8], [68], [-26], [-62], [6], [-72], [-30], [52], [44], [28], [24], [-68], [-68], [-82], [14], [-44], [78], [-6], [46], [80], [18], [-24], [62], [-54], [80], [-88], [2], [-38], [-74], [80], [-62], [-60], [-58], [-32], [-10], [-32], [18], [-6], [60], [-38], [2], [58], [-72], [-34], [-12], [42], [-28], [-66], [-72], [-64], [4], [54], [-14], [-4], [12], [-22], [72], [-34], [92], [64], [20], [-30], [-54], [-30], [-72], [-32], [38], [24], [-98], [-46], [46], [-66], [6], [18], [-28], [48], [22], [0], [-8], [-14], [72], [-24], [66], [58], [-80], [0], [88], [12], [12], [30], [-34], [52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8624_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8624_2_a_f();".
// 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_8624_2_a_f(:prec:=1)
    chi := MakeCharacter_8624_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_8624_2_a_f();".
// 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_8624_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8624_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![0, 1]>,<13,R![2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;