
// Make newform 3120.2.a.o in Magma, downloaded from the LMFDB on 25 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_o();".
// 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_o();".
// 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], [8], [6], [-6], [-4], [0], [-2], [-2], [4], [0], [-10], [-4], [-10], [-12], [8], [-8], [-8], [-12], [-14], [-16], [-16], [12], [-12], [12], [20], [-4], [-10], [-6], [8], [-10], [0], [22], [16], [4], [22], [10], [10], [0], [-4], [-18], [0], [20], [2], [4], [4], [24], [-16], [2], [-6], [12], [-2], [-24], [-16], [-10], [-6], [4], [-26], [-12], [-24], [6], [-30], [-10], [14], [24], [-28], [-14], [24], [36], [-2], [18], [12], [8], [14], [30], [-26], [-10], [-8], [0], [-2], [32], [-16], [-6], [8], [-6], [26], [-28], [-32], [-26], [-42], [-38], [-10], [18], [-26], [-36], [-8], [20], [2], [-24], [6], [-40], [12], [-36], [22], [-24], [22], [28], [-10], [-6], [10], [-12], [18], [-44], [30], [42], [-2], [48], [26], [-30], [44], [14], [32], [-4], [-36], [-16], [38], [-30], [-12], [16], [-6], [22], [10], [38], [32], [-10], [2], [38], [-10], [52], [12], [-10], [40], [46], [4], [-36], [12], [-14], [-18], [28], [2], [-52], [20], [-40], [-30], [18], [-34], [-4], [24], [14], [-38], [30], [52], [0], [-26], [-14], [54], [40], [20], [20], [52], [-32], [-50], [62], [-18], [-32], [26], [-10], [-10], [-54], [38], [-4], [-12], [2], [52], [-50], [-8], [58], [-20], [-28], [-2], [8], [-4], [-18], [54], [38], [14], [-14], [0], [-54], [-34], [-12], [-30], [56], [0], [38], [-20], [-58], [-12], [48], [-44], [-64], [-46], [-24], [6], [-12], [-14], [-62], [60], [30], [14], [48], [18], [-16], [4], [8], [-28], [70], [44], [16], [38], [-36], [-28], [14], [-30], [26], [-20], [4], [-68], [32], [-64], [46], [0], [-22], [-72], [-26], [54], [-26], [66], [-48], [-6], [54], [24], [30], [-72], [38], [-48], [-64], [28], [44], [-22], [-2], [8], [-54], [42], [36], [74], [-22], [32], [-12], [-40], [-30], [6], [-12], [16], [-34], [74], [-66], [-4], [-18], [28], [-44], [60], [-50], [6], [52], [-6], [-36], [12], [-18], [40], [-30], [-24], [-72], [66], [38], [-8], [-42], [22], [64], [-32], [12], [44], [-44], [-22], [56], [-70], [10], [-54], [14], [-76], [-16], [-6], [-28], [-40], [16], [82], [24], [56], [48], [-54], [-34], [-34], [2], [68], [-42], [70], [80], [-62], [56], [-12], [-34], [84], [-8], [18], [-26], [28], [-6], [-84], [54], [8], [-66], [46], [-26], [76], [40], [18], [10], [-92], [-42], [-72], [34], [16], [-56], [84], [-32], [-80], [42], [58], [22], [-24], [-52], [22], [-18], [-14], [-22], [30], [64], [4], [-44], [-32], [2], [-40], [80], [-36], [-42], [46], [70], [-30], [-62], [62], [42], [24], [-52], [50], [60], [6], [26], [-6], [44], [-52], [96], [-74], [-64], [18], [-44], [26], [80], [-92], [26], [70], [42], [16], [-34], [-86], [-56], [-40], [14], [-26], [72], [-80], [2], [44], [-72], [-100], [-74], [22], [52], [-22], [-32], [-14], [56], [-62], [44], [90], [34], [-64]];
    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_o();".
// 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_o(: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_o();".
// 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_o( : 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![-8, 1]>,<19,R![-6, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;