
// Make newform 8624.2.a.k in Magma, downloaded from the LMFDB on 17 April 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_k();".
// 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_k();".
// 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], [-1], [0], [0], [1], [-1], [-6], [-2], [6], [9], [4], [2], [-6], [4], [6], [0], [3], [11], [-11], [0], [2], [-5], [6], [-18], [-13], [-15], [16], [-12], [2], [9], [7], [-18], [-9], [-8], [6], [19], [-4], [-17], [-3], [-21], [15], [2], [-6], [14], [-3], [-14], [10], [-26], [-18], [-16], [-6], [9], [26], [12], [3], [9], [12], [-29], [29], [18], [10], [-30], [-20], [-24], [17], [12], [-35], [14], [0], [2], [-18], [3], [10], [-31], [-23], [36], [12], [-22], [-33], [-4], [12], [-28], [3], [-34], [1], [-12], [-18], [-22], [21], [22], [-12], [-15], [-20], [-42], [40], [33], [-6], [-42], [16], [-25], [28], [-18], [18], [-36], [-32], [-7], [-9], [36], [0], [2], [40], [-10], [21], [-20], [-20], [9], [-5], [12], [18], [0], [14], [-28], [6], [21], [-11], [39], [26], [42], [-14], [-25], [-50], [36], [4], [-46], [-18], [-22], [24], [-14], [-18], [-48], [-20], [-15], [10], [-30], [-52], [-6], [-46], [36], [37], [-30], [17], [33], [-29], [-3], [28], [6], [-32], [39], [8], [15], [-24], [-36], [40], [-15], [-18], [6], [40], [-34], [-4], [-30], [-6], [-22], [27], [-46], [46], [-39], [16], [54], [16], [-46], [-8], [-6], [-16], [12], [24], [42], [5], [-17], [8], [-3], [2], [24], [13], [-36], [0], [39], [-40], [14], [-6], [-51], [9], [37], [-28], [14], [15], [-48], [-32], [18], [36], [25], [38], [-6], [16], [-12], [-15], [-7], [31], [24], [-60], [54], [-7], [7], [15], [34], [-30], [17], [9], [-36], [55], [18], [32], [70], [-23], [-18], [-5], [21], [50], [-51], [15], [-60], [-48], [-56], [76], [-58], [6], [24], [22], [-57], [-14], [-9], [-19], [36], [6], [14], [-9], [54], [-58], [4], [54], [-52], [-8], [12], [2], [65], [-3], [61], [48], [33], [58], [42], [-22], [-68], [29], [-44], [32], [-80], [-3], [23], [38], [-66], [33], [-62], [-21], [-7], [34], [78], [-61], [-78], [-14], [-12], [-42], [39], [-36], [66], [-31], [-39], [-14], [60], [42], [-14], [44], [33], [-11], [-57], [-17], [29], [75], [-40], [30], [-34], [63], [42], [-24], [4], [3], [32], [-39], [-12], [74], [0], [-80], [23], [27], [-56], [12], [-19], [64], [16], [3], [-33], [50], [54], [1], [54], [-26], [-45], [-28], [-72], [-37], [-32], [-34], [66], [75], [-62], [-6], [6], [20], [-17], [0], [0], [-68], [-94], [-90], [73], [74], [54], [24], [-18], [54], [-36], [47], [66], [24], [-36], [-65], [-91], [66], [4], [14], [-63], [-56], [45], [6], [67], [80], [69], [24], [98], [66], [-28], [84], [-102], [1], [-6], [-8], [6], [22], [-94], [-56], [66], [53], [36], [99], [-53], [0], [68], [-59], [-15], [-5], [51], [-13], [-81], [-41], [-3], [3], [64], [74], [12], [-44], [-51], [44], [-54], [69], [58], [-64], [-6], [-3], [-8], [21], [33], [54], [23], [-66], [-48], [-7], [-36], [-69], [-90], [-65], [-57]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8624_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![0, 1]>,<13,R![1, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;