
// Make newform 1824.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1824_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_1824_2_a_d();".
// 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_1824_2_a_d();".
// 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_1824_a();"
function MakeCharacter_1824_a()
    N := 1824;
    order := 1;
    char_gens := [799, 229, 1217, 97];
    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_1824_a_Hecke(Kf)
    return MakeCharacter_1824_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], [0], [0], [-2], [-1], [-6], [10], [-10], [8], [6], [4], [6], [-6], [-12], [-10], [-8], [-8], [-2], [-6], [-12], [-6], [-2], [8], [2], [12], [-4], [-10], [-2], [-4], [-2], [-12], [20], [-6], [2], [4], [4], [6], [-12], [-20], [10], [-26], [12], [-16], [20], [22], [-20], [6], [18], [22], [-18], [-4], [-14], [6], [-10], [0], [22], [6], [28], [22], [0], [22], [22], [-18], [-24], [-6], [8], [-14], [18], [26], [16], [-8], [24], [8], [8], [-22], [-2], [38], [16], [-8], [12], [-34], [-10], [24], [10], [-30], [36], [-24], [-36], [10], [26], [-28], [28], [6], [-42], [-22], [40], [-14], [16], [-12], [-12], [18], [4], [18], [-16], [-30], [-36], [34], [-22], [18], [2], [-4], [-44], [-14], [4], [-6], [-36], [28], [-40], [-2], [42], [20], [-20], [-24], [22], [-2], [-4], [-22], [12], [20], [-22], [26], [22], [-18], [-10], [28], [38], [30], [-16], [-32], [4], [-20], [44], [-36], [14], [42], [-4], [-4], [-8], [-54], [-52], [-36], [8], [12], [20], [-42], [38], [-42], [20], [-58], [-16], [36], [46], [32], [-34], [10], [54], [36], [4], [-20], [-42], [22], [-46], [2], [-4], [-12], [-18], [6], [8], [-60], [44], [-26], [14], [12], [-36], [4], [22], [-14], [-30], [-12], [16], [-6], [-16], [-66], [-34], [34], [-54], [50], [-14], [26], [0], [10], [24], [-32], [44], [52], [-6], [-52], [34], [-40], [64], [-28], [28], [-50], [-52], [30], [-60], [-8], [4], [-74], [-14], [60], [-28], [-34], [10], [-68], [-14], [0], [42], [-12], [58], [-6], [-44], [-22], [-50], [-16], [44], [-40], [36], [-20], [68], [28], [-30], [6], [-28], [-12], [28], [58], [50], [50], [46], [18], [28], [16], [-50], [-48], [-14], [-42], [-26], [-68], [74], [-36], [-18], [-36], [18], [50], [76], [24], [36], [68], [14], [40], [58], [44], [52], [-60], [18], [0], [-16], [-64], [-10], [44], [60], [66], [-38], [-42], [64], [-66], [52], [12], [-74], [-44], [4], [8], [-22], [-4], [28], [-44], [-62], [34], [-76], [-12], [60], [-42], [60], [-20], [-50], [-58], [18], [-60], [82], [-44], [-42], [10], [12], [-40], [-14], [18], [-64], [-62], [-30], [70], [38], [-34], [-64], [28], [8], [-12], [34], [18], [4], [64], [-84], [-28], [32], [54], [-58], [52], [-64], [-18], [10], [14], [58], [36], [2], [80], [12], [-48], [16], [-2], [36], [78], [-24], [-86], [-34], [-12], [-98], [-72], [-22], [42], [36], [36], [28], [-94], [40], [14], [-74], [28], [-52], [26], [-6], [-48], [38], [-76], [42], [38], [-10], [46], [-6], [86], [16], [66], [20], [-84], [44], [-82], [-28], [-60], [-38], [58], [-68], [-76], [-24], [-66], [-26], [-42], [-48], [-8], [-12], [14], [-82], [-30], [-66], [28], [-66], [90], [84], [-36], [34], [-8], [68], [-76], [-38], [-48], [-36], [-22], [-2], [-60], [18], [40], [-82], [76], [-18], [38], [28], [-30], [-92], [-46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1824_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1824_2_a_d();".
// 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_1824_2_a_d(:prec:=1)
    chi := MakeCharacter_1824_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_1824_2_a_d();".
// 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_1824_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1824_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![0, 1]>,<23,R![6, 1]>],Snew);
    return Vf;
end function;