
// Make newform 7440.2.a.g in Magma, downloaded from the LMFDB on 23 April 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_g();".
// 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_g();".
// 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], [-4], [-2], [2], [0], [0], [6], [0], [-1], [-2], [-10], [4], [-4], [6], [4], [0], [-4], [16], [4], [-4], [-8], [6], [14], [-10], [12], [-12], [-2], [2], [-18], [-4], [-12], [6], [-2], [-4], [18], [12], [-14], [-14], [2], [-20], [0], [-10], [-6], [-20], [4], [10], [20], [-20], [-22], [8], [-22], [-2], [18], [-10], [-24], [0], [-2], [6], [4], [-2], [28], [0], [28], [-6], [-18], [-32], [-24], [-14], [36], [-24], [-26], [6], [16], [-6], [20], [18], [30], [-18], [-36], [-30], [-40], [-20], [-24], [-12], [6], [8], [0], [18], [20], [-16], [-10], [-6], [-22], [-16], [-4], [-42], [20], [6], [-36], [18], [12], [30], [26], [-6], [24], [-34], [48], [-10], [16], [-42], [30], [-34], [-8], [10], [-44], [18], [30], [28], [14], [-40], [42], [20], [28], [-38], [-32], [12], [-48], [-14], [-2], [-26], [-16], [-18], [-26], [14], [-2], [40], [38], [-6], [-32], [24], [-18], [56], [4], [-16], [6], [-46], [2], [-30], [18], [22], [36], [28], [-28], [-32], [-16], [30], [30], [28], [36], [-12], [-26], [-48], [-18], [-10], [-28], [6], [-38], [22], [50], [44], [-16], [-2], [0], [6], [-44], [10], [-16], [-20], [-32], [34], [-26], [-44], [-32], [-36], [-42], [44], [54], [40], [-4], [12], [42], [48], [60], [-16], [38], [66], [-42], [-36], [38], [8], [14], [10], [-56], [2], [-32], [-36], [42], [-20], [52], [4], [-48], [28], [36], [26], [4], [-6], [-62], [46], [-72], [56], [66], [36], [44], [-8], [-18], [-48], [10], [-8], [58], [-12], [-64], [-2], [-72], [18], [30], [-22], [66], [8], [52], [34], [-70], [56], [0], [40], [-46], [-10], [50], [-24], [-2], [18], [-22], [-26], [-14], [0], [-18], [-12], [26], [-2], [0], [44], [4], [-50], [48], [68], [-18], [-2], [-28], [-22], [-54], [-48], [76], [68], [-26], [-56], [4], [24], [62], [-70], [60], [-16], [-8], [54], [4], [16], [4], [10], [64], [-30], [50], [68], [76], [-16], [38], [-20], [-80], [-6], [-14], [-48], [34], [26], [24], [-20], [46], [14], [-12], [-74], [-36], [30], [6], [-4], [-62], [32], [12], [14], [90], [12], [-2], [-22], [90], [-36], [-58], [32], [10], [18], [-40], [44], [-78], [6], [46], [-58], [16], [20], [-44], [-68], [-24], [90], [-42], [-58], [-86], [84], [0], [-52], [-78], [-84], [-90], [-64], [-40], [-10], [-26], [40], [90], [-46], [38], [0], [-60], [38], [40], [-48], [-82], [-6], [-66], [-36], [52], [-64], [-50], [58], [46], [-36], [16], [-44], [-50], [-48], [-2], [-54], [-80], [74], [10], [44], [-80], [12], [98], [36], [-30], [44], [0], [-74], [-32], [34], [-54], [-14], [44], [-52], [8], [26], [56], [14], [-30], [100], [36], [10], [-76], [-18], [4], [24], [62], [30], [-64], [26], [-12], [22], [36], [-66], [34], [42], [-100], [-52], [22], [-8], [44], [82], [-70], [12], [-54], [-6], [-44], [-6], [-90], [-12], [86], [12], [54], [104], [-10], [-6], [78], [-64], [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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7440_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![2, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;