
// Make newform 7440.2.a.y in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7440_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_7440_2_a_y();".
// 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_7440_2_a_y();".
// 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_7440_a();"
function MakeCharacter_7440_a()
    N := 7440;
    order := 1;
    char_gens := [6511, 1861, 4961, 2977, 5521];
    v := [1, 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_7440_a_Hecke(Kf)
    return MakeCharacter_7440_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], [1], [-2], [4], [-4], [-6], [0], [0], [4], [-1], [4], [-6], [8], [12], [-2], [-6], [10], [10], [14], [4], [8], [4], [-8], [-2], [14], [-14], [16], [18], [-6], [12], [-14], [-18], [12], [14], [16], [-6], [-6], [0], [6], [4], [18], [10], [-2], [2], [0], [-16], [-16], [-20], [-26], [6], [-20], [10], [28], [-2], [-24], [-12], [24], [-8], [6], [-14], [26], [-2], [-6], [-4], [26], [28], [8], [-4], [2], [10], [-30], [0], [18], [8], [-24], [36], [-18], [-4], [6], [30], [-30], [10], [-4], [-24], [12], [20], [-4], [24], [-12], [24], [14], [28], [0], [4], [12], [0], [18], [0], [-30], [-46], [-18], [-8], [-20], [28], [42], [12], [-46], [-6], [-26], [-14], [-24], [26], [-28], [8], [36], [-40], [-32], [-22], [22], [-18], [-16], [-34], [-4], [4], [50], [-10], [-8], [30], [-26], [44], [-8], [-16], [-12], [36], [30], [6], [-20], [-6], [20], [40], [12], [4], [4], [2], [34], [30], [42], [-28], [24], [-18], [-56], [20], [-24], [-58], [-8], [36], [24], [-2], [12], [28], [-50], [40], [6], [-30], [-24], [-8], [18], [-6], [-22], [0], [-50], [62], [46], [0], [-44], [28], [-54], [50], [-26], [-22], [36], [50], [6], [64], [52], [6], [18], [-42], [14], [4], [16], [-12], [32], [40], [14], [-2], [30], [-30], [40], [-66], [-8], [-26], [-14], [-26], [-54], [-44], [-60], [-10], [-32], [-68], [12], [62], [44], [48], [70], [34], [-38], [56], [-34], [14], [-44], [-58], [-6], [-68], [-74], [-42], [-16], [-44], [34], [-12], [-24], [52], [72], [-60], [-24], [-54], [-18], [-32], [-36], [-56], [20], [-44], [-46], [50], [-50], [16], [40], [-36], [64], [2], [-14], [-40], [-26], [-50], [-18], [-66], [-72], [10], [14], [-6], [12], [38], [-34], [-66], [-52], [38], [-42], [-50], [6], [-22], [-24], [76], [-32], [-26], [46], [24], [18], [38], [8], [24], [28], [48], [-26], [2], [32], [-8], [-2], [28], [72], [-30], [-36], [18], [-2], [64], [-44], [-24], [-6], [16], [72], [82], [46], [-72], [-60], [-76], [-82], [-4], [-34], [-28], [-34], [-56], [72], [-10], [32], [-28], [50], [-20], [4], [-2], [-52], [60], [-36], [-34], [-38], [-42], [-30], [52], [-34], [64], [66], [14], [66], [-52], [60], [-40], [32], [54], [78], [-58], [76], [64], [42], [20], [-16], [14], [-22], [-54], [0], [60], [42], [20], [28], [-62], [28], [94], [66], [-84], [-48], [38], [-60], [-98], [96], [-72], [-66], [54], [-92], [-62], [52], [38], [70], [64], [60], [-34], [-72], [-64], [60], [-18], [-14], [18], [-32], [80], [-66], [16], [-38], [-60], [28], [12], [-88], [-8], [32], [-90], [-66], [18], [74], [94], [74], [-32], [30], [-76], [-12], [-18], [-18], [26], [22], [-56], [20], [-98], [-88], [76], [-56], [-24], [94], [36], [60], [58], [-54], [16], [74], [14], [-4], [60], [-76], [-80], [38], [34], [90], [60], [-32], [-12], [-24], [-42], [-50], [76], [-52], [44], [36], [-102], [82], [-52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7440_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7440_2_a_y();".
// 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_7440_2_a_y(:prec:=1)
    chi := MakeCharacter_7440_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(3067) - 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_7440_2_a_y();".
// 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_7440_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7440_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-4, 1]>,<13,R![4, 1]>,<17,R![6, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;