
// Make newform 8624.2.a.ba 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_ba();".
// 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_ba();".
// 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], [-2], [0], [1], [0], [-4], [4], [4], [2], [-2], [-6], [-4], [4], [2], [2], [-6], [-4], [0], [12], [-16], [8], [-12], [-10], [2], [0], [-6], [-12], [-14], [18], [16], [8], [-10], [-12], [-2], [8], [6], [-16], [-24], [-8], [-20], [10], [-16], [-6], [6], [-22], [-20], [-6], [8], [-14], [14], [-8], [20], [22], [30], [-24], [-6], [20], [-2], [6], [-16], [-12], [16], [18], [-26], [30], [-20], [-10], [12], [4], [-18], [-24], [-22], [38], [0], [6], [-2], [-34], [-6], [-24], [2], [10], [-24], [2], [-12], [4], [-26], [-38], [-12], [28], [-18], [28], [-4], [12], [-40], [-20], [18], [18], [-16], [22], [20], [22], [28], [6], [4], [18], [22], [24], [-28], [-28], [-8], [34], [-42], [-18], [16], [46], [22], [30], [-34], [28], [22], [46], [0], [12], [26], [10], [-26], [-18], [-10], [-4], [-4], [16], [52], [-50], [0], [-16], [26], [4], [54], [42], [24], [6], [16], [36], [-50], [6], [56], [24], [46], [16], [26], [-46], [-52], [36], [-8], [44], [-40], [2], [-24], [-60], [-52], [-14], [48], [-2], [-6], [-30], [-36], [8], [42], [26], [20], [-18], [0], [-2], [-6], [-14], [4], [-54], [52], [56], [32], [-52], [46], [44], [-48], [18], [-34], [-34], [6], [0], [34], [-12], [16], [6], [20], [54], [-50], [-46], [8], [-68], [-10], [-24], [10], [-64], [54], [6], [26], [60], [-46], [-62], [-42], [-30], [-16], [-54], [44], [30], [0], [40], [-36], [46], [42], [-20], [14], [-24], [16], [-46], [42], [40], [28], [12], [-14], [56], [-40], [-14], [-26], [-60], [-14], [54], [-16], [30], [-8], [20], [-58], [34], [-12], [-8], [-2], [2], [12], [-8], [62], [-8], [44], [-18], [-52], [36], [74], [-4], [6], [-36], [-8], [60], [-36], [56], [-18], [14], [58], [-70], [-36], [-54], [38], [8], [10], [72], [72], [2], [-20], [-26], [-22], [36], [-48], [4], [-12], [-48], [68], [-36], [30], [-26], [-30], [60], [18], [22], [-34], [-44], [14], [20], [-18], [58], [12], [-28], [-56], [42], [0], [-72], [24], [14], [36], [86], [-88], [-10], [-40], [-34], [70], [-52], [-8], [56], [-26], [-40], [50], [-18], [32], [2], [-12], [-64], [2], [-74], [-40], [-50], [0], [66], [6], [66], [40], [76], [-44], [14], [-62], [92], [-26], [-44], [6], [54], [96], [16], [-34], [60], [-22], [80], [40], [-62], [20], [-2], [78], [-60], [30], [-12], [-30], [32], [-54], [24], [-58], [36], [0], [-12], [-66], [38], [24], [96], [-86], [88], [-70], [-72], [-50], [40], [-78], [70], [72], [-44], [28], [68], [62], [-82], [-88], [18], [44], [-6], [-56], [-50], [4], [58], [62], [26], [72], [30], [8], [38], [-20], [-58], [-16], [66], [-12], [50], [-48], [-42], [-8], [56], [-34], [58], [66], [42], [24], [18], [-92], [-28], [-18], [78], [-64], [-98], [6], [-32], [-18], [-44], [-84], [-24], [30], [-78], [-16], [30], [2], [-20]];
    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_ba();".
// 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_ba(: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_ba();".
// 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_ba( : 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![2, 1]>,<13,R![0, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;