
// Make newform 3120.2.a.v in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3120_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_3120_2_a_v();".
// 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_3120_2_a_v();".
// 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_3120_a();"
function MakeCharacter_3120_a()
    N := 3120;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 2641];
    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_3120_a_Hecke(Kf)
    return MakeCharacter_3120_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], [-1], [0], [-2], [2], [4], [-4], [-2], [-6], [4], [-8], [-2], [-4], [-2], [-8], [-8], [-4], [-8], [-12], [6], [4], [8], [4], [-20], [0], [-12], [20], [-2], [6], [0], [18], [-12], [-10], [-4], [12], [-10], [2], [2], [8], [-16], [10], [8], [12], [10], [4], [8], [-8], [-8], [10], [-6], [-20], [-18], [16], [28], [22], [-10], [28], [-6], [16], [0], [-18], [-34], [22], [14], [-8], [0], [-34], [-32], [28], [-18], [18], [-12], [24], [30], [-14], [-10], [-18], [20], [16], [-2], [-16], [-24], [30], [12], [-2], [-14], [20], [24], [-10], [6], [18], [6], [-26], [-18], [20], [-12], [-20], [-10], [0], [-34], [16], [0], [-12], [-14], [-24], [38], [-28], [22], [-18], [26], [0], [2], [-40], [46], [42], [14], [4], [-14], [-22], [-12], [14], [16], [48], [20], [-40], [14], [34], [28], [16], [42], [18], [-38], [-22], [-28], [-18], [42], [30], [-30], [-4], [36], [-10], [16], [-50], [-12], [-28], [4], [26], [54], [-28], [-22], [20], [12], [24], [-34], [-22], [18], [12], [-24], [-18], [-6], [10], [-36], [-24], [38], [-22], [-26], [40], [32], [12], [-48], [16], [30], [-10], [-30], [-24], [34], [-26], [-50], [26], [2], [-12], [4], [-62], [32], [62], [24], [-54], [-44], [-52], [-62], [48], [-52], [54], [6], [58], [6], [-26], [24], [-22], [-50], [20], [-38], [68], [-24], [-54], [4], [-10], [36], [-40], [28], [48], [18], [-48], [22], [-36], [-26], [18], [-16], [-62], [-34], [48], [26], [-8], [52], [-32], [-44], [6], [36], [-44], [22], [4], [12], [-50], [6], [34], [-60], [-60], [52], [24], [-76], [-54], [-40], [-30], [-24], [30], [-18], [-50], [-66], [-72], [-70], [6], [56], [-42], [28], [-18], [28], [-8], [12], [-64], [26], [2], [64], [14], [-42], [32], [-6], [10], [-52], [16], [-72], [58], [-18], [-12], [-76], [6], [-70], [6], [-64], [-74], [-8], [0], [12], [-74], [42], [16], [50], [-36], [-52], [-30], [-24], [34], [72], [-64], [74], [-42], [-4], [78], [18], [-16], [-16], [20], [8], [60], [-54], [-48], [10], [-54], [30], [38], [4], [24], [-6], [4], [32], [36], [34], [16], [84], [64], [10], [90], [-66], [-22], [-24], [22], [86], [12], [-62], [8], [-28], [-74], [-68], [-4], [14], [70], [68], [50], [60], [66], [8], [-22], [-90], [-2], [-80], [-48], [-6], [-30], [-32], [26], [-32], [50], [0], [40], [-60], [32], [-8], [-46], [-22], [-10], [72], [44], [70], [-6], [34], [-38], [-90], [-24], [-4], [36], [-32], [34], [24], [96], [32], [6], [38], [-30], [-26], [-22], [50], [50], [-48], [-72], [-78], [72], [6], [-46], [-18], [28], [28], [-24], [14], [100], [18], [-20], [-42], [-12], [-4], [18], [42], [-66], [48], [-10], [-34], [-52], [-8], [6], [-82], [56], [24], [42], [-68], [64], [84], [34], [78], [44], [82], [-72], [42], [-12], [-10], [-84], [14], [10], [72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3120_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3120_2_a_v();".
// 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_3120_2_a_v(:prec:=1)
    chi := MakeCharacter_3120_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_3120_2_a_v();".
// 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_3120_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3120_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![4, 1]>,<17,R![0, 1]>,<19,R![2, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;