
// Make newform 4624.2.a.f in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4624_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_4624_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_4624_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_4624_a();"
function MakeCharacter_4624_a()
    N := 4624;
    order := 1;
    char_gens := [4047, 1157, 4049];
    v := [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_4624_a_Hecke( : Kf := Kf)
    return MakeCharacter_4624_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]];
    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], [2], [-6], [0], [-4], [4], [0], [-8], [4], [-6], [-8], [8], [10], [0], [-12], [-8], [12], [-2], [-4], [-16], [10], [18], [10], [0], [-18], [0], [6], [-8], [-2], [-10], [6], [6], [-8], [-2], [-2], [-8], [24], [12], [20], [-16], [10], [-20], [-4], [2], [-24], [-10], [6], [14], [0], [-22], [24], [-26], [16], [8], [16], [-8], [6], [-22], [-26], [-12], [24], [2], [-20], [-32], [22], [30], [10], [6], [-24], [-28], [-14], [34], [0], [6], [12], [34], [-10], [14], [10], [28], [34], [-28], [-8], [-14], [-6], [-22], [32], [12], [4], [12], [-12], [-38], [12], [-34], [-14], [16], [-4], [22], [10], [8], [2], [-10], [14], [-36], [34], [8], [14], [-48], [14], [-34], [-10], [24], [-14], [2], [24], [-8], [-28], [-14], [-38], [-36], [-46], [-14], [42], [32], [-36], [-32], [10], [4], [40], [4], [-34], [22], [-2], [-18], [22], [10], [2], [-38], [-20], [4], [2], [6], [48], [8], [50], [-48], [48], [-16], [-18], [-32], [36], [-30], [32], [0], [-14], [38], [20], [-10], [34], [56], [8], [10], [-36], [28], [-4], [-46], [44], [-4], [10], [48], [-6], [-64], [-10], [10], [-4], [-32], [-50], [-8], [-26], [52], [42], [24], [10], [-12], [-20], [58], [-56], [22], [30], [20], [38], [-30], [46], [38], [4], [-2], [0], [12], [-20], [-34], [-26], [-44], [-42], [-24], [44], [54], [-12], [42], [14], [-40], [12], [32], [18], [-40], [2], [-24], [18], [70], [24], [-18], [40], [-48], [-46], [6], [-12], [-8], [30], [58], [-6], [40], [-6], [-56], [64], [-14], [-60], [6], [-24], [54], [-12], [-64], [-10], [6], [-64], [-56], [-10], [32], [56], [22], [38], [48], [22], [-2], [-24], [60], [-22], [-28], [22], [-8], [-12], [-20], [-16], [-46], [64], [-46], [46], [-34], [30], [-16], [-60], [26], [-56], [-18], [-16], [-48], [-38], [-22], [0], [-8], [40], [32], [-22], [46], [-72], [2], [-12], [32], [-30], [48], [42], [78], [-30], [-28], [-56], [-8], [-6], [-34], [-8], [-14], [54], [8], [-14], [2], [2], [-60], [-76], [-32], [6], [-28], [-24], [-78], [-52], [0], [-58], [-44], [56], [-26], [-90], [-50], [26], [-82], [64], [-54], [-22], [-18], [-78], [-36], [52], [-8], [12], [4], [-4], [-2], [-6], [-38], [30], [-10], [-88], [-38], [-46], [-60], [-32], [-6], [-18], [-52], [8], [-72], [-28], [-8], [-30], [30], [8], [54], [-10], [-72], [-14], [-38], [8], [20], [14], [72], [30], [-28], [-74], [-48], [-40], [-38], [-8], [50], [-44], [-62], [40], [-64], [-38], [48], [54], [46], [42], [80], [54], [36], [-18], [-34], [-48], [56], [-42], [2], [-24], [14], [84], [-100], [-4], [-40], [74], [-32], [-10], [-22], [-30], [-4], [-18], [-72], [-22], [-66], [-80], [-50], [-2], [-64], [-70], [6], [38], [-100], [-66], [-66], [12], [-84], [80], [-38], [56], [-30], [-32], [-12], [60], [98], [54], [-18], [-82], [-8], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4624_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4624_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_4624_2_a_f(:prec:=1)
    chi := MakeCharacter_4624_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_4624_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_4624_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4624_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<5,R![0, 1]>,<7,R![0, 1]>,<13,R![6, 1]>],Snew);
    return Vf;
end function;