
// Make newform 5760.2.a.h in Magma, downloaded from the LMFDB on 15 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5760_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_5760_2_a_h();".
// 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_5760_2_a_h();".
// 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_5760_a();"
function MakeCharacter_5760_a()
    N := 5760;
    order := 1;
    char_gens := [2431, 901, 641, 3457];
    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_5760_a_Hecke(Kf)
    return MakeCharacter_5760_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], [0], [-1], [-2], [6], [2], [6], [2], [6], [-6], [-4], [6], [2], [-4], [-10], [-2], [10], [-10], [4], [16], [-6], [0], [-8], [-6], [2], [10], [18], [-16], [-10], [6], [18], [10], [22], [6], [-14], [8], [-2], [8], [10], [18], [-10], [-10], [12], [10], [-10], [20], [-10], [-10], [28], [-10], [14], [16], [-14], [6], [6], [-2], [-30], [-20], [22], [6], [20], [-22], [-20], [16], [-14], [6], [10], [-22], [8], [-18], [6], [-4], [-22], [-26], [-26], [18], [2], [10], [2], [22], [-26], [30], [-12], [-14], [-4], [36], [30], [-6], [-38], [22], [12], [-8], [-2], [6], [2], [-26], [10], [6], [28], [-34], [-28], [2], [-24], [-6], [26], [34], [8], [-34], [12], [-38], [18], [34], [-34], [-34], [-8], [30], [24], [18], [14], [-10], [22], [-6], [6], [36], [30], [18], [-10], [0], [-2], [30], [-30], [6], [36], [22], [6], [38], [14], [4], [54], [22], [26], [-22], [-14], [8], [6], [36], [34], [30], [6], [-6], [-34], [-30], [-40], [-14], [-24], [-4], [-28], [-18], [26], [-14], [-12], [54], [6], [-34], [-10], [30], [-52], [-6], [-10], [46], [-30], [-2], [40], [34], [8], [-6], [-14], [18], [26], [-10], [2], [-30], [50], [-26], [26], [-62], [14], [-56], [22], [36], [-46], [-20], [22], [66], [-12], [-26], [30], [66], [-50], [-66], [-14], [12], [-6], [18], [-54], [18], [-32], [-24], [-54], [-38], [2], [2], [-14], [-48], [-44], [30], [-46], [-30], [-38], [-66], [-26], [-44], [26], [38], [-68], [-2], [-18], [-64], [-2], [-18], [-46], [-38], [20], [-26], [-44], [-18], [18], [46], [10], [56], [-24], [2], [-14], [38], [22], [-28], [26], [42], [46], [58], [-22], [-18], [-54], [26], [18], [-42], [-2], [-8], [-70], [-6], [14], [60], [-50], [-2], [62], [-30], [-46], [18], [-36], [26], [22], [-12], [-62], [64], [-14], [2], [64], [46], [10], [-6], [26], [-24], [66], [-26], [8], [44], [-22], [-26], [-60], [-38], [-22], [-36], [-18], [30], [14], [-14], [-20], [-38], [-6], [12], [-54], [18], [64], [-64], [-78], [-38], [24], [-50], [52], [6], [-6], [-54], [18], [-32], [-30], [-26], [6], [-60], [-38], [-6], [-18], [10], [-46], [-2], [14], [-78], [74], [76], [-18], [-34], [-2], [-14], [8], [48], [58], [-72], [-26], [-50], [-22], [42], [-26], [-26], [42], [16], [66], [86], [22], [-80], [-68], [-42], [70], [2], [-6], [-10], [-74], [-58], [40], [-18], [-42], [-90], [-22], [82], [54], [18], [-20], [-22], [-42], [-6], [-34], [18], [-26], [-6], [42], [24], [26], [-26], [68], [98], [-66], [82], [-46], [30], [-90], [62], [42], [-98], [-68], [22], [-52], [-42], [38], [-22], [-46], [-44], [8], [-38], [-40], [22], [82], [-30], [-42], [46], [2], [-58], [74], [96], [-86], [82], [-56], [30], [-38], [42], [76], [14], [74], [-78], [64], [-82], [62], [-82], [-54], [-70], [-2], [26], [-6], [-14], [32], [-26], [-70], [84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5760_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5760_2_a_h();".
// 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_5760_2_a_h(:prec:=1)
    chi := MakeCharacter_5760_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_5760_2_a_h();".
// 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_5760_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5760_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-6, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<29,R![6, 1]>],Snew);
    return Vf;
end function;